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\LRbibentry{1}{AlvarezBecherMereb2023}{N. \'Alvarez, V. Becher, and M. Mereb. ``{Poisson generic sequences}''. In: \emph{International Mathematics Research Notices}~2023.24 (2023), pp.~20970--20987. doi: \href{https://doi.org/10.1093/imrn/rnac234}{\nolinkurl{10.1093/imrn/rnac234}}. arXiv: \href{https://arxiv.org/abs/2202.01632}{\nolinkurl{2202.01632}}.}
\LRbibentry{2}{AlvarezEtAlPoissonGenericity}{N. \'{A}lvarez, V. Becher, E. Cesaratto, M. Mereb, Y. Peres, and B. Weiss. ``{Poisson genericity in numeration systems with exponentially mixing probabilities}''. In: \emph{Transactions of the American Mathematical Society}~379.5 (2026), pp.~3403--3423. doi: \href{https://doi.org/10.1090/tran/9592}{\nolinkurl{10.1090/tran/9592}}. arXiv: \href{https://arxiv.org/abs/2411.04116}{\nolinkurl{2411.04116}}.}
\LRbibentry{3}{ArratiaGordonWaterman1990}{R. Arratia, L. Gordon, and M. S. Waterman. ``{The Erd\H{o}s--R\'{e}nyi law in distribution, for coin tossing and sequence matching}''. In: \emph{Annals of Statistics}~18.2 (1990), pp.~539--570. doi: \href{https://doi.org/10.1214/aos/1176347615}{\nolinkurl{10.1214/aos/1176347615}}.}
\LRbibentry{4}{ArratiaGoldsteinGordon1989}{R. Arratia, L. Goldstein, and L. Gordon. ``{Two moments suffice for {Poisson} approximations: The {Chen--Stein} method}''. In: \emph{Annals of Probability}~17.1 (1989), pp.~9--25. doi: \href{https://doi.org/10.1214/aop/1176991491}{\nolinkurl{10.1214/aop/1176991491}}.}
\LRbibentry{5}{ArratiaGoldsteinGordon1990}{R. Arratia, L. Goldstein, and L. Gordon. ``{Poisson approximation and the {Chen--Stein} method}''. In: \emph{Statistical Science}~5.4 (1990), pp.~403--424. doi: \href{https://doi.org/10.1214/ss/1177012015}{\nolinkurl{10.1214/ss/1177012015}}.}
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\LRbibentry{8}{BasakNathZaharescu2024}{D. Basak, K. Nath, and A. Zaharescu. ``{{Poisson} behavior for chains of small quadratic non-residues}''. In: \emph{International Mathematics Research Notices}~2024.4 (2024), pp.~3356--3390. doi: \href{https://doi.org/10.1093/imrn/rnad168}{\nolinkurl{10.1093/imrn/rnad168}}.}
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\title{\textbf{Long runs and rare patterns}\\[1mm]
\large of a random completely multiplicative function}
\author{Brice Pouly}
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\begin{document}
\begin{refsection}

\maketitle
\vspace{-0.6em}
\noindent\begin{minipage}{\textwidth}\footnotesize
This research received substantial assistance from AI systems; see the
\hyperref[sec:ai-assistance]{declaration at the end of the article}.
\end{minipage}\medskip
\begin{abstract}
Let $f$ be a random completely multiplicative function with independent
symmetric signs at the primes. We study rare constant runs and prescribed
words at logarithmic lengths, despite long-range multiplicative identities.
A uniform weighted two-window square-relation estimate, with critical
bound $O_\varepsilon(N^{5/3+\varepsilon})$, separates exact rational
relations from residual components.

At critical intensity, this estimate gives Poisson comparisons retaining
positions, exact excess lengths and signs. For dictionaries, uniform
bounds under size and overlap conditions are complemented by typical
bounds obtained by averaging the distance after fixing the dictionary.
In a quantitatively controlled regime of diverging intensity, an exact
one-word coupling by largest-odd-prime pivots compares the full signed
run field under prescribed small-prime conditioning. For fixed
admissible window parameters and almost every fixed realization of $f$,
run-start counts in prescribed near-macroscopic windows satisfy an
empirical Poisson law along dyadic scales.

In the exceptionally rare regime, a relative marked comparison separates
the boundary event from bulk occurrences. Conditional on existence, the
first location, sign and overshoot have a two-source lattice law with
scale-dependent weights. The two overshoots use different clocks: prime
rank at the boundary and integer distance in the bulk.
\end{abstract}

\medskip
\noindent\textbf{Keywords.} Random multiplicative functions; long runs;
rare patterns; Poisson approximation; square-product relations; boundary effects.
\par\smallskip
\noindent\textbf{MSC 2020.} 11N64; 60F05; 60C05; 11D45; 94B05.
\clearpage
\tableofcontents
% BEGIN ORIGINAL SOURCE: sections/01_introduction.tex
\section{Introduction}\label{sec:introduction}
In a sequence of independent fair signs, a long constant run is rare for a
simple reason: each additional sign must agree with the preceding one.
A left-maximal run of length at least $L$ starts with probability $2^{-L}$,
so $N$ potential starts have natural intensity $N2^{-L}$. The transition
between many runs and no runs occurs at $L=\log_2N+O(1)$; recording starts
rather than all constant windows removes the local clustering within a run
\cite{ArratiaGoldsteinGordon1990,ArratiaGordonWaterman1990,Godbole1991}.

Logarithmic blocks also underlie the original Erd\H{o}s--R\'enyi law
of large numbers \cite{ErdosRenyi1970}; the distributional and marked
run laws used here are the later results cited above.

Now impose complete multiplicativity. Choose the signs $f(p)$ independently
at the primes and set
\begin{equation}\label{eq:model}
 f(1)=1,\qquad f(n)=\prod_p f(p)^{v_p(n)},\qquad
 \PP(f(p)=1)=\PP(f(p)=-1)=\tfrac12.
\end{equation}
In particular $f(n^2)=1$. This is the completely multiplicative, sometimes
called \emph{extended Rademacher}, model, not the squarefree-supported model. Random
multiplicative models go back to Wintner \cite{Wintner1944}; the
completely multiplicative convention here is fixed by \eqref{eq:model}.
The identity $f(2n)=f(2)f(n)$ persists at every scale. Distant blocks therefore cannot simply be treated as independent: the
effect of these identities must be counted. Nevertheless, the critical run
statistics have the independent-sign Poisson form.

There is a second, quite different mechanism at the left boundary. Requiring
$f(1),\ldots,f(L)$ to be equal is exactly the requirement that every prime
sign up to $L$ be positive. Its probability is $2^{-\pi(L)}$, not $2^{-L}$.
Thus a sufficiently exceptional run can be produced either by many potential
starts in the bulk or by a single arithmetically cheaper event at the boundary.
Our results quantify both mechanisms and their competition.

The same approach also distinguishes many prescribed words at once. Two
sources of dependence must be separated: words can overlap because a
suffix matches a prefix, and distant occurrences can interact because a
product of their integer arguments is a square. The first is a property
of the dictionary; the second is a property of the arithmetic sequence.
Keeping them separate is the organizing principle of the paper.

\subsection{Main results}
For $x\ge2$ and $L\ge1$, let
\begin{equation}\label{eq:start}
 J_{x,L}=\1_{\{f(x-1)=-f(x),\ f(x)=\cdots=f(x+L-1)\}}.
\end{equation}
Write $I_N=[N,2N)\cap\Z$, $Z_{N,L}=\sum_{x\in I_N}J_{x,L}$ and
$\lambda_N=N2^{-L}$. On a start, let $\ell_x$ be the maximal rightward run
length. Every such length is almost surely finite. Define
\[
 K_{x,e}=\1_{\{J_{x,L}=1,\ \ell_x=L+e\}},\qquad
 K^s_{x,e}=K_{x,e}\1_{\{f(x)=s\}},\quad e\in\NN,\ s\in\{\pm1\}.
\]
Let $\nu_{\rm geo}\{e\}=2^{-e-1}$ and
$\nu_\pm=(\delta_{-1}+\delta_{+1})/2$.
A \emph{word} is a prescribed finite sign sequence, a \emph{dictionary}
is a set of such words, and \emph{pattern} is the broader term for the
corresponding occurrence observable. All logarithms without an indicated
base are natural. All total-variation distances below use the convention
$\dTV(\mu,\nu)=\sup_A|\mu(A)-\nu(A)|$.

\begin{theorem}[Critical sign-resolved run field]\label{thm:poisson-main}
Fix $C>0$. Uniformly for integer lengths $|L-\log_2N|\le C$,
\begin{equation}\label{eq:poisson-main}
 \dTV\left(\Law((K^s_{x,e})_{x,e,s}),
       \bigotimes_{x,e,s}\Pois(2^{-L-e-2})\right)
 \le \exp\left[-\left(\frac1{\sqrt2}+o_C(1)\right)
                    \sqrt{\log N\log\log N}\right].
\end{equation}
Here $(x,e,s)$ ranges over $I_N\times\NN\times\{\pm1\}$, and the
comparison is on the common countable lattice. In particular the same
upper bound applies to $\dTV(\Law(Z_{N,L}),\Pois(\lambda_N))$.
\end{theorem}
Since $L$ is integral, changing it by one divides $\lambda_N$ by two;
$\lambda_N$ need not converge. This is why the approximation retains
its moving parameter. For continuous positions, the consequence is a weak limit. Along a
subsequence with $\lambda_N\to\lambda\in(0,\infty)$,
\[
 \sum_{x\in I_N}\sum_{e\ge0}\sum_{s\in\{\pm1\}}K^s_{x,e}\delta_{(x/N,e,s)}
 \ \Longrightarrow\ {\rm PPP}(\lambda\,dt\otimes\nu_{\rm geo}\otimes\nu_\pm)
\]
weakly on finite point configurations on $[1,2]\times\NN\times\{\pm1\}$.
The nonempty configurations on a deterministic spatial grid and those of a
diffuse Poisson process are singular. Consequently the last convergence is
weak, not in total variation. The lattice comparison is stronger in the
appropriate prelimit space: a single coupling controls positions, excesses,
signs, threshold counts and order statistics simultaneously. In the bulk,
positive and negative long runs have equal limiting intensities, despite
the deterministic positivity of every square.

\paragraph{A complete prefix field at growing intensity.}
There is also a comparison beyond bounded critical intensity. Let
$L=\lfloor\log_2M\rfloor-a_M$, with $a_M\ge0$ integral and
$a_M=o(\log M)$, and put $\Lambda=(M-L)2^{-L}\to\infty$.
For an event of cost $I_M=-\log\PP(\mathcal A_M)$ measurable in the
hard prime field $\mathcal F_{Y_M^\sharp}$, the budget
$I_M+\log\Lambda\le V_M-c\nu_M$ gives
\[
 \dTV\!\left(\Law((K^s_{x,e})_{2\le x\le M-L+1,e\ge0,s}\mid\mathcal A_M),
       \bigotimes_{x,e,s}\Pois(2^{-L-e-2})\right)
 \ll e^{-c'\nu_M}+M^{-1/3+\varepsilon}
\]
for $0<c'<c$ and $0<\varepsilon<1/3$.
The exact hard scales are defined in \eqref{eq:saddle-definitions}.
\Cref{thm:micro-run-tv} retains every site, exact excess and actual sign;
it requires only the base window to be contained and excludes the run
from $1$. Its selected-pivot coupling does not replace the finer critical
rate above or the relative low-intensity estimates below. The full proof
of \cref{thm:micro-run-tv} is given in the companion,
Appendix~\ref{supp:sec:microscopic-pivots}.
Restricting the prefix at $4N$ gives the same one-factor hard-field
budget on $[N,2N)$ (\cref{cor:dyadic-micro}); no change of rescaled
grids or enlargement of the conditioning field is involved.
As a consequence, for each fixed pair $0<\alpha<1$ and $\tau>0$,
\cref{cor:empirical-poisson} fixes almost every realization of $f$ and
obtains a spatial empirical $\Pois(\tau)$ law for run-start counts along
dyadic scales, in windows of $M\exp\{-\alpha V_M+O(1)\}=M^{1-o(1)}$
start sites.

Let $R_M$ be the longest constant run in $(f(1),\ldots,f(M))$.
\begin{theorem}[Longest run and competing rare-event sources]
\label{thm:global-summary}
At critical lengths $|L-\log_2M|\le C$,
\begin{equation}\label{eq:intro-prefix}
 \PP(R_M<L)=\exp(-M2^{-L})+o_C(1).
\end{equation}
Moreover,
\begin{equation}\label{eq:intro-as}
 R_M=\log_2M+O(\log\log M)\qquad\hbox{almost surely}.
\end{equation}
Suppose instead that $L\to\infty$, $L=O(\log M)$, and
$\lambda_M=M2^{-L}\to0$. Then
\begin{equation}\label{eq:intro-two-source}
 \PP(R_M\ge L)=2^{-\pi(L)}+M2^{-L}
                  +o(2^{-\pi(L)}+M2^{-L}).
\end{equation}
Let $X_{M,L}$ be the leftmost start of a length-$L$ constant window contained
in $[1,M]$, on the event $R_M\ge L$. If
$L-\log_2M-\pi(L)\to s\in\mathbb R$, then
\begin{equation}\label{eq:intro-location}
 \Law(X_{M,L}/M\mid R_M\ge L)
 \ \Longrightarrow\
 \frac1{1+2^{-s}}\delta_0+
 \frac{2^{-s}}{1+2^{-s}}\Leb_{[0,1]}.
\end{equation}
The boundary branch is more precise than its atom at macroscopic position
zero: with conditional probability tending to one its start is exactly $1$.
\end{theorem}
The centring in \eqref{eq:intro-location} compares two exact
costs, $L-\log_2M$ and $\pi(L)$, rather than replacing the prime count by a
first-order asymptotic at a precision where that replacement is invalid.
\Cref{thm:two-clock} strengthens the location law to a lattice comparison
with moving source weights, sign and overshoot. Its geometric clock counts
integers in the bulk and prime ranks at the boundary; no phase-convergent
subsequence is needed for that prelimit comparison.

The third result treats many words at once, with their absolute signs
prescribed. For a dictionary $\mathcal W_N\subset\{\pm1\}^{B_N}$ of
$m_N\ge1$ distinct words, define its occurrence field by
\[
 I_{x,w}=\1_{\{f(x-1+j)=w_j\ (0\le j<B_N)\}},\qquad
                     x\in I_N,\ w\in\mathcal W_N.
\]
Write $w\sim_d v$ if the suffix $(w_d,\ldots,w_{B_N-1})$ equals the
prefix $(v_0,\ldots,v_{B_N-d-1})$, and set
\begin{equation}\label{eq:intro-overlap}
 \Omega(\mathcal W_N)=\frac1{m_N}\sum_{w,v\in\mathcal W_N}
           \sum_{d=1}^{B_N-1}2^{-d}\1_{\{w\sim_d v\}}.
\end{equation}
\Needspace{14\baselineskip}
\begin{theorem}[Uniform and typical dictionaries of rare words]\label{thm:patterns-summary}
\textup{(i) Uniform dictionaries.} Fix $C>0$ and $0<\eta<1/2$.
For deterministic dictionaries, suppose
\[
 m_N\le N^{1/2-\eta},\qquad
 |B_N-\log_2(Nm_N)|\le C,\qquad
 \Omega(\mathcal W_N)\longrightarrow0.
\]
Then
\begin{equation}\label{eq:intro-word-field}
 \dTV\left(\Law((I_{x,w})_{x\in I_N,w\in\mathcal W_N}),
       \bigotimes_{x\in I_N,w\in\mathcal W_N}\Pois(2^{-B_N})\right)
                          \longrightarrow0.
\end{equation}
The same conclusion holds after any deterministic restriction of the
site--word index set. The rare-word field is also asymptotically equivalent
in total variation to the field formed from independent fair signs.

\smallskip\noindent
\textup{(ii) Typical dictionaries.} Fix a positive logarithmic band for
$B_N$ and a compact subinterval of $(0,\infty)$ for $Nm_N2^{-B_N}$.
Choose $\mathcal W_N$ uniformly among the size-$m_N$ dictionaries,
independently of $f$. The two distances in part \textup{(i)} tend to zero
in probability over this choice, with the dictionary held fixed when
forming the laws. No restriction $m_N\le N^{1/2-\eta}$ is required in
this part.
\end{theorem}
The total target intensity is $Nm_N2^{-B_N}$, while each word has mean
$N2^{-B_N}$. The uniform assertion is proved quantitatively in
\cref{thm:word-field}; it includes explicit marker dictionaries with no
overlaps (\cref{cor:marker-dictionaries}). The typical assertion, proved
in \cref{thm:typical-dictionaries}, averages the full error rather than
only the overlap weight. It permits $m_N\asymp N$ at
$B_N=2\log_2N+O(1)$, or any fixed polynomial dictionary size in a
suitable logarithmic band. In each case the expected total number of
occurrences remains bounded even though the number of possible types
grows. Both position and word identity are retained.

The same typical conclusion holds for uniformly sampled affine cosets
in $\mathbb F_2^{B_N}$ (\cref{cor:affine-dictionaries}). A matrix of
$O((\log N)^2)$ bits can therefore describe an admissible random
candidate containing order $N$ words. This is a high-probability result
over the description, not a deterministic certificate for an arbitrary
matrix.

For a single word counted up to sign, take $\mathcal W=\{a,-a\}$.
The run-start word $a=(-1,+1,\ldots,+1)$ has overlap weight $2^{-L}$.
For a specified dictionary, nonvanishing weighted overlap is outside
the small-error regime of the uniform bound;
short-period words illustrate the possible local clusters.
The compound-Poisson law in \cref{cor:window-clusters} describes the
clusters produced by counting every constant window rather than only
run starts.

\Needspace{6\baselineskip}
The results share arithmetic profiles, but use different comparisons.
The raw profile controls run fields; a marginal cap controls dictionaries
uniformly, while averaging the word selection gives the typical theorem.
For a growing prefix run field, the extra reciprocal-pivot estimate
controls the changes actually made by one conditional forcing.
The relative comparison resolves the competition between the exact
boundary event and the bulk. The distinction between uniform and typical
concerns the choice of observable, not a change of multiplicative model.

\subsection{The mechanism and the arithmetic estimate}
\begin{figure}[H]
\centering
\begin{tikzpicture}[node distance=9mm and 9mm]
\node[flowblue,text width=60mm] (a) {Affine word systems (Section~2)\\
\small relative signs and prescribed values};
\node[flowrust,text width=60mm,right=of a] (b) {Rational code and residual graph\\(Section~3)\\
\small CRT, Runge, Pell and kernel energy};
\node[flowblue,text width=60mm,below=of a] (c) {Condition on small primes\\
\small exact dependency graph, or\\selected-pivot forcing (Appendix F)};
\node[flowrust,text width=60mm,right=of c] (d) {Poisson comparisons (Sections~4--7)\\
\small word fields; critical and\\growing signed run fields};
\draw[flowarrow] (a)--(b);
\draw[flowarrow] (b)--(d);
\draw[flowarrow] (a)--(c);
\draw[flowarrow] (c)--(d);
\end{tikzpicture}
\caption{The arithmetic and probabilistic interfaces. The weighted two-window
estimate bounds the separated-pair term. Private prime coordinates supply
correct marginals and an exact conditional graph. For the growing prefix
run field, selected pivots instead construct a full one-word conditional
law; the directed change sets need not form a dependency graph. Local
overlap is checked separately in both routes.}\label{fig:proof-flow}
\end{figure}

Write prime signs as $f(p)=(-1)^{X_p}$ with independent uniform
$X_p\in\mathbb F_2$. A pattern up to sign prescribes $L$ affine linear
conditions on these prime coordinates. For two separated windows, let
$\rho_L(x,y)$ be the dimension of the homogeneous relation space of their
stacked systems. It depends on the windows, not on the prescribed signs.
The affine Fourier identity gives
\[
 \left|\PP(I_x=I_y=1)-2^{-2L}\right|
 \le 2^{-2L}(2^{\rho_L(x,y)}-1).
\]
Thus the useful arithmetic object counts \emph{all nonzero relations},
with their full multiplicity, rather than merely the pairs that carry one.
For prescribed absolute words the relevant kernel omits the two block-parity
conditions. Its dimension increases by at most two. A host count valid
without parity restrictions shows that the same weighted bound survives;
this is the first bridge from relative signs to the dictionary theorem.

\Needspace{4\baselineskip}
The second bridge is probabilistic: at a fixed site the dictionary words are
mutually exclusive, so the chance of a joint occurrence cannot exceed a
single-site probability. Capping the relation weight at this ceiling,
before summing over pairs, is what permits dictionaries of size
$N^{1/2-\eta}$.

The uniform estimate proved in \cref{sec:two-window-anatomy} is, for fixed
$0<\delta<1$ and $B=L+1\asymp\log M$,
\begin{equation}\label{eq:intro-master-profile}
 \sum_{\substack{M^\delta\le x,y<M\\|x-y|>L}}
 (2^{\rho_L(x,y)}-1)
 \ll_\varepsilon M^\varepsilon
 \left(M^{3/2}2^{B/6}+M2^{2B/3}+M^{2/3}2^B\right).
\end{equation}
The fixed logarithmic band and $\delta$ are understood in the implicit
constant. This single geometry includes the dyadic and bounded-ratio
estimates by restriction. At criticality the bound is $M^{5/3+\varepsilon}$.

Two devices make the proof work. Exact rational coincidences are first
removed as a systematic binary code. The remaining relations are resolved
by the graph of prime coordinates larger than $B$. For the hardest terminal
population, determinants force many small odd kernels into the same short
window. Counting the kernels separately is wasteful; a shifted pair-energy
estimate detects the necessary concentration. Finally one conditions on
prime signs below a larger cutoff $Y$. Most starts then have exact
conditional marginal $2^{-L}$, and disjoint prime supports give an exact
conditional dependency graph. Chen--Stein is applied \emph{after} this
arithmetic separation, not as a substitute for it. For the prefix coupling,
one forces a single prescribed word and charges only the prime coordinates
actually modified. A prime shared by two supports need not be selected by
that forcing; the finite example in \cref{supp:pivot:sec:forcing} makes
this distinction explicit. The reciprocal-pivot estimate reduces this
cost without discarding the two-window profile. Its analytic bound is
also a consequence of the classical largest-prime-factor reciprocal
asymptotics of Erd\H{o}s--Ivi\'c--Pomerance
\cite{ErdosIvicPomerance1986}; Appendix F gives both the direct uniform
Rankin proof and the square-part reduction to that antecedent.
The contribution of the coupling is its exact realization of the
conditional law and the arithmetic control of the resulting changes,
not a new general Palm--Stein principle or a new reciprocal asymptotic.



\subsection{Relation to earlier work}
The independent-sign theory supplies the run-start normalization, the
geometric excess law and the lattice extreme-value comparison
\cite{Foldes1979,GordonSchillingWaterman1986,ArratiaGordonWaterman1990}.
Dependency-graph Poisson approximation is classical
\cite{ArratiaGoldsteinGordon1989,ArratiaGoldsteinGordon1990,ChenRollin2013}.
Poisson approximation for occurrences of prescribed words and the role of
self-overlap are also classical \cite{Godbole1991,GodboleSchaffner1993}.
For several words, Reinert--Schbath \cite{ReinertSchbath1998} give
Poisson-process and compound-Poisson approximations for clumps in Markov
chains; Roquain--Schbath \cite{RoquainSchbath2007} treat overlapping rare
word families. Thus a joint approximation with word types is not itself
a new probabilistic principle.
The contribution here is to verify joint, site-labelled comparisons in a
nonstationary arithmetic sequence with exact multiplicative identities,
uniformly for logarithmic word length and a growing dictionary.

Najnudel proves eventual independence of fixed-length blocks and almost-sure
equidistribution of fixed words in roots-of-unity models \cite{Najnudel2020}.
For Steinhaus values, the stronger threshold of Turk reported in
\cite[arXiv version 2, Remark~2.3]{Najnudel2020} already permits blocks of
length at most any fixed multiple of the logarithm of their starting
point. Logarithmic length alone is therefore not the distinction here.
In the binary model, values at squares are deterministic, so the
Steinhaus criterion cannot be read as independence of fair signs in every
block. Our question instead concerns simultaneous, site-labelled fields
of many rare occurrences and the weighted interactions between their
moving windows. The uniform theorem makes explicit which part of that
control is insensitive to the affine right-hand side; the typical
theorem uses the additional averaging supplied by a random choice of
those sides. Fixed-pattern and sign-complexity questions also arise for
deterministic multiplicative functions
\cite{ButtkewitzElsholtz2011,MatomakiRadziwillTao2016,Mangerel2024}.

An arithmetic predecessor for consecutive rare patterns is the work of
Basak--Nath--Zaharescu \cite{BasakNathZaharescu2024} on chains of small
quadratic non-residues. Their published abstract states Poisson behaviour
in initial ranges of size $(\log p)^A$, $A>2$, for almost all prime moduli
in short intervals. This varies a quadratic character's prime modulus,
rather than drawing independent prime signs as here. This comparison is
limited to the stated model and observable; no inclusion or exhaustive
separation of the chain-length regimes is asserted.

The random-origin perspective itself has precedents.
Pandey--Wang--Xu \cite[arXiv version 4, Theorem~1.2]{PandeyWangXu2024} obtain complex
Gaussian limits for normalized short-interval sums of typical Steinhaus
functions. Their good sets have measure $1-o(1)$ at each size; the
common full-probability set along the prescribed dyadic scales in
\cref{cor:empirical-poisson} uses summability in addition to a field
comparison. Its observable is a rare binary run-start count and its
mode of convergence is total variation to a finite-mean Poisson law.
Poisson-generic sequences \cite{AlvarezBecherMereb2023} instead concern
the distribution over long words of their occurrence counts in a fixed
sequence. The extensions to invariant exponentially mixing measures
\cite{AlvarezEtAlPoissonGenericity} do not directly apply to the nonstationary
multiplicative sign sequence. These distinctions concern the observable
and the quantifiers, not the invention of empirical observation.

Square-product paucity and Diophantine rigidity are important tools in random
multiplicative functions. Related applications include short-interval central
limit theorems and random Chowla problems
\cite{ChatterjeeSoundararajan2012,SoundararajanXu2023,KlurmanShkredovXu2023,ChinisShala2025}. For fixed polynomial values, de la Bret\`eche, Wang and Xu
use square-product counts and Evertse--Silverman or Pell estimates
\cite{deLaBretecheWangXu2026}. Munsch--Shparlinski--Sun--Xiao
\cite[Theorem~1.3]{MunschShparlinskiSunXiao2026} prove an unconditional
square-product bound, uniform in the translation of one interval, for
each fixed number of factors. The constants depend on that fixed number;
the result does not directly control all relation weights in two moving
windows of logarithmic length. Tao's study of products of consecutive integers
with unusual anatomy provides a particularly close comparison for the
smooth-number and Diophantine mechanisms \cite{Tao2026}.
These works do not provide the growing-window profile
\eqref{eq:intro-master-profile} as an immediate specialization.
The properly normalized short-interval sums of Harper, Soundararajan and Xu
\cite{HarperSoundararajanXu2026} concern a different observable and, in their
stated model, Steinhaus prime values rather than binary run indicators.

We use Runge's method in an explicitly proved split-product form
\cite{Runge1887,Walsh1992,Walsh1995Corrections}; for fixed-degree component counts, bounded-height square-class localization
and a Pell equation suffice (\cref{lem:split-bounded}). The uniform
polynomial-height Pell count is already in
\cite[arXiv text, Lemma~4]{CillerueloGaraev2011} and
\cite[Lemma~2.10 and Remark~2.12]{Tao2026}; its proof is included to
make the required uniformity explicit. Uniform integral-point
theorems such as Evertse--Silverman~\cite{EvertseSilverman1986} provide
stronger, height-free information, but are not inputs to our proof.
The microscopic analysis uses
the uniform prime-divisor lower bound of Laishram--Shorey
\cite{LaishramShorey2004} and the pointwise result of
Balasubramanian--Shorey \cite{BalasubramanianShorey1993}.
The geometric marked target, its thinning chain and its Gaussian limits are
probabilistic consequences once the arithmetic comparison is established;
they are not presented as new properties of Poisson processes.

\subsection{Reading guide and scope}
Section~\ref{sec:affine-one-window} develops the affine encoding and the
one-window bounds. Section~\ref{sec:two-window-anatomy} proves the final
uniform two-window estimate and its extension to prescribed values.
Sections~\ref{sec:poisson-transfer}--\ref{sec:conditional-laws} extract scalar,
pattern, marked and conditional consequences. Section~\ref{sec:global-prefix}
handles the finite prefix, the selected-pivot signed comparison, the
boundary and the relative crossover.
A reader primarily interested in probability may take the uniform,
prescribed-value and capped two-window profiles as arithmetic inputs
(\cref{thm:master-profile,prop:absolute-profile,prop:capped-profile}), then
continue with Sections~\ref{sec:poisson-transfer}--\ref{sec:global-prefix}.
The two-window proof includes its Diophantine counting arguments, using
standard ideal and unit theory for the Pell step. Section~\ref{sec:global-prefix}
adds the explicitly stated prime-divisor and smooth-product inputs needed
near the boundary. The companion expands the technical steps called from
the article, rather than giving a second proof of the relation estimate.
The exact prime-cutoff definitions in Section~\ref{sec:poisson-transfer}
are not needed to read the three numbered introductory statements; they
are needed for the quantitative growing-intensity comparison.

We do not assert optimality of the exponent $5/3$ or of the prime-cutoff
ranges. The one-factor hard-cutoff comparison is proved for the
base-contained signed run field when $\Lambda\to\infty$, not for an
arbitrary word dictionary, a larger conditioning field or every intensity
regime. The size exponent in the uniform dictionary theorem is likewise a
sufficient bound, not an optimality statement; the typical theorem has
different quantifiers.

% END ORIGINAL SOURCE: sections/01_introduction.tex

% BEGIN ORIGINAL SOURCE: sections/02_affine.tex
\section{Affine events and private prime coordinates}\label{sec:affine-one-window}
A run event is a linear system over $\F$. Its probability is determined by
rank and compatibility; its dependence on a second event is determined by
relations between the two systems. This elementary observation lets us keep
arithmetic and probability separate.
For $n\ge1$, let
\[
 v(n)=(v_p(n)\bmod2)_p\in\F^{(\mathcal P)}.
\]
Write the prime signs additively: $X=(X_p)_{p\in\mathcal P}$
denotes the sequence of independent Bernoulli variables, uniform on
$\F$, defined by $f(p)=(-1)^{X_p}$. Then
\[
 f(n)=(-1)^{\langle X,v(n)\rangle}.
\]

For a start $x$, introduce the $L$ rows
\begin{align*}
 a_{x,0}&=v(x-1)+v(x), & b_{x,0}&=1,\\
 a_{x,j}&=v(x)+v(x+j), & b_{x,j}&=0,
 \qquad 1\le j<L.
\end{align*}
Let $A_x$ be the matrix with rows $a_{x,j}$ and $b_x=(1,0,\dots,0)$. Then
\[
 J_{x,L}=\1_{\{A_xX=b_x\}}.
\]
For a tuple $\mathbf x=(x_1,\dots,x_k)$, stack the matrices and the right-hand sides. Set
\[
 \calR(\mathbf x)=\ker(A_{\mathbf x}^{\mathsf T}),
 \qquad
 \rho(\mathbf x)=\dim\calR(\mathbf x),
\]
and
\[
 q_{\mathbf x}(c)=\langle c,b_{\mathbf x}\rangle.
\]

\begin{lemma}[Affine Fourier identity]\label{lem:fourier}
For every tuple $\mathbf x=(x_1,\dots,x_k)$,
\[
 2^{kL}\PP(J_{x_1,L}=\cdots=J_{x_k,L}=1)
 =\sum_{c\in\calR(\mathbf x)}(-1)^{q_{\mathbf x}(c)}.
\]
Consequently,
\[
 \PP(J_{x_1,L}=\cdots=J_{x_k,L}=1)
 =2^{-kL}\eta(\mathbf x)2^{\rho(\mathbf x)},
\]
where $\eta(\mathbf x)=1$ if $q_{\mathbf x}$ vanishes on $\calR(\mathbf x)$, and $0$ otherwise. Finally,
\begin{equation}\label{eq:affine-abs}
\left|\eta(\mathbf x)2^{\rho(\mathbf x)}-1\right|
 \le2^{\rho(\mathbf x)}-1.
\end{equation}
\end{lemma}

\begin{proof}
For a matrix with $m=kL$ rows,
\[
 \1_{\{AX=b\}}
 =2^{-m}\sum_{c\in\F^m}(-1)^{\langle c,AX-b\rangle}.
\]
Taking expectations, the terms with $A^{\mathsf T}c\ne0$ vanish. A character sum over a vector space equals the cardinality of that space if the character is trivial, and $0$ otherwise. If $\eta=0$, then $q$ is nontrivial, hence $\rho\ge1$, and $1\le2^\rho-1$.
\end{proof}


\subsection{The boundary map of a tree}
Put $B=L+1$, $I_L=\{-1,0,\ldots,L-1\}$ and
$V_x=\{x+i:i\in I_L\}$. The rows of $A_x$ are the edge vectors of the
star with edges $\{x-1,x\}$ and $\{x,x+j\}$, $1\le j<L$.
For a finite tree, the map taking an edge set to the vertices of odd degree
is a bijection onto the even vertex subsets. Thus a relation among the
rows of $A_x$ selects an even subset of $V_x$ whose product is a square.

\begin{lemma}[Square-product interpretation]\label{lem:even-subsets}
For disjoint windows $V_{x_1},\ldots,V_{x_k}$, the relation space is in
bijection with tuples of even subsets $S_i\subset V_{x_i}$ such that
$\prod_i\prod_{n\in S_i}n$ is a square. The affine character is
$\sum_i\1_{\{x_i-1\in S_i\}}$ modulo two.
More generally, the edge relations of any tree are identified with its
square-product even vertex subsets.
\end{lemma}
\begin{proof}
The sum of the selected edge vectors is the sum of the valuation vectors
at the odd-degree vertices. It vanishes exactly when their product is a
square. The boundary map is injective because a tree has no nonempty even
degree edge subgraph, and its image has dimension $|V|-1$, the dimension of
the even-subset space. The affine character records the root leaf edges.
\end{proof}

\subsection{One-window arithmetic}
For $H\ge2$ define the odd large-prime kernel
\[
 K_H(n)=\prod_{\substack{p>H\\v_p(n)\ \text{odd}}}p.
\]
Call $n$ \emph{$H$-defective} when $K_H(n)=1$. If an interval has diameter
less than $H$, every nondefective vertex has a prime coordinate appearing
there and nowhere else in the interval. Such a coordinate is a private
pivot. Notice that the definition uses odd valuations, not merely the
presence of a large prime divisor.

\begin{lemma}[Split-product Runge bound]\label{lem:runge}
There is an absolute $C_0$ such that the following holds. Let $d=2k\ge2$,
let $\gamma_1,\ldots,\gamma_d$ be distinct integers with
$|\gamma_i|\le R$, where $R\ge1$, and set $G(T)=\prod_i(T+\gamma_i)$.
If $U\in\Z$, $U\ge2R$, and $G(U)$ is an integer square, then
\begin{equation}\label{eq:runge-bound}
 U\le(C_0dR)^{2d}.
\end{equation}
\end{lemma}
\begin{proof}
Expand $\sqrt{G(T)}$ at infinity and let $P(T)$ be its polynomial part.
Its coefficients lie in $2^{-d}\Z$ and, for $U>4R$,
$|\sqrt{G(U)}-P(U)|\le(CdR)^d/U$.
If the difference is nonzero, integrality of $U$ and $\sqrt{G(U)}$ makes
its modulus at least $2^{-d}$. Otherwise the nonzero integer polynomial
$2^{2d}(G-P^2)$ vanishes at $U$; its coefficient height is at most
$(CdR)^{2d}$, and the integral-root bound applies. The range $2R\le U\le4R$
is immediate. All constants and the growing-degree coefficient estimates
are given in \cref{supp:lem:runge-calculation}. This is a direct
split-product implementation of Runge's method, not an unqualified
invocation of a fixed-degree theorem \cite{Runge1887,Walsh1992,Walsh1995Corrections}.
\end{proof}

\begin{proposition}[Few defects, with their exponential weight]
\label{prop:defects}
Fix $0<c_1<c_2$. Uniformly for integers $c_1\log X\le H\le c_2\log X$, an
interval of $H+1$ integers in $[X/2,3X]$ contains
$O_{c_1,c_2}(\log X/\log\log X)$ $H$-defective integers. If $m_u$ counts
these defects in $[u,u+H]$, then
\begin{equation}\label{eq:defect-weight}
 \sum_{X\le u<2X}(2^{m_u}-1)\le X^{1/2+o_{c_1,c_2}(1)}.
\end{equation}
\end{proposition}
\begin{proof}
Write each defective integer as $sa^2$, where $s$ is squarefree and
$H$-smooth. Associate to the $m$ defects in a window the columns
$((v_p(s))_{p\le H},1)\in\F^{\pi(H)+1}$. A nonzero kernel word selects
an even number $d$ of distinct integers with square product. The Runge
bound gives $X/2\le(CdH)^{2d}$; consequently its Hamming weight is at
least $c\log X/\log H$ for a fixed $c>0$.
Let $t$ be an integer of order $\log X/\log H$ with
$2t+1\le c\log X/\log H$. If the kernel is zero, $m\le\pi(H)+1$.
If $m<2t$, the desired bound is again immediate. Otherwise disjoint
Hamming balls about its words give
\[
 \sum_{j\le t}\binom mj\le2^{\pi(H)+1},\qquad
 (m/t)^t\le2^{\pi(H)+1}.
\]
Since $(\pi(H)+1)/t=O_{c_1,c_2}(1)$, it follows that
$m=O_{c_1,c_2}(\log X/\log\log X)$.

On the other hand the number of defective integers up to $3X$ is at most
\[
 \sqrt{3X}\sum_{\substack{s\ \text{squarefree}\\P^+(s)\le H}}s^{-1/2}
 =\sqrt{3X}\prod_{p\le H}(1+p^{-1/2})=X^{1/2+o(1)}.
\]
Each belongs to at most $H+1=X^{o(1)}$ windows. Only
$X^{1/2+o(1)}$ windows therefore have a defect, and the pointwise bound
makes each weight $2^{m_u}$ equal to $X^{o(1)}$. This proves the sum.
\end{proof}

For $0<\delta<1$, write $U_{M,\delta}=[\lceil M^\delta\rceil,M)\cap\Z$.
If $B\asymp\log M$, a dyadic slice $X\le x<2X$ in this interval has
$B\asymp_\delta\log X$. Applying \cref{prop:defects} on these slices gives
\begin{equation}\label{eq:macro-defects}
 \sum_{x\in U_{M,\delta}}(2^{m_B(x)}-1)\le M^{1/2+o(1)},\qquad
 m_B(x)=\#\{n\in V_x:K_B(n)=1\}.
\end{equation}
The same assertion holds for windows whose width and defect threshold are
any two comparable quantities in a fixed logarithmic band: use an enclosing
window of width equal to a fixed multiple of the threshold and subdivide
it into a bounded number of windows before the coding argument. In the
applications below the threshold can always be chosen at least the width.

\begin{corollary}[Uniform first moment and an affine bound]
\label{cor:first-moment}
Put $p=2^{-L}$ and $m_B^\circ(x)=\#\{n\in V_x\setminus\{x-1\}:K_B(n)=1\}$.
For any right-hand side $b\in\F^L$,
\begin{equation}\label{eq:one-affine-error}
 \left|\PP(A_xX=b)-p\right|
 \le p(2^{m_B^\circ(x)}-1),\qquad
 \PP(A_xX=b)\le2^{-L+\max(m_B(x)-1,0)}.
\end{equation}
For every deterministic $A\subset I_N$ and $B$ in a fixed logarithmic band,
\[
 \EE\sum_{x\in A}J_{x,L}=|A|p+O(pN^{1/2+o(1)}).
\]
The analogous error over any subset of $U_{M,\delta}$ is $O(pM^{1/2+o(1)})$.
\end{corollary}
\begin{proof}
A private pivot excludes its vertex from any relation. An even subset of
$m$ defective vertices has at most $\max(m-1,0)$ degrees of freedom.
Thus $\rho(x)\le\max(m_B(x)-1,0)\le m_B^\circ(x)$.
Apply \cref{lem:fourier}, which is unchanged by the affine right-hand side,
and sum \cref{prop:defects} or \eqref{eq:macro-defects}. Restricting a
nonnegative error sum to a mask can only decrease it.
\end{proof}


For $Y\ge2$ write
$\mathcal F_Y=\sigma(f(p):p\text{ prime},\ p\le Y)$ for the information
carried by the small prime signs.

\begin{corollary}[Prescribed values in one window]\label{cor:absolute-marginal}
Let $M_x$ have rows $v(n)$, $n\in V_x$, and let $w\in\{\pm1\}^B$.
Write $I_{x,w}=\1_{\{(f(x-1+j))_{0\le j<B}=w\}}$ and $p_B=2^{-B}$.
Then
\begin{equation}\label{eq:absolute-marginal}
 |\PP(I_{x,w}=1)-p_B|\le p_B(2^{m_B(x)}-1).
\end{equation}
If, for every vertex $n\in V_x$, there is a prime $p>Y$, where $Y>B$,
such that $v_p(n)$ is odd, then
$\PP(I_{x,w}=1\mid\mathcal F_Y)=p_B$ for every realization of
$\mathcal F_Y$.
Fix $0<c_1<c_2$. Uniformly for $c_1\log N\le B\le c_2\log N$,
every deterministic start mask $A\subset I_N$ and dictionary
$\mathcal W\subset\{\pm1\}^B$ satisfy
\[
 \EE\sum_{x\in A}\sum_{w\in\mathcal W}I_{x,w}
 =|A|\,|\mathcal W|\,2^{-B}
 +O_{c_1,c_2}\!\left(|\mathcal W|\,2^{-B}N^{1/2+o_{c_1,c_2}(1)}\right).
\]
The expectation is unconditional. The threshold and error are uniform in
$B,A,\mathcal W$, which may vary with $N$; empty masks and dictionaries
are allowed. The mask restricts starts, not vertices: every occurrence
uses all $B$ values $x-1,\ldots,x+B-2$, without truncation at the mask or
at the boundary of $I_N$.
\end{corollary}
\begin{proof}
For a fixed window of positive integers, take a prime cutoff at least
as large as its largest vertex, and at least $Y$ when conditioning.
The word event in the infinite product
is the inverse image of the corresponding finite-cylinder event, so their
probabilities agree exactly. Only the prime signs are independent; the
integer values retain all their multiplicative relations.
In a relation among the rows of $M_x$, every nondefective vertex is
excluded by its private coordinate. Hence the nullity is at most $m_B(x)$,
without the even-subset restriction used for relative signs. Finite Fourier
inversion gives \eqref{eq:absolute-marginal}. On a good support, the
projection of $M_x$ onto primes above $Y$ has an identity minor of order
$B$, so every prescribed value vector has probability $2^{-B}$ on each
positive-mass assignment atom of $\mathcal F_Y$. Sum the nonnegative
bound using \cref{prop:defects}, then use linearity of expectation.
The same threshold works for every mask and every dictionary; no
independence between occurrences is needed.
\end{proof}

\begin{lemma}[Private pivots on a graph]\label{lem:private-tree}
Let $G$ be a finite graph. In each connected component suppose all vertices
except possibly one have distinct private prime coordinates in the whole
graph. The kernel of its edge vectors, projected onto these coordinates,
is contained in its cycle space. In particular its rank is at least
$|E(G)|-\beta_1(G)$, where $\beta_1(G)=|E(G)|-|V(G)|+c(G)$.
For a tree this rank is $|E(G)|$.
\end{lemma}
\begin{proof}
A vanishing edge combination has even degree at every pivoted vertex.
The sum of degrees in a connected component is even, so its possible
unpivoted vertex has even degree as well. The selected edge set is therefore
in the cycle space, of dimension $\beta_1(G)$.
\end{proof}

\begin{lemma}[Local run pairs]\label{lem:local-pairs}
If $0<|x-y|<L$, then $J_{x,L}J_{y,L}=0$. Over ordered pairs with
$|x-y|=L$ in a dyadic block or in $U_{M,\delta}$, the homogeneous weights
$2^{\rho(x,y)}$ have sum at most $N^{1+o(1)}$ or $M^{1+o(1)}$, respectively.
\end{lemma}
\begin{proof}
An overlapping start requires two consecutive signs within the first
constant run to be opposite. At distance $L$, the two start trees form a
tree on an interval of diameter $2L$. With threshold $2L+1$, private
pivots and \cref{prop:defects} bound its relation dimension by
$O(\log N/\log\log N)$ on a slice. There are $O(N)$ such pairs;
summing slices gives the macroscopic assertion.
\end{proof}

% END ORIGINAL SOURCE: sections/02_affine.tex

% BEGIN ORIGINAL SOURCE: sections/03_arithmetic.tex
\Needspace{12\baselineskip}
\section{A uniform two-window relation estimate}\label{sec:two-window-anatomy}
The transfer in \cref{thm:transfer} separates four costs: weighted
one-window defects, supports removed at the prime cutoff, shared-prime
graph edges, and the weighted relations estimated here. Only the last
requires the two-window arithmetic. We sum $2^{\rho(x,y)}-1$, not merely
count pairs with a relation, by separating exact rational identities from
residual prime relations in one macroscopic geometry. Word overlaps are
treated separately in \cref{thm:word-field}.

Fix $0<\delta<1$ and $0<\beta_-<\beta_+<\infty$. Throughout this section,
\begin{equation}\label{eq:uniform-band}
 U=U_{M,\delta},\qquad \beta_-\log M\le B=L+1\le\beta_+\log M,
 \qquad Q_B=2^B.
\end{equation}
A factor $M^{o(1)}$ is uniform in this band, with dependence only on the
fixed parameters (and on an explicitly fixed component size when one is
used). All these parameters are fixed before $M$. Every occurrence lies in
\begin{equation}\label{macro:eq:occ-range}
 M^\delta-1\le n\le M+B\le2M.
\end{equation}

\begin{theorem}[Uniform weighted relation profile]\label{thm:master-profile}
For every $\varepsilon>0$,
\begin{align}
 R_{2,\delta}(M,L)
 &:=\sum_{\substack{x,y\in U\\|x-y|>L}}(2^{\rho_L(x,y)}-1)\notag\\
 &\ll_{\delta,\beta_-,\beta_+,\varepsilon}
 M^\varepsilon\left(M^{3/2}Q_B^{1/6}+MQ_B^{2/3}+M^{2/3}Q_B\right)
 \label{eq:master-profile}\\
 &\ll_{\delta,\beta_-,\beta_+,\varepsilon}
 M^\varepsilon\left(M^{5/3}+M^{2/3}Q_B\right).
 \label{eq:master-envelope}
\end{align}
The defining sum is over ordered pairs. In particular
$R_2(N,L)\ll_{C,\varepsilon}N^{5/3+\varepsilon}$ when
$|L-\log_2N|\le C$ in a dyadic block. The same conclusion, with a constant
depending on the fixed upper ratio, holds on $[N,\kappa N)$.
\end{theorem}
The last assertions follow by restriction from \eqref{eq:master-envelope}
at $M=2N$ or $M=\kappa N$ and, for example, $\delta=1/2$; the lower endpoint
is then at least $M^\delta$ for large $N$. No sum of independent block laws
is being taken. The proof of the estimates occupies the rest of this section.

There are three obstacles to counting the relations. Exact rational
identities may create many of them for one identifiable reason; we remove
that subspace first. Residual relations are then controlled by combining
rarity of their hosts with a bound on their dimension. In the terminal
case neither resource is sufficient separately: determinants force
several small kernels into one window, and it is this simultaneous
concentration, rather than the size of the candidate set, that must be
counted. The partition below distinguishes which of these resources is
available.


\subsection{Rational identities and the residual graph}
For a separated pair, use the even-subset description in
\cref{lem:even-subsets}. An exact rational relation can be present before
any primes are exposed. For example, with $x=100$, $y=67$ and $L=3$, the
windows contain the two units $(99,66)$ and $(102,68)$ of ratio $3:2$.
Their product is $6732^2$. Each selected subset in a window has two vertices,
so this gives a nonzero homogeneous relation; its two root occurrences also
make the affine character even. In fact the individual ranks are three and
the stacked rank is five, giving joint probability $1/32$, rather than
$1/64$ for independent starts.

\begin{figure}[htbp]\centering
\begin{tikzpicture}[x=1.5cm,y=1.1cm]
\foreach \i/\t in {0/99,1/100,2/101,3/102}{
 \node[occurrence] (a\i) at (\i,1) {};
 \node[above=2mm of a\i,font=\small] {$\t$};}
\foreach \i/\t in {0/66,1/67,2/68,3/69}{
 \node[occurrence] (b\i) at (\i,0) {};
 \node[below=2mm of b\i,font=\small] {$\t$};}
\draw[eventedge] (a0)--(a1)--(a2);\draw[eventedge] (a1) to[bend left=20] (a3);
\draw[eventedge] (b0)--(b1)--(b2);\draw[eventedge] (b1) to[bend right=20] (b3);
\draw[channel] (a0)--(b0);\draw[channel] (a3)--(b2);
\node[align=left,font=\small,anchor=west] at (3.6,.5)
 {$2\cdot99=3\cdot66$\\$2\cdot102=3\cdot68$};
\end{tikzpicture}
\caption{An exact two-unit relation. The thick edges are rational units;
the thin edges encode the affine start trees. Even subsets of the units
produce relations. This macroscopic example also illustrates why counting
all prime coincidences without quotienting the rational subspace loses
information.}\label{fig:rational-example}
\end{figure}
For coprime $a,b\geq1$, set $h=by-ax$ and
\[
 \mathcal M_{a,b}(h)=\{(i,j)\in I_L^2:ai-bj=h\},
 \qquad m(a,b,h)=|\mathcal M_{a,b}(h)|.
\]
A member of $\mathcal M_{a,b}(h)$ is an exact unit, because
$a(x+i)=b(y+j)$.  Even subsets of the exact units form a subspace
$\mathcal S_{a,b,h}\subset\Rspace(x,y)$.

\begin{lemma}[Rational code and height]\label{macro:lem:rational-code}
If $m=m(a,b,h)\geq1$, then
$\dim\mathcal S_{a,b,h}=m-1$.  If $m\geq2$, then
\[
 \max(a,b)\leq L<B.
\]
This conclusion is independent of the ratio between the endpoints
$x$ and $y$.
\end{lemma}

\begin{proof}
For an exact unit, coprimality gives integers $t$ with
$x+i=bt$ and $y+j=at$.  An even collection $E$ of units satisfies
\[
 \prod_{(i,j)\in E}(x+i)(y+j)
 =(ab)^{|E|}\left(\prod_{(i,j)\in E}t\right)^2,
\]
which is a square.  The even subsets of an $m$-element set have
dimension $m-1$.  Distinct solutions of $ai-bj=h$ differ by
$(b,a)$; since both offsets remain in an interval of diameter $L$,
one has $a,b\leq L$.
\end{proof}

Fix once and for all $A=3$.  Call $(x,y)$ aligned if there are
coprime $a,b\leq B^A$ such that
\begin{equation}\label{macro:eq:aligned}
 |by-ax|<(a+b)B.
\end{equation}

\begin{lemma}[Uniqueness of the macroscopic channel]\label{macro:lem:channel-unique}
For $M$ sufficiently large in terms of
$(\delta,\beta_-,\beta_+)$, an aligned pair has at most one reduced
fraction $(a,b)$ satisfying \eqref{macro:eq:aligned}.  Every channel with at
least two exact units is aligned and therefore has this unique
fraction.
\end{lemma}

\begin{proof}
If $(a,b)$ and $(a',b')$ both satisfy \eqref{macro:eq:aligned}, then
\[
 |(a'b-ab')x|<4B^{2A+1}.
\]
But $x\geq M^\delta$ and $B=O(\log M)$, so the right-hand side is
$o(M^\delta)$.  The integer $a'b-ab'$ is therefore zero for large
$M$, and reducedness gives $(a,b)=(a',b')$.  If a channel has two
units, Lemma~\ref{macro:lem:rational-code} gives $a,b\leq L$, and for any
unit $h=ai-bj$ satisfies $|h|<(a+b)B$.
\end{proof}

When the canonical aligned channel has $m\geq2$, define
\[
 \Srat=\mathcal S_{a,b,h},\qquad \sigma=m-1.
\]
Otherwise define $\Srat=0$ and $\sigma=0$.  In all cases put
\[
 \tau=\rho-\sigma.
\]
Thus no contraction is made when $m_{\rm ex}\leq1$.

\begin{remark}[Disparate scales]\label{macro:rem:disparate}
Suppose, for example, that $x\asymp M^\delta$ and $y\asymp M$.
If there were two exact units, then $a,b\leq L$ and
$x+i=bt$, $y+j=at$, so
\[
 y\leq L(x+B)+B=o(M),
\]
a contradiction.  Hence every genuinely disparate-scale pair has
$m_{\rm ex}\leq1$.  It remains in the ordinary large-prime graph and
is treated by the small-product, shallow-core or deep-core bounds below;
no illegitimate valuation identity or code contraction is used.
\end{remark}

Now form the graph $G_{>B}(x,y)$ on the occurrence set
$\cO=V_x\sqcup V_y$.  For a prime $p>B$, let $\cO_p$ be the
occurrences at which $v_p$ is odd.  Each block has diameter $B-1<p$,
so $|\cO_p|\leq2$, with at most one occurrence in each block.  If
$|\cO_p|=1$, pin that vertex to zero; if $|\cO_p|=2$, join the two
vertices; parallel equations are deduplicated.  A vertex with no
large-prime label is defective.  Let $D$ be the number of defective
vertices and $c$ the number of nontrivial unpinned components.

\begin{lemma}[Resolution and square class of components]\label{macro:lem:graph-resolution}
The solution space of the $p>B$ equations is canonically isomorphic,
after one representative is chosen in each component, to
\[
 \F^D\oplus\F^c.
\]
In particular, $D+2c\leq2B$.  Every nontrivial unpinned component
$C$ meets both blocks and has a unique representation
\[
 \prod_{n\in C}n=d_Cz_C^2,
\]
where $d_C$ is positive, squarefree, and $B$-smooth.
\end{lemma}

\begin{proof}
A pinned component is zero.  All variables in an unpinned component
containing an edge are equal and contribute one free coordinate.  An
isolated defective vertex contributes one free coordinate.  Every
nontrivial component uses at least two occurrences, giving
$D+2c\leq2B$.

Every edge is bipartite.  For each $p>B$, an unpinned component
contains either both odd occurrences of $p$ or none; a single odd
occurrence would pin it.  Hence the total $p$-valuation of the product
of the component is even for every $p>B$.  Its squarefree part is
therefore supported on $p\leq B$, and uniqueness is immediate.
\end{proof}

Assume $m\geq2$.  An exact unit has the form
$x+i=bt$, $y+j=at$ with $a,b<B$.
For every $p>B$, $p\nmid ab$ and therefore
\[
 v_p(x+i)\equiv v_p(t)\equiv v_p(y+j)\pmod2.
\]
If $K_B(t)=1$, the two occurrences are defective; otherwise they form
an isolated unpinned component of size two.  Distinct exact units use
disjoint occurrences.  Let $d$ be the number of exact units of the
first type and $n=m-d$ the number of the second type, and define
\[
 D^\#=D-d,
 \qquad c^\#=c-n,
\]
removing the $n$ exact components from the residual component set
$\Cres$.  If $m<2$, set $D^\#=D$, $c^\#=c$, and let $\Cres$ be all
nontrivial unpinned components.  Notice that $D^\#$ is a dimensional
correction, not literally a number of remaining defective vertices.

\begin{lemma}[Quotient core]\label{macro:lem:quotient-core}
Uniformly for all separated pairs,
\[
 \tau\leq D^\#+c^\#,
 \qquad D^\#+2c^\#\leq2B,
 \qquad D^\#=O(B/\log B).
\]
\end{lemma}

\begin{proof}
For $m\geq2$, the $m$ exact-unit vectors are independent in the
large-prime solution space and the even code has dimension $m-1$.
The parity functional of the first block descends to the quotient and
is nonzero there; hence the even-first-block subspace has dimension
$D+c-m$.  The full relation quotient embeds in that subspace, and the
small-prime equations can only decrease dimension.  Therefore
\[
 \tau\leq D+c-m=(D-d)+(c-n)=D^\#+c^\#.
\]
For $m<2$, the estimate is the direct inclusion
$\Rspace\subset W_{>B}$.  The inequality
$D^\#+2c^\#\leq2B$ follows from $D+2c\leq2B$.  The defect estimate follows from \cref{prop:defects}, applied to each
window on its own dyadic slice; \eqref{eq:uniform-band} makes the constants
uniform.
\end{proof}

From each component of $\Cres$, choose canonically its smallest
large-prime label and then the lexicographically first edge carrying
that label.  This gives
\[
 (i_s,j_s,p_s),\qquad 1\leq s\leq c^\#,
\]
with distinct offsets in each block, distinct primes $p_s>B$, and
$p_s\mid x+i_s$, $p_s\mid y+j_s$.  When $m\geq2$, these cells are not
exact; when $m=1$, only the at most two components meeting the two
occurrences of the unique exact unit can be exceptions.  Put
\[
 P^\#=\prod_{s=1}^{c^\#}p_s,
\]
with the empty product equal to one.


The dimension decomposition gives the exact identity
\begin{equation}\label{eq:weight-split}
 2^\rho-1=(2^\sigma-1)+2^\sigma(2^\tau-1).
\end{equation}
We first sum the rational term. The residual term will be divided into
exhaustive geometric sectors.

\subsection{Hosts and rational mass}
The elementary prime and divisor estimates used below are
\begin{align}
 \pi(B)&\ll B/\log B,&
 \sum_{p\le B}p^{-1/2}+B\sum_{p>B}p^{-3/2}
 &\ll \sqrt B/\log B,\label{macro:eq:prime-half}\\
 \sum_{p>B}p^{-2}&\ll(B\log B)^{-1},&
 \omega(n)+\log\tau(n)&\ll_K\log M/\log\log M
 \quad(1\le n\le M^K).
 \label{macro:eq:omega-tau}
\end{align}
Here $\tau(n)$ in \eqref{macro:eq:omega-tau} is the divisor function,
not the residual dimension. The last bound is used only for fixed $K$.
Partial summation of the prime number theorem proves the prime estimates;
the divisor estimate is standard \cite{Tenenbaum2015,NicolasRobin1983}.
In particular,
\begin{equation}\label{macro:eq:euler-subpoly}
 \prod_{p\le B}(1+c p^{-1/2})=M^{o(1)},\qquad 2^{\pi(B)}=M^{o(1)}
 \quad(c\hbox{ fixed}),
\end{equation}
while a squarefree $B$-smooth integer has size at most
\begin{equation}\label{macro:eq:Bsmooth-height}
 \prod_{p\le B}p=\exp(O(B))=M^{O_{\beta_+}(1)}.
\end{equation}
Let $H_{2,\delta}(M,L)$ count separated ordered pairs in $U^2$ with
$\rho_L(x,y)>0$. We also need $H^{\rm val}_{2,\delta}(M,B)$, the number of
such pairs for which some nonempty subset of $V_x\sqcup V_y$ has square
product, with no parity requirement in either window.
\begin{proposition}[Host bound]\label{macro:prop:host}
Uniformly in the fixed band,
\[
 H_{2,\delta}(M,L)\le H^{\rm val}_{2,\delta}(M,B)\le M^{3/2+o(1)}.
\]
\end{proposition}

\begin{proof}
For each pair counted by $H^{\rm val}_{2,\delta}$, choose a nonempty
square-product subset and its first selected occurrence, ordering first the $x$-block, then the $y$-block, and then
the offsets increasingly.  Suppose first that this occurrence is
$n=x+i$; the other orientation costs a factor two.  Put $K=K_B(n)$.
For every $p\mid K$, the parity of the selected square product forces
an occurrence of odd $p$-valuation in the other block.  It is unique
there because $p>B$ and the block diameter is $B-1$.  Forgetting
compatibility between primes, there are at most $B^{\omega(K)}$
assignments of the primes of $K$ to offsets of the other block.

A fixed assignment determines one residue class for $y$ modulo every
$p\mid K$, hence one class modulo $K$.  The interval $\UM$ has length
less than $M$, so it contains at most $M/K+1$ representatives.  By
\eqref{macro:eq:occ-range}, $K\leq n\leq2M$, and therefore
\[
 M/K+1\leq3M/K,
\]
including $K=1$.  The orientation and selected offset cost at most
$2B$, and hence
\begin{equation}\label{macro:eq:host-raw}
 H^{\rm val}_{2,\delta}(M,B)
 \ll BM\sum_{n\leq2M}\frac{B^{\omega(K_B(n))}}{K_B(n)}.
\end{equation}
Write uniquely $n=a^2ur$, where $u$ is squarefree and $B$-smooth and
$r=K_B(n)$ is squarefree with all prime factors $>B$.  Summing over
$a$ first gives
\begin{align*}
 \sum_{n\leq2M}\frac{B^{\omega(K_B(n))}}{K_B(n)}
 &\leq \sqrt{2M}
 \sum_u u^{-1/2}\sum_r B^{\omega(r)}r^{-3/2}\\
 &\leq \sqrt{2M}
 \prod_{p\leq B}(1+p^{-1/2})
 \prod_{p>B}(1+Bp^{-3/2}).
\end{align*}
For this host estimate, elementary integer sums suffice:
\[
 \sum_{p\le B}p^{-1/2}\le\sum_{n=1}^B n^{-1/2}\le2\sqrt B,
 \qquad B\sum_{p>B}p^{-3/2}
 \le B\int_B^\infty t^{-3/2}\,dt=2\sqrt B.
\]
Since $\log(1+t)\le t$, the weighted kernel sum is at most
$\sqrt{2M}\exp(4\sqrt B)$. Under $B\le\beta_+\log M$,
$B\exp(4\sqrt B)=M^{o(1)}$, uniformly in the length. Substitution in
\eqref{macro:eq:host-raw} proves the claim. The sharper prime-sum bounds
are not needed at this step.
\end{proof}
\begin{proposition}[Rational mass]\label{macro:prop:systematic}
The contributions with canonical height $q=\max(a,b)$ satisfy
\begin{equation}\label{eq:systematic}
 \sum_{q\ge3}(2^\sigma-1)\le MQ_B^{1/3}M^{o(1)},\qquad
 \sum_{q=2}(2^\sigma-1)\le MQ_B^{1/2}M^{o(1)}.
\end{equation}
The geometry-only sum, without translation parameters, satisfies
\begin{equation}\label{macro:eq:geometry-sum}
 \sum_{\substack{(a,b)=1,\ \max(a,b)\ge2,\ h\in\Z\\m(a,b,h)\ge2}}2^{m(a,b,h)-1}\le Q_B^{1/2}M^{o(1)}.
\end{equation}
\end{proposition}
\begin{proof}
At height $q$ there are $O(q)$ reduced pairs $(a,b)$ and $O(qB)$
possible nonempty values of $h$. A unit $(i_0,j_0)$ gives
$x=bt-i_0$, $y=at-j_0$, with $O(M/q)$ choices of $t$.
Units are spaced by $(b,a)$, so $m-1\le B/q$, and $m\ge2$ implies
$q\le L$. Summing $O(MqB2^{B/q})$ for $q\ge3$ gives the first bound;
$q=2$ gives the second. At $q=1$, separation excludes a unit.
Omitting the $O(M/q)$ translation factor and summing the same geometries
proves \eqref{macro:eq:geometry-sum}.
\end{proof}
\begin{remark}[Why the dyadic interval helps, and what it does not remove]\label{rem:dyadic-edge}
The starts $(x,2x)$ cannot both belong to $[N,2N)$. This removes the
volumetric $2:1$ family, not all ratio-two occurrences in two windows:
$(N-1,2N-2)$ can still occur when the starts are $N$ and $2N-1$.
Our master estimate keeps the volumetric branch and therefore also applies
to the finite prefix. In a dyadic restriction, a more precise channel count
improves this particular term, but that improvement is not needed for
\eqref{eq:master-envelope}.
\end{remark}
\subsection{The partition and the shallow sectors}

The partition balances two resources: the number of host pairs and the
weight carried by the relation space at each host. The rational mass has
already been separated. For the residual mass, a small dimension makes
many hosts affordable; when the dimension is large, a bounded component
or a defect must instead make the hosts rare. The first tests exploit
these alternatives, with Runge excluding the aligned deep case. In the
terminal sector neither a one-kernel host count nor the full relation
weight suffices separately: several small kernels in one short window
must be counted together. The assembly in \S\ref{sec:profile-assembly}
records the cost obtained from each branch.

For a nonaligned pair let $\widetilde k$ be the rank of the small-prime
and two block-parity equations after solving all large-prime equations.
Then
\begin{equation}\label{macro:lem:exact-rank}
 \tau=D^\#+c^\#-\widetilde k.
\end{equation}
Set $T_B=\lfloor(B-2)/3\rfloor$. After removing $2^\sigma-1$ in
\eqref{eq:weight-split}, apply the following tests successively:
\begin{center}\small
\begin{tabularx}{\textwidth}{@{}cX@{}}\toprule
Sector & Defining test, after the preceding tests have failed\\\midrule
1 & $P^\#\le M$.\\
2 & $\sigma>0$ and the canonical channel has $q\le\sqrt{\log B}$.\\
3 & $c^\#\le B/6$.\\
4 & The pair is aligned.\\
5 & $c^\#\le2B/3$.\\
6 & $D^\#\ge3$.\\
7 & Write $c^\#=B-s$; then $s+\widetilde k>T_B$.\\
8 & $s+\widetilde k\le T_B$.\\\bottomrule
\end{tabularx}
\end{center}
The tests are complementary at each step. In sectors 5--8 the pair is
nonaligned, $\sigma=0$, and $D^\#=D$, $c^\#=c$. In sectors 7--8 one
also has $c^\#>2B/3$ and $D^\#\le2$. Thus this is a disjoint partition
of all residual mass, including pairs with very different heights.

\begin{proposition}[Direct bounds in sectors 1--3]\label{prop:shallow-final}
Up to a common factor $M^{o(1)}$, the residual masses of sectors 1, 2 and 3
are bounded, respectively, by
\[
 M^{3/2}+MQ_B^{1/2},\qquad
 M^{3/2}+MQ_B^{1/2}+MQ_B^{1/3},\qquad
 M^{3/2}Q_B^{1/6}.
\]
\end{proposition}
\begin{proof}
In sector 1 the canonical certificate contains exactly $c^\#$ distinct
primes, all greater than $B$. Hence
\begin{equation}\label{eq:growing-moment} 
 (B+1)^{c^\#}\le P^\#\le M,\qquad
 c^\#\le\frac{\log M}{\log(B+1)},\qquad 2^{c^\#}=M^{o(1)}.
\end{equation}
The last estimate is uniform in the fixed positive logarithmic band.
Together with $D^\#=O(B/\log B)$ and
$\tau\le D^\#+c^\#$, it gives $2^\tau=M^{o(1)}$ pointwise.
When $\sigma=0$, only relational hosts contribute, and their number is
$M^{3/2+o(1)}$ by \cref{macro:prop:host}. When $\sigma>0$, use
\[
 2^\sigma(2^\tau-1)
 \le2(2^\sigma-1)M^{o(1)}
\]
and sum the rational mass in \cref{macro:prop:systematic}.
The two branches give
$M^{o(1)}(M^{3/2}+MQ_B^{1/2})$, without a critical-scale substitution
or any restriction on the ratio between the starts.

In sector 2, write the canonical channel as $h=by-ax$.
If a prime $p>B$ joins a nonexact cell $(i,j)$, it divides
$a(x+i)-b(y+j)=ai-bj-h\ne0$. The existence of an exact unit bounds
this integer by $O(qB)$. Thus every residual certificate prime is at
most a fixed multiple of $qB$.
Consequently $c^\#\le\pi(CqB)=o(B)$ uniformly for
$q\le\sqrt{\log B}$. Also $D^\#=o(B)$, so $\tau=o(B)$.
Summing the systematic weights by \eqref{eq:systematic} bounds this
sector. The displayed $M^{3/2}$ term is harmless and also covers a
partition convention that sends a zero-dimensional rational code here.

In sector 3, $\sigma=0$ or $q>\sqrt{\log B}$, hence $\sigma=o(B)$.
The quotient estimate gives
$2^\sigma(2^\tau-1)\le Q_B^{1/6}M^{o(1)}$.
Only relational hosts contribute. Multiply by their $M^{3/2+o(1)}$ count.
\end{proof}

The untreated pairs now have a large certificate product and more than
$B/6$ residual components. The next step either rules out their alignment
or extracts a bounded component whose square class supplies a rigid
Diophantine equation.

\subsection{Diophantine counting and deep cores}
A bounded component forces a product of a fixed number of shifted values
to lie in one square class. We need to count solutions only in a polynomial
height range. Two of the factors already suffice: after localizing their
square classes, they satisfy a Pell equation. This bounded-height argument
keeps the uniformity we need without a height-free integral-point theorem.
The uniform polynomial-height Pell estimate is classical: see
\cite[arXiv text, Lemma~4]{CillerueloGaraev2011} and
\cite[Lemma~2.10 and Remark~2.12]{Tao2026}.
We retain a self-contained proof with the explicit height dependence
required here; the lemma is not a separate priority claim.
\begin{lemma}[Uniform Pell count]\label{macro:lem:pell}
Fix $K_0>0$.  Let $A,C$ be positive squarefree integers with
$A/C\notin\mathbb Q^{\times2}$, and let $e\neq0$.  If
$A,C,|e|\leq M^{K_0}$, then the number of integer solutions of
\[
 Az^2-Cw^2=e,
 \qquad |z|,|w|\leq M^{K_0},
\]
is
\[
 \exp\!\left(O_{K_0}\!\left(\frac{\log M}{\log\log M}\right)\right)
 =M^{o(1)}.
\]
\end{lemma}

\begin{proof}The detailed reduction and uniformity checks are given in \cref{supp:lem:pell}.\end{proof}
\begin{lemma}[Split products in a polynomial range]\label{lem:split-bounded}
Fix $d\ge2$ and $K>0$. Let $h_1,\ldots,h_d$ be distinct integers and
$e$ a positive squarefree integer with $|h_i|,e\le M^K$.
The number of integer pairs $(X,Y)$ satisfying
\[
 |X|\le M^K,\qquad X+h_i>0\ (1\le i\le d),\qquad
              \prod_{i=1}^d(X+h_i)=eY^2
\]
is at most $\exp(O_{d,K}(\log M/\log\log M))$.
\end{lemma}
\begin{proof}
Put $\Delta=\prod_{i<j}(h_i-h_j)\ne0$ and
$S=\{p:p\mid e\Delta\}$. If $p\notin S$, it divides at most one
factor $X+h_i$ and has even valuation in the product. Hence the
squarefree part of every factor is supported on $S$.
In particular
\[
 X+h_1=su^2,\qquad X+h_2=tv^2,
 \qquad su^2-tv^2=h_1-h_2\ne0,
\]
where $s,t$ are positive squarefree integers supported on $S$.
There are at most $4^{|S|}$ choices of $(s,t)$. We need retain only
$s,t\le2M^K$, and $u,v\le(2M^K)^{1/2}$.
For $s\ne t$, their quotient is not a rational square, so
\cref{macro:lem:pell} bounds the solutions uniformly. For $s=t$,
the equation $s(u-v)(u+v)=h_1-h_2$ is bounded by the divisor estimate.
Each pair $(u,v)$ determines $X$, and then at most two values of $Y$.
Finally $|e\Delta|\le M^{O_{d,K}(1)}$, so
$|S|=O_{d,K}(\log M/\log\log M)$; the divisor and Pell bounds have the
same exponential scale. Multiplying the bounds proves the claim.
\end{proof}
This argument counts bounded-height solutions, not all integral points.
The distinction is precisely what makes a single Pell equation sufficient.
The square-class and height bookkeeping is recorded in
\cref{supp:sec:split-localization}.

\begin{lemma}[Normalization of a component]\label{lem:component-normalization}
Let $I,J$ be two nonempty sets of offsets, with $|I|+|J|\le K_0$, and let $d$ be a positive squarefree $B$-smooth integer. If
\begin{equation}\label{eq:component-square}
 P_I(x)Q_J(y)=dz^2,
 \qquad
 P_I(X)=\prod_{i\in I}(X+i),
 \quad
 Q_J(Y)=\prod_{j\in J}(Y+j),
\end{equation}
then, after fixing $x,I,J,d$, there is a positive squarefree integer $e\le M^{O_{K_0}(1)}$ such that
\begin{equation}\label{eq:normalized-mobile}
 Q_J(y)=ev^2.
\end{equation}
The integer $e$ is the squarefree part of $dP_I(x)$.
\end{lemma}

\begin{proof}
In the group $\mathbb Q^\times/\mathbb Q^{\times2}$, the equality \eqref{eq:component-square} gives
\[
 [Q_J(y)]=[dP_I(x)].
\]
The positive squarefree representative of the right-hand class is $e$; the squarefree part of $Q_J(y)$ is therefore $e$. Since $|I|\le K_0$, we have $dP_I(x)\le M^{O_{K_0}(1)}$.
\end{proof}
\begin{lemma}[One-sided count of a component]\label{lem:one-sided-component}
Fix $K_0$. For $x,I,J,d$ fixed as in \cref{lem:component-normalization}, the number of $y\in U_{M,\delta}$ satisfying \eqref{eq:component-square} is
\[
 \begin{cases}
 O(M^{1/2}),&|J|=1,\\
 M^{o_{K_0}(1)},&|J|\ge2.
 \end{cases}
\]
The same assertion holds after exchanging the two blocks.
\end{lemma}

\begin{proof}
\Cref{lem:component-normalization} gives $Q_J(y)=ev^2$.
For $|J|=1$, the equation $y+j=ev^2$ leaves at most
$1+\sqrt{3M/e}=O(M^{1/2})$ values.
For $|J|\ge2$, use \cref{lem:split-bounded}. Its degree is at most
$K_0$, all shifted factors are positive, $y\le M$, and
$e\le M^{O_{K_0}(1)}$. Thus a single fixed polynomial-height exponent
covers every coefficient and variable. The symmetric statement follows
by interchanging the blocks.
\end{proof}
\begin{lemma}[Two singleton components]\label{lem:singleton-pair}
Fix two offsets $i,j$. The number of triples $(x,y,d)$, with $x,y\in U_{M,\delta}$, $d$ positive, squarefree and $B$-smooth, satisfying
\[
 (x+i)(y+j)=dz^2
\]
is
\begin{equation}\label{eq:size2-hosts}
 M\exp\!\left(O\!\left(\frac{\sqrt B}{\log B}\right)\right).
\end{equation}
\end{lemma}

\begin{proof}
Set $X=x+i$, $Y=y+j$. A prime-by-prime analysis of $XY=dz^2$ gives a unique parametrization
\[
 X=ecu^2,
 \qquad
 Y=(d/e)cv^2,
\]
where $e\mid d$, where $c$ is squarefree and $(c,d)=1$. For fixed $e$, the conditions $1\le X,Y\le3M$ force
\[
 c\le C_e:=\frac{3M}{\max(e,d/e)}.
\]
Dropping the ``squarefree'' and ``coprime to $d$'' conditions, which only increases the count, the number of triples $(u,v,c)$ is at most
\begin{align*}
 \sum_{c\le C_e}
 \left(1+\sqrt{\frac{3M}{ec}}\right)
 \left(1+\sqrt{\frac{3Me}{dc}}\right)
 &\ll
 C_e
 +\sqrt{\frac Me}\sqrt{C_e}
 +\sqrt{\frac{Me}d}\sqrt{C_e}
 +\frac M{\sqrt d}\sum_{c\le C_e}\frac1c\\
 &\ll
 \frac{M\log M}{\sqrt d}.
\end{align*}
Here we used $\max(e,d/e)\ge\sqrt d$. After summing over $e\mid d$, the number of solutions for fixed $d$ is
\[
 \ll\frac{M\tau(d)\log M}{\sqrt d}.
\]
Finally
\[
 \sum_{\substack{d\text{ squarefree}\\p\mid d\Rightarrow p\le B}}
 \frac{\tau(d)}{\sqrt d}
 =\prod_{p\le B}\left(1+\frac2{\sqrt p}\right)
 =\exp\!\left(O\!\left(\frac{\sqrt B}{\log B}\right)\right).
\]
The factor $\log M$ is absorbed into the exponential $M^{o(1)}$.
\end{proof}
\begin{lemma}[Hosts of a bounded component]\label{lem:bounded-component}
Fix $K_0$. The number of separated pairs $(x,y)$ whose non-aligned residual graph contains a nontrivial unpinned component meeting both blocks and of size at most $K_0$ is
\begin{equation}\label{eq:bounded-hosts}
 M^{1+o_{K_0}(1)}.
\end{equation}
If this component has exactly two vertices, one in each block, we have the sharper bound
\[
 M\exp\!\left(O\!\left(\frac{\sqrt B}{\log B}\right)\right).
\]
\end{lemma}

\begin{proof}
Let $I,J$ be the offset sets of the component.
\Cref{macro:lem:graph-resolution} provides, with a unique coefficient
$d$, equation \eqref{eq:component-square}. There are only
$B^{O_{K_0}(1)}=M^{o(1)}$ choices of the two offset sets. We separate
three branches.

\paragraph{Two degrees at least two.}
If $|I|,|J|\ge2$, fix $x$, the offsets and $d$.
\Cref{lem:one-sided-component} leaves $M^{o_{K_0}(1)}$ values of
$y$. Summing over the $M$ values of $x$ and the
$2^{\pi(B)}=M^{o(1)}$ coefficients $d$ gives
$M^{1+o_{K_0}(1)}$ hosts.

\paragraph{A single degree equal to one.}
If $|I|=1<|J|$, the same argument fixes $x$ and uses the mobile side of
degree $|J|\ge2$. If $|J|=1<|I|$, we exchange the two blocks. We again
obtain $M^{1+o_{K_0}(1)}$ after summing over $d$.

\paragraph{Two singletons.}
If $|I|=|J|=1$, \cref{lem:singleton-pair} applies directly. It already
counts the triples $(x,y,d)$ and, in particular, performs the
simultaneous summation
\[
 \sum_{d\ B\text{-smooth, squarefree}}\frac{\tau(d)}{\sqrt d}.
\]
No factor $2^{\pi(B)}$ must be reintroduced in this branch.
After the union over the $B^2$ offset pairs, absorbed into the
exponential, the sharp bound remains
\[
 M\exp\!\left(O\!\left(\frac{\sqrt B}{\log B}\right)\right).
\]
The three branches give both assertions.
\end{proof}
\begin{proposition}[No aligned residual core of linear density]\label{macro:prop:aligned}
For every fixed $\alpha>0$, no aligned pair in $\UM^2$ satisfies
$c^\#\geq\alpha B$ once $M$ is sufficiently large in terms of the
fixed parameters and $\alpha$.
\end{proposition}

\begin{proof}
Let $m_{\rm ex}$ be the number of exact units of the canonical
channel.  If $m_{\rm ex}\geq2$, the nondefective exact components have
already been removed in $\Cres$.  If $m_{\rm ex}=1$, make no height or
valuation assumption and remove directly the at most two graph
components containing its two occurrences.  If $m_{\rm ex}=0$, remove
nothing.  In every case, from $c^\#\geq\alpha B$ there remain at least
$\alpha B/2$ nontrivial components containing no exact unit.  Their
total size is at most $2B$, so at least $m\geq\alpha B/4$ have size at
most $K_\alpha=\lceil8/\alpha\rceil$.

For each such component $C_t$ form the column
\[
 \Phi(C_t)=\left(
 (v_p(\prod_{n\in C_t}n)\bmod2)_{p\leq B},
 |C_t\cap V_x|\bmod2,
 |C_t\cap V_y|\bmod2
 \right)
 \in\F^{\pi(B)+2}.
\]
The kernel of the matrix of these $m$ columns has codimension at most
$r=\pi(B)+2$.  Choose
\[
 t=\left\lceil C_\alpha\frac{B}{\log B\log\log B}\right\rceil
\]
with $C_\alpha$ large.  If no nonzero kernel word had weight at most
$2t$, the Hamming bound would give
$\sum_{j\leq t}\binom{m}{j}\leq2^r$.  Since $m\gg_\alpha B$ and
$t=o(m)$,
\[
 \log \binom{m}{t}
 \geq t\log(m/t)-O(t)
 =(C_\alpha+o_\alpha(1))\frac{B}{\log B},
\]
which exceeds $r\log2$ when $C_\alpha$ is fixed sufficiently large.
Thus a nonzero word uses
\[
 O_\alpha\!\left(\frac{B}{\log B\,\log\log B}\right)
\]
components.

Let $I,J$ be the unions of their offsets.  The last two coordinates of
$\Phi$ make $|I|$ and $|J|$ even; the remaining coordinates, together
with the large-prime equations internal to each component, show that
\[
 \prod_{i\in I}(x+i)\prod_{j\in J}(y+j)
\]
is a square.  Put $d_0=|I|+|J|$; then
\[
 d_0\ll_\alpha\frac{B}{\log B\log\log B}.
\]
For the canonical channel $(a,b,h)$ define
\[
 G(T)=\prod_{i\in I}(T+ai)
      \prod_{j\in J}(T+h+bj).
\]
There are no root collisions: one would be an exact unit, and all
components containing such a unit were removed.  At $T=ax$,
\[
 G(ax)=a^{|I|}b^{|J|}
       \prod_{i\in I}(x+i)\prod_{j\in J}(y+j)
\]
is an integer square.  All roots have modulus $O(B^{A+1})$, while
$ax\geq x\geq M^\delta$; in particular $ax\geq2R$ for the root
radius $R=O(B^{A+1})$ once $M$ is large.  \Cref{lem:runge} gives
\[
 ax\leq(Cd_0B^{A+1})^{2d_0}.
\]
The logarithm of the right-hand side is
$O_\alpha(B/\log\log B)=o(B)$, whereas
\[
 \log(ax)\geq\delta\log M
 \geq(\delta/\beta_+)B.
\]
This contradiction closes the aligned deep sector directly at scale
$M$.
\end{proof}
We now work in the nonaligned complement. A component of size at most
$2/\alpha$ exists whenever $c^\#\ge\alpha B$, since the total number of
occurrences is $2B$. The preceding component count therefore gives
$M^{1+o(1)}$ hosts in every fixed positive-density deep sector.
For sector 5, $\tau\le D^\#+c^\#\le2B/3+o(B)$, so its mass is
$MQ_B^{2/3}M^{o(1)}$.
\begin{lemma}[Two defects in one block]\label{macro:lem:two-defects}
The number of starts $x\in\UM$ whose window contains two distinct
$B$-defective occurrences is $M^{o(1)}$.
\end{lemma}

\begin{proof}
Write
$x+i_1=d_1u^2$ and $x+i_2=d_2v^2$, with $d_1,d_2$ squarefree and
$B$-smooth.  If $d_1\neq d_2$, then $d_1/d_2$ is not a square and
\[
 d_1u^2-d_2v^2=i_1-i_2\neq0
\]
is covered by Lemma~\ref{macro:lem:pell}; all coefficients and variables are
in fixed polynomial ranges by \eqref{macro:eq:Bsmooth-height} and
\eqref{macro:eq:occ-range}.  If $d_1=d_2=d$, then $d\mid i_1-i_2$ and
\[
 (u-v)(u+v)=(i_1-i_2)/d,
\]
which has $M^{o(1)}$ solutions by the divisor bound.  There are
$4^{\pi(B)}B^2=M^{o(1)}$ choices of coefficients and offsets.
\end{proof}
\begin{proposition}[Sector 6]\label{macro:prop:sector6}
For nonaligned deep-core pairs with $c^\#>2B/3$ and $D^\#\geq3$,
\[
 R_{\rm res}\leq M^{1/2}\QB M^{o(1)}.
\]
\end{proposition}

\begin{proof}
One block contains at least two defects, so its start has
$M^{o(1)}$ choices by Lemma~\ref{macro:lem:two-defects}.  Since
$c^\#>2B/3$, a residual component has bounded size.  Fix its offsets
and square class at a cost $M^{o(1)}$.  The start in the other block is
then counted by Lemma~\ref{lem:one-sided-component}: at most $M^{1/2}$ choices
when its component degree is one, and $M^{o(1)}$ otherwise.  Hence the
number of hosts is $M^{1/2+o(1)}$.

Finally $D^\#+2c^\#\leq2B$ and
$D^\#=O(B/\log B)$ imply
$D^\#+c^\#\leq B+o(B)$.  Therefore
$2^\tau\leq\QB M^{o(1)}$, and multiplication proves the result.
\end{proof}
For sector 7, integrality gives $s+\widetilde k\ge T_B+1$.
Thus \eqref{macro:lem:exact-rank} and $D^\#\le2$ imply
$\tau\le B+1-T_B=2B/3+O(1)$. The same deep-host count bounds its mass
by $MQ_B^{2/3}M^{o(1)}$.

\subsection{Terminal kernels: count clustering, not just candidates}

The remaining issue is concentration within a window. On a slice with
larger start of size $X$, one kernel below a constant multiple of
$\sqrt{XB}$ allows $X^{3/4+o(1)}$ candidate starts
(\cref{lem:terminal-partners}); this count is too large when paired with
the full relation weight $Q_B$. If the component constraints force two
such kernels in the same window, the shifted energy
(\cref{lem:kernel-energy}) reduces the count to the
$X^{2/3+o(1)}$ scale. If only one small kernel is forced, the residual
dimension is smaller, and the one-kernel container is sufficient.
\Cref{macro:lem:determinant} supplies the small kernels;
\cref{prop:terminal-final} combines the two populations. The gain comes
from counting their concentration, not from shrinking the container.


It remains to consider sector 8. Write $j=s+\widetilde k\le T_B$.
If a component has size $2+e_t$, then
$\sum_t e_t\le2B-2c^\#=2s$. At most $2s$ components have size greater
than two, so at least $B-3s\ge B-3j\ge2$ are isolated size-two
components. For such a component $\{x+i_t,y+j_t\}$ put
$R_t=K_B(x+i_t)$.
\begin{lemma}[Terminal kernels and determinants]\label{macro:lem:determinant}
For every isolated size-two component,
\[
 R_t=K_B(x+i_t)=K_B(y+j_t)>1.
\]
Kernels belonging to distinct components are coprime.  For $s\neq t$,
\[
 R_sR_t\mid
 \Delta_{s,t}:=(x+i_s)(y+j_t)-(x+i_t)(y+j_s),
\]
and
\[
 0<|\Delta_{s,t}|\leq4MB
\]
for large $M$.
\end{lemma}

\begin{proof}
An isolated unpinned component contains exactly the two occurrences
carrying every one of its $p>B$ labels, so the full kernels agree and
are nontrivial.  A prime cannot label two components, because each
block contains at most one odd occurrence of that prime; hence the
kernels are coprime.  Divisibility of the determinant follows term by
term.

If the determinant vanished, the two occurrence ratios would be equal.
Writing their common reduced ratio as $b/a$ gives
$x+i_s=bu_s$, $y+j_s=au_s$ and similarly with $u_t$.  Subtraction yields
\[
 i_s-i_t=b(u_s-u_t),
 \qquad j_s-j_t=a(u_s-u_t).
\]
Distinct components have distinct offsets, so $u_s\neq u_t$, and
$a,b\leq L$.  The two cells would therefore be two units of an aligned
exact channel, contradicting nonalignment.  Finally
\[
 |\Delta_{s,t}|
 \leq|x+i_s||j_t-j_s|+|y+j_s||i_s-i_t|
 \leq4MB
\]
by \eqref{macro:eq:occ-range}.
\end{proof}
Slice terminal pairs by $X\le\max(x,y)<2X$, where
$M^\delta/2\le X\le M$. For counting, call the larger coordinate $z$
and the smaller one its partner; keep each pair's original canonical data.
The two possible orientations cost only a factor two, and no symmetry
of the canonical selection is assumed. On this slice
$R_sR_t\ll XB$. Set $T_X=(C XB)^{1/2}$ with an absolute constant large
enough for this inequality. At most one isolated kernel exceeds $T_X$.

\begin{lemma}[Partners and a one-dimensional container]\label{lem:terminal-partners}
On a terminal slice, a fixed value of either start has $X^{o(1)}$
possible partners. The number of possible values of either coordinate,
and hence of the larger start, is at most $X^{3/4+o(1)}$.
All constants are uniform in the slice.
\end{lemma}
\begin{proof}
Fix either start and write it as $x$, with partner $y$, for this count
only; the original canonical data of the pair are unchanged.
Choose two isolated components at a cost $O(B^4)=X^{o(1)}$.
Their distinct coprime kernels $R,S>1$ are fixed by $x$. The other
occurrences have the unique forms
$y+j=Ruz^2$, $y+v=Su'w^2$ with independent squarefree $B$-smooth
coefficients $u,u'$.
Thus
\[
 Ruz^2-Su'w^2=j-v\ne0.
\]
The coefficients $Ru,Su'$ are squarefree, of size $O(X)$, and their
quotient is nonsquare because the large-prime supports of $R,S$ are
disjoint. The Pell bound gives $X^{o(1)}$ partners per pair of smooth
coefficients, of which there are $4^{\pi(B)}=X^{o(1)}$.
This argument needs only $y\ll X$, not $y\asymp X$.

At least one isolated kernel is at most $T_X$. Integers with such a
kernel have the unique representation $n=ua^2r$, where $u$ is squarefree
and $B$-smooth and $r=K_B(n)\le T_X$. Therefore
\begin{equation}\label{eq:kernel-container}
 \#\{n\le3X:K_B(n)\le T_X\}
 \le\sqrt{3X}\prod_{p\le B}(1+p^{-1/2})\sum_{r\le T_X}r^{-1/2}
 \ll X^{3/4+o(1)}.
\end{equation}
Each such integer lies in at most $B=X^{o(1)}$ windows, whether the
coordinate is the first or the second start. Their union also bounds the
larger starts. Since
$B/\log X$ stays in a fixed compact interval, all constants are uniform.
\end{proof}

\begin{lemma}[Shifted small-kernel energy]\label{lem:kernel-energy}
For $T\ll(XB)^{1/2}$ put
$A_T(x)=\#\{i\in I_L:K_B(x+i)\le T\}$. Uniformly on every slice,
\begin{equation}\label{eq:kernel-energy}
 \sum_{X\le x<2X}\binom{A_T(x)}2\ll_\varepsilon X^{2/3+\varepsilon}B^2.
\end{equation}
\end{lemma}
\begin{proof}
Let $W_B=\prod_{p\le B}(1+p^{-1/2})=X^{o(1)}$.
Kernels $r\le R_0=X^{1/3}$ occur at at most
\[
 \sqrt{3X}W_B\sum_{r\le R_0}r^{-1/2}=X^{2/3+o(1)}
\]
integers in $[X/2,3X]$. Each can anchor at most $O(B^2)$ pairs of
occurrences and windows, so these integers give the required bound.

For the remaining pairs fix $1\le h\le B$ and dyadic ranges
$r=K_B(n)\asymp R$, $s=K_B(n+h)\asymp S$, with
$R,S\ge X^{1/3}$ and $R,S\ll(XB)^{1/2}$.
Set $q=n/r$, $q'=(n+h)/s$. Both are a square times a squarefree
$B$-smooth integer, with $q\asymp X/R$ and $q'\asymp X/S$.
To count their choices explicitly, put
\[
 \mathscr Q_B(T)=\{a^2u\le T:u\text{ squarefree},\ P^+(u)\le B\}.
\]
The squarefree decomposition is unique, and therefore
\[
 |\mathscr Q_B(T)|
 \le\sum_{\substack{u\le T\text{ squarefree}\\P^+(u)\le B}}
                       \left\lfloor\sqrt{T/u}\right\rfloor
 \le\sqrt T\,W_B.
\]
For suitable absolute constants in the height bounds, the number of
cofactor pairs $(q,q')$ is thus at most
\[
 C\sqrt{X/R}\,W_B\sqrt{X/S}\,W_B
       =\frac{CXW_B^2}{\sqrt{RS}}.
\]
Fix such a pair and write $g=(q,q')$. The equation
\[
 qr+h=q's
\]
requires $g\mid h$. When it has a solution $(r_0,s_0)$, all solutions are
\[
 r=r_0+\frac{q'}g t,\qquad s=s_0+\frac qg t,\qquad t\in\Z.
\]
The box $r\asymp R$, $s\asymp S$ contains at most
\[
 1+C\min\{Rg/q',Sg/q\}\le1+CBRS/X
\]
solutions, since $g\le h\le B$. Dropping any remaining squarefreeness
or coprimality restrictions only enlarges this upper bound. Multiplying
the number of cofactor choices by the number of solutions for each gives
\[
 X^{o(1)}\left(\frac{X}{\sqrt{RS}}+B\sqrt{RS}\right)
 \le X^{2/3+o(1)}.
\]
Here $X/\sqrt{RS}\le X^{2/3}$, while
$B\sqrt{RS}\ll X^{1/2}B^{3/2}=X^{1/2+o(1)}$, as $B\asymp\log X$.
There are $X^{o(1)}$ pairs of dyadic ranges. Sum over $h$ and note
that a shifted pair lies in at most $B$ windows. This proves
\eqref{eq:kernel-energy}.
\end{proof}

\begin{proposition}[Terminal mass]\label{prop:terminal-final}
Sector 8 contributes at most
\[
 M^\varepsilon\left(M^{2/3}Q_B+M^{3/4}Q_B^{2/3}\right).
\]
\end{proposition}
\begin{proof}
On a fixed larger-start slice put $j=s+\widetilde k$ and
$m_j=B-3j-1$. The sector test gives $m_j\ge1$.
By \cref{macro:lem:exact-rank},
$\tau+j=B+D^\#$ with $D^\#\le2$.
There are at least $B-3j$ isolated kernels, all but at most one below
$T_X$, so each window has at least $m_j$ small kernels.

First collect all pairs with $m_j\ge2$, without separating the values
of $j$. Their larger start $z$ satisfies
$\binom{A_{T_X}(z)}2\ge1$. The energy estimate counts at most
$X^{2/3+o(1)}B^2$ such starts, and the uniform partner bound contributes
only $X^{o(1)}$. Since $2^\tau-1\le4Q_B$, the total mass is at most
$X^{2/3+o(1)}Q_B$; the factor $B^2$ is subpolynomial.

In the complementary population, $m_j\le1$ implies $3j\ge B-2$.
Consequently $3\tau\le2B+8$, and
$2^\tau-1\le8Q_B^{2/3}$. At least one small kernel remains, so the
one-dimensional container and partner bound give at most
$X^{3/4+o(1)}$ pairs. Their mass is at most
$X^{3/4+o(1)}Q_B^{2/3}$.

The dyadic slices of the larger coordinate exhaust the terminal pairs.
Their logarithmic bands are uniform, and geometric summation proves the
claim. The partner estimate uses only its upper height bound, never
comparability with $X$.
\end{proof}

\Needspace{15\baselineskip}
\subsection{Assembly and lower bounds}\label{sec:profile-assembly}
\begin{proof}[Proof of \cref{thm:master-profile}]
The completed estimates are as follows; a factor $M^{o(1)}$ is suppressed.
\begin{center}\small
\begin{tabularx}{\textwidth}{@{}lXl@{}}\toprule
Population & Mechanism & Weighted mass\\\midrule
Rational & Exact-unit enumeration & $MQ_B^{1/3}+MQ_B^{1/2}$\\
1 & Small product and rational mass & $M^{3/2}+MQ_B^{1/2}$\\
2 & Small channel height & $M^{3/2}+MQ_B^{1/2}+MQ_B^{1/3}$\\
3 & Host count and rank & $M^{3/2}Q_B^{1/6}$\\
4 & Runge and a short codeword & $0$\\
5 & Bounded components & $MQ_B^{2/3}$\\
6 & Two defects and a component & $M^{1/2}Q_B$\\
7 & Exact rank deficit & $MQ_B^{2/3}$\\
8 & Shifted kernel energy and Pell & $M^{2/3}Q_B+M^{3/4}Q_B^{2/3}$\\\bottomrule
\end{tabularx}
\end{center}
Every entry is bounded by one of the three monomials in
\eqref{eq:master-profile}. The partition is exhaustive and
\eqref{eq:weight-split} is exact, so summation proves that profile.
Finally
\[
 M^{3/2}Q_B^{1/6}=(M^{5/3})^{5/6}(M^{2/3}Q_B)^{1/6},\qquad
 MQ_B^{2/3}=(M^{5/3})^{1/3}(M^{2/3}Q_B)^{2/3};
\]
weighted arithmetic--geometric mean proves \eqref{eq:master-envelope}.
\end{proof}

The exponent $5/3$ is an upper bound, not an optimality claim. In a dyadic
block, the ordered pairs $x=2t,y=3t$, $N/2\le t<2N/3$, and their reversals
give
\begin{equation}\label{eq:rational-lower-dyadic}
 R_2(N,L)\ge\left(\frac N3+O(1)\right)
       \left(2^{\lfloor(B+1)/3\rfloor-1}-1\right).
\end{equation}
On $U_{M,\delta}$, the pairs $x=t,y=2t$ and their reversals give
\begin{equation}\label{eq:rational-lower-macro}
 R_{2,\delta}(M,L)\ge(M-2M^\delta+O(1))
       \left(2^{\lfloor B/2\rfloor-1}-1\right).
\end{equation}
These follow simply by counting the even code of exact units, regardless
of any extra residual relations. At criticality the lower exponents are
$4/3$ and $3/2$, respectively. The one-dimensional container
\eqref{eq:kernel-container} is itself of order $X^{3/4+o(1)}$ when
$T\asymp\sqrt{XB}$: choose $r$ prime near $T$ and $n=ra^2\asymp X$.
What improves the terminal estimate is not a smaller container, but the
fact that its elements cannot cluster sufficiently often in one short
window, as quantified by \eqref{eq:kernel-energy}.


The weighted relation estimate is now available. A Poisson comparison
still requires the conditional marginal probabilities, an exact dependency
graph and the local overlap counts. Before constructing that graph, we
record two forms of the estimate adapted to prescribed words.

\subsection{From relative signs to prescribed values}\label{sec:absolute-relations}
There is little extra arithmetic in prescribing the sign of a word rather
than identifying it with its negative. What changes is the parity
restriction on square-product subsets.

For separated windows let $\rho^{\rm val}_B(x,y)$ be the nullity of the
transpose of the matrix with all $2B$ valuation rows $v(n)$,
$n\in V_x\sqcup V_y$. Thus its kernel consists of all square-product
subsets, without requiring even cardinality in each block. Write
$R^{\rm val}_{2,\delta}(M,B)$ for the sum of
$2^{\rho^{\rm val}_B(x,y)}-1$ over the same separated pairs as $R_{2,\delta}$,
and $R^{\rm val}_2(N,B)$ for its dyadic analogue.

\begin{proposition}[Prescribing the global signs]\label{prop:absolute-profile}
Uniformly under the hypotheses of \cref{thm:master-profile},
\begin{align}
 R^{\rm val}_{2,\delta}(M,B)
 &\le4R_{2,\delta}(M,B-1)+3H^{\rm val}_{2,\delta}(M,B),
       \label{eq:absolute-profile-comparison}\\
 R^{\rm val}_{2,\delta}(M,B)
 &\ll_\varepsilon M^\varepsilon
   \left(M^{3/2}Q_B^{1/6}+MQ_B^{2/3}+M^{2/3}Q_B\right)
 \ll_\varepsilon M^\varepsilon(M^{5/3}+M^{2/3}Q_B).
       \label{eq:absolute-profile}
\end{align}
The same conclusion holds on a dyadic block and on bounded-ratio
subintervals, with the corresponding ambient scale.
\end{proposition}
\begin{proof}
Imposing the two block-parity equations on the full square-relation
kernel gives exactly the relative-sign kernel, by the tree boundary
bijection. Therefore
\[
 \rho_{B-1}(x,y)\le\rho^{\rm val}_B(x,y)
                         \le\rho_{B-1}(x,y)+2.
\]
If the full kernel is zero both weights vanish. Otherwise
\[
 2^{\rho^{\rm val}_B}-1
 \le4(2^{\rho_{B-1}}-1)+3.
\]
The host term includes every separated ordered pair admitting a
nonempty square-product subset of the full vertex occurrences, with no
even-cardinality condition in either block. In particular it may count a
pair whose relative-sign kernel is zero. Sum the inequality using exactly
this unrestricted host estimate from \cref{macro:prop:host}. Its $M^{3/2+o(1)}$ term is absorbed by the first
monomial in \eqref{eq:master-profile}. Restriction proves the other
geometries. The extra two affine constraints therefore cost only a fixed
factor and a sparse set of new relational pairs, not an exponential
factor in the window length.
\end{proof}


\subsection{Capping the relation weight}\label{sec:capped-relations}
The full relation weight is useful when estimating moments. For the
probability that one of many mutually exclusive words occurs, however,
its largest values can be wasteful: a joint probability cannot exceed
either marginal. We record a version of the arithmetic estimate that
preserves this elementary ceiling.

For $T\ge1$ define
\[
 R_{2,\delta}^{[T]}(M,L)
   =\sum_{\substack{x,y\in U_{M,\delta}\\|x-y|>L}}
                         \min\{T,2^{\rho_L(x,y)}-1\}.
\]
The notation $R_{2,\delta}^{\mathrm{val},[T]}(M,B)$ has the same meaning
with the full-value nullity $\rho_B^{\rm val}$; omission of $\delta$
denotes the corresponding dyadic sum.
\begin{proposition}[Capped two-window profile]\label{prop:capped-profile}
Under the fixed-band hypotheses of \cref{thm:master-profile}, uniformly
also for $T\ge1$,
\begin{align}
 R_{2,\delta}^{[T]}(M,L)
 &\ll_\varepsilon M^\varepsilon
 \left(M^{3/2}Q_B^{1/6}+MQ_B^{2/3}
                    +M^{2/3}\min\{T,Q_B\}\right),
       \label{eq:capped-profile}\\
 R_{2,\delta}^{\mathrm{val},[T]}(M,B)
 &\ll_\varepsilon M^\varepsilon
 \left(M^{3/2}Q_B^{1/6}+MQ_B^{2/3}
                    +M^{2/3}\min\{T,Q_B\}\right).
       \label{eq:capped-value-profile}
\end{align}
The same bounds hold on dyadic and bounded-ratio subintervals. The
implicit constants are independent of $T$.
\end{proposition}
\begin{proof}
Write $2^\rho-1=(2^\sigma-1)+2^\sigma(2^\tau-1)$ and use
$\min(T,a+b)\le a+\min(T,b)$ for $a,b\ge0$.
The rational contribution and sectors 1--5 and 7 are left uncapped;
the assembly table bounds their sum by
$M^{o(1)}(M^{3/2}Q_B^{1/6}+MQ_B^{2/3})$.
In sector 6, the proof of its host bound gives $M^{1/2+o(1)}$ pairs,
and the pointwise weight is at most $Q_BM^{o(1)}$. Here the factor is
chosen independently of $T$. The inequality
$\min(T,EQ_B)\le E\min(T,Q_B)$ for $E\ge1$ therefore bounds this
contribution by $M^{1/2+o(1)}\min(T,Q_B)$.

For sector 8, use the two populations in the proof of
\cref{prop:terminal-final}. On a larger-start slice, the first population
has at most $X^{2/3+o(1)}$ pairs and pointwise weight at most $4Q_B$.
All geometric and partner bounds were fixed before $T$, so
\[
 \min\{T,2^\tau-1\}\le4\min\{T,Q_B\}
\]
gives capped mass $X^{2/3+o(1)}\min\{T,Q_B\}$ with a common
subpolynomial factor independent of $T$. Leave the complementary
population uncapped: its mass
$X^{3/4+o(1)}Q_B^{2/3}$ is absorbed by $X^{1+o(1)}Q_B^{2/3}$.
Geometric summation proves \eqref{eq:capped-profile}.

Finally, the two parity constraints give the pointwise inequality
\[
 \min\{T,2^{\rho_B^{\rm val}}-1\}
 \le4\min\{T,2^{\rho_{B-1}}-1\}
                       +3\1_{\{\rho_B^{\rm val}>0\}}.
\]
The last indicator has sum $M^{3/2+o(1)}$ by the full-value host
bound. This proves \eqref{eq:capped-value-profile}; restriction proves
the other geometries.
\end{proof}
The cap changes only the dense terminal contribution. Its benefit will
come from choosing $T$ according to a probability, rather than from
improving the uncapped exponent $5/3$.

% END ORIGINAL SOURCE: sections/03_arithmetic.tex

% BEGIN ORIGINAL SOURCE: sections/04_poisson.tex
\section{From arithmetic relations to Poisson laws}\label{sec:poisson-transfer}
The arithmetic estimate is now available independently of the probability
argument. The remaining difficulty is not a lack of pair information, but
the absence of a sparse unconditional dependency graph. Conditioning on the
small prime signs supplies exactly such a graph.

\subsection{An explicit transfer inequality}
Let $\mathcal F_Y=\sigma(X_p:p\le Y)$, with $Y>2B$. Let $D_Y$ be the
starts in $I_N$ for which some vertex of $V_x$ is $Y$-defective, and put
$G_Y=I_N\setminus D_Y$. We deliberately use the whole support in this
removal rule, so that the same convention works for marks and overlaps.
For $x\in G_Y$, the private-pivot lemma gives, for every environment,
\begin{equation}\label{eq:conditional-marginal}
 \PP(A_xX=b\mid\mathcal F_Y)=2^{-L}
 \quad\hbox{for every }b\in\F^L.
\end{equation}
After conditioning, an event at $x$ depends only on the prime coordinates
\[
 \Pi_x(Y)=\{p>Y:\exists n\in V_x,\ v_p(n)\hbox{ odd}\}.
\]
Join starts whose coordinate sets intersect. Disjoint collections with no
edge between them use disjoint independent coordinates. This is an exact
conditional dependency graph, not an approximate decorrelation assertion.

Write $E_Y(A)$ for its ordered edge count restricted to a mask $A$, and set
\[
 \mathcal M_B(A)=\sum_{x\in A}(2^{m_B(x)}-1),\qquad
 D_Y(A)=A\cap D_Y,\qquad \mathcal M_B=\mathcal M_B(I_N).
\]
All defect counts use the complete support, including $x-1$.
By \cref{cor:first-moment}, the actual probability mass removed from $A$
is at most $p(\mathcal M_B(A)+|D_Y(A)|)$, where $p=2^{-L}$.
Deleting the corresponding Poisson target coordinates costs at most
$p|D_Y(A)|$.
The edge count obeys
\begin{equation}\label{eq:edge-count}
 E_Y(I_N)\ll B^2\left(\frac{N^2}{Y\log Y}
                   +N\log\log N+\frac N{\log N}\right).
\end{equation}
Indeed a prime $p>Y$ can occur in at most $O(B(N/p+1))$ windows;
sum the square of this quantity over $Y<p\le3N$ and use partial summation.

We use the dependency-graph form of the Arratia--Goldstein--Gordon theorem
\cite{ArratiaGoldsteinGordon1989,ArratiaGoldsteinGordon1990}.
For indicators $(I_\alpha)$ with neighbourhoods $B_\alpha$ containing
$\alpha$, let
\[
 b_1=\sum_\alpha\sum_{\beta\in B_\alpha}p_\alpha p_\beta,
 \qquad
 b_2=\sum_\alpha\sum_{\substack{\beta\in B_\alpha\\\beta\ne\alpha}}
                  \EE(I_\alpha I_\beta),\qquad \mu=\sum_\alpha p_\alpha.
\]
An exact dependency graph has no third error term. The scalar
Poisson distance is at most $C(1\wedge\mu^{-1})(b_1+b_2)$; the joint
indicator-to-independent-Poisson distance is at most $C(b_1+b_2)$.
These assertions also apply on every fibre of $\mathcal F_Y$.

\begin{theorem}[Masked arithmetic-to-Poisson transfer]\label{thm:transfer}
For every deterministic $A\subset I_N$, put
$\mu_A=|A\cap G_Y|p$ and
$R_2(A)=\sum_{x,y\in A,\ |x-y|>L}(2^{\rho(x,y)}-1)$.
Then
\begin{align}
 &\EE\,\dTV\left(\Law\left(\sum_{x\in A}J_{x,L}\mid\mathcal F_Y\right),
                         \Pois(|A|p)\right)\notag\\
 &\quad\ll p(\mathcal M_B(A)+2|D_Y(A)|)
 +(1\wedge\mu_A^{-1})p^2\bigl(|A|+E_Y(A)+R_2(A)\bigr).
 \label{eq:transfer}
\end{align}
The multiplier is interpreted as one if $\mu_A=0$. Replacing it by one
gives a valid joint-field bound on the same finite site lattice.
\end{theorem}
\begin{proof}
Delete bad actual sites and their target coordinates. The natural coupling
costs at most $p(\mathcal M_B(A)+2|D_Y(A)|)$. On each remaining fibre, the
marginals are exactly $p$, and the graph above is a dependency graph.
Thus $b_1\le p^2(|A|+E_Y(A))$.
Overlapping starts are incompatible. At distance $L$, the union is a tree
with all its vertices possessing private coordinates above $Y$, since its
diameter is $2L<Y$. Its joint probability is therefore $p^2$.
For separated pairs the affine Fourier identity gives, after averaging,
\[
 \EE b_2\ll p^2\bigl(|A|+E_Y(A)+R_2(A)\bigr).
\]
Apply the scalar or process theorem on each fibre, average, and add the
deletions. The target is deterministic, so mixing also gives the same
unconditional bound.
\end{proof}
The denominator uses the mask's own mean. For a sparse mask it cannot be
replaced by the ambient intensity $Np$. At bounded ambient intensity,
all nonnegative sums are bounded by their full-block values, so the critical
estimate is uniform over all deterministic masks.

\subsection{The two prime-cutoff scales}\label{sec:cutoff-definition}
The leading scales are already visible without any special functions.
Set $S_N=\sqrt{\log N\log\log N}$ and $Y=e^{aS_N}$. The normalized number of bad
starts has leading upper exponent $1/(2a)$, while the normalized edge count
has exponent $a$. Their balance is $a=1/\sqrt2$.
For rare conditioning events and growing intensities we retain the smaller
second-order terms. The following definitions keep these terms explicit.

For $\nu\ge e$, let $u(\nu)\ge1$ solve $e^{u}/u=\nu$, and define
\[
 D(\nu)=\nu u(\nu)-\operatorname{Ei}(u(\nu)),\qquad
 H_N=\log N.
\]
For sufficiently large $N$, define $V_N$ and $\widehat V_N$ by
\begin{equation}\label{eq:saddle-definitions}
 V_N=D(H_N/V_N),\qquad
 \widehat V_N=2D(H_N/\widehat V_N),
 \qquad \nu_N=H_N/V_N,\quad\widehat\nu_N=H_N/\widehat V_N.
\end{equation}
The relevant solutions have $H_N/V_N,H_N/\widehat V_N\ge e$.
Since $D'(\nu)=u(\nu)>0$, each equation has exactly one such solution
for large $N$. Put $Y_N^\sharp=\lfloor e^{V_N}\rfloor$ and
$\widehat Y_N=\lfloor e^{\widehat V_N}\rfloor$.

\begin{proposition}[Uniform cutoff estimates]\label{prop:cutoffs}
Let $w\asymp\sqrt{H_N\log H_N}$, $\nu_w=H_N/w$ and $Y=\lfloor e^w\rfloor$.
Uniformly in a fixed logarithmic band,
\begin{equation}\label{eq:free-cutoff}
 \frac{|D_Y|}{N}\le e^{-D(\nu_w)+o(\nu_w)},\qquad
 \frac{E_Y(I_N)}{N^2}\le e^{-w+o(\nu_w)}+N^{-1+o(1)}.
\end{equation}
The maximum degree $\Delta_Y$ of the graph on all starts, including bad
ones, satisfies the same second bound after normalization by $N$.
Moreover,
\begin{align}
 V_N^2&=\frac{H_N}{2}
       \bigl(\log H_N+\log\log H_N-\log2-2+o(1)\bigr),
       \label{eq:hard-expansion}\\
 \widehat V_N^2&=H_N
       \bigl(\log H_N+\log\log H_N-\log4-2+o(1)\bigr).
       \label{eq:soft-expansion}
\end{align}
In particular $\widehat V_N\sim\sqrt2 V_N$ and
$\log\log N=o(\nu_N)$.
\end{proposition}
\begin{proof}
If $A(X,Y)$ counts $Y$-defective integers up to $X$, their unique
square--smooth factorization gives, for $0<\zeta<1/2$,
\[
 A(X,Y)/X\le X^{-\zeta}\prod_{p\le Y}(1+p^{-1+\zeta}).
\]
Choose $\zeta=u(\nu_w)/w$. The prime number theorem and partial summation
give $\sum_{p\le e^w}\log(1+p^{-1+\zeta})
=\operatorname{Ei}(u(\nu_w))+o(\nu_w)$.
This proves the first bound, because $|D_Y|\le B A(3N,Y)$ and
$\log B=o(\nu_w)$. The edge bound follows from \eqref{eq:edge-count}.
Each window uses $O(BH_N/w)$ primes above $Y$, each meeting
$O(B(N/Y+1))$ other windows, which proves the maximum-degree bound.
The expansions follow by substituting the asymptotic expansion of
$\operatorname{Ei}(u)$ into \eqref{eq:saddle-definitions}.
\Cref{supp:prop:cutoff-calculation} gives the uniform remainder calculation
and both expansions in detail.
\end{proof}

\begin{theorem}[Hard deletion and soft retention]\label{thm:scalar-rates}
Fix $0<\beta_-<\beta_+<\infty$. Uniformly when
$\beta_-\log N\le L+1\le\beta_+\log N$, for every $\varepsilon>0$,
\begin{align}
 \dTV(\Law(Z_{N,L}),\Pois(\lambda_N))
 &\ll\min\left\{1,\lambda_N
       \left(e^{-V_N+o(\nu_N)}+N^{-1/3+\varepsilon}\right)\right\},
       \label{eq:hard-rate}\\
 \dTV(\Law(Z_{N,L}),\Pois(\lambda_N))
 &\ll\min\left\{1,
       e^{-(\widehat V_N-\log^+\lambda_N)/2+o(\widehat\nu_N)}
                   +N^{-1/3+\varepsilon}\right\}.
       \label{eq:soft-rate}
\end{align}
The first estimate also bounds the mean conditional distance given
$\mathcal F_{Y_N^\sharp}$, and the second given
$\mathcal F_{\widehat Y_N}$. Neither saddle is asserted to be optimal for
the distribution itself.
\end{theorem}
\begin{proof}
At the hard cutoff, $|D_Y|=o(N)$ and $\mu_{I_N}\asymp\lambda_N$.
The master estimate and $Q_B=2N/\lambda_N$ give
\[
 R_2(N,L)/N^2\ll N^{-1/3+\varepsilon}(1+\lambda_N^{-1}).
\]
Its contribution to \eqref{eq:transfer} is
$O(\lambda_NN^{-1/3+\varepsilon})$. The defect mass contributes
$\lambda_NN^{-1/2+o(1)}$ and the cutoff terms contribute
$\lambda_Ne^{-V_N+o(\nu_N)}$. This proves \eqref{eq:hard-rate}.

For the second estimate retain the exceptional indicators in the scalar
sum instead of coupling all of them to zero. The scalar Stein equation
has factors $\|f\|_\infty\ll1\wedge\lambda^{-1/2}$ and
$\|\Delta f\|_\infty\le1\wedge\lambda^{-1}$
\cite{Krokowski2017,Rollin2012SteinFactors}.
Telescope only the good-start terms, using their exact conditional
marginal. Bound the exceptional part by its total first moment times
$\|f\|_\infty$. Exceptional neighbours of a good start stay in the
neighbour sums; their mean is controlled after averaging, not pointwise.
Set $\ell=\log^+\lambda$ and $K=\max(1,\lambda)=e^\ell$.
The same two Stein factors, at the same cutoff, give for every $\lambda>0$
the upper bound
\begin{equation}\label{eq:soft-free-rate}
 e^{\ell/2-D(H_N/w)+o(H_N/w)}
 +e^{\ell-w+o(H_N/w)}+K N^{-1/3+\varepsilon/2}.
\end{equation}
Here a smaller arithmetic exponent absorbs the subpolynomial terms.
The detailed calculation is in \cref{supp:sec:soft-application}.
At $w=\widehat V_N$ choose a common nonnegative remainder
$\eta_N\widehat\nu_N$, $\eta_N\to0$, in the two exponentials and put
$q=\exp(-(w-\ell)/2+\eta_N\widehat\nu_N)$.
If $q+N^{-1/3+\varepsilon}\ge1$, the bound $\dTV\le1$ suffices.
Otherwise $q<1$ implies $\ell\le w$, the second exponential is at most
$q$, and $K\le e^w\le N^{\varepsilon/2}$ for all sufficiently large $N$.
This proves \eqref{eq:soft-rate}. In particular intensities below one
are covered directly, without switching to a different conditioning
argument. All thresholds precede the length in the fixed band; the
fibrewise proof gives the claimed mean distance for the full
$\mathcal F_{\widehat Y_N}$.
\end{proof}
The hard estimate is more accurate at bounded intensity. The soft estimate
permits larger intensities: $\log^+\lambda_N\le\widehat V_N-c\widehat\nu_N$
for fixed $c>0$ suffices. In particular the hard-cutoff endpoint
$\log\lambda_N=V_N+O(1)$ is covered, but no convergence at the exact
new endpoint $\log\lambda_N=\widehat V_N$ is inferred.

Write $(z)_2=z(z-1)$ for the second falling factorial.
\begin{corollary}[First two moments]\label{cor:moments}
At critical lengths,
\begin{align*}
 \EE Z_{N,L}&=\lambda_N+O(N^{-1/2+o(1)}),\\
 \EE(Z_{N,L})_2&=\lambda_N^2+O_\varepsilon(N^{-1/3+\varepsilon}),\\
 \operatorname{Var}Z_{N,L}&=\lambda_N+O_\varepsilon(N^{-1/3+\varepsilon}).
\end{align*}
\end{corollary}
\begin{proof}
The first assertion is \cref{cor:first-moment}. Expand the second factorial
moment over ordered distinct starts. Overlaps cost $O(NLp^2)$;
touching pairs cost $p^2N^{1+o(1)}$ by \cref{lem:local-pairs}.
Separated pairs differ from $p^2$ by at most
$p^2(2^{\rho(x,y)}-1)$, whose sum is
$O_\varepsilon(N^{-1/3+\varepsilon})$. The variance identity concludes.
This argument proves the moments directly: total variation alone would
not imply convergence of unbounded moments.
\end{proof}

% END ORIGINAL SOURCE: sections/04_poisson.tex

% BEGIN ORIGINAL SOURCE: sections/05_patterns_fields.tex
\section{Rare word fields, exact marks and the Poisson staircase}
\label{sec:patterns-fields}
The two-window estimate controls dependence between distant words. Their
local dependence has a different origin: a suffix of one word may coincide
with a prefix of another. Keeping these two issues separate gives a joint
law for a growing dictionary of prescribed words, and then a sign-resolved
law for long runs.

\subsection{A dictionary of prescribed words}\label{sec:word-dictionaries}
Joint word and clump approximations are classical in Markov sequences
\cite{ReinertSchbath1998}. Here the local overlap condition is kept
separate from the arithmetic estimate for separated windows.
Let $\mathcal W\subset\{\pm1\}^B$ be a nonempty deterministic set of distinct
words, and write $m=|\mathcal W|$, $p_B=2^{-B}$ and $\Lambda=Nmp_B$.
All signs, including the first one, are prescribed. Define
\[
 I_{x,w}=\1_{\{f(x-1+j)=w_j\ (0\le j<B)\}},
              \qquad x\in I_N,\quad w\in\mathcal W.
\]
The first letter is at $x-1$; this indexing matches the left-border
convention for runs.
For $1\le d<B$ write $w\mathrel{\sim_d}v$ when
$w_{d+j}=v_j$ for $0\le j<B-d$. The directed overlap weight is
\begin{equation}\label{eq:dictionary-overlap}
 \Omega(\mathcal W)=\frac1m\sum_{w,v\in\mathcal W}
                   \sum_{d=1}^{B-1}2^{-d}\1_{\{w\sim_d v\}}.
\end{equation}
The normalization makes $\Lambda\Omega$ the expected scale of ordered
local pairs, up to an absolute factor. It includes self-overlaps and
cross-overlaps; controlling each word separately would miss the latter.
For example, $(+,+,-)$ followed one site later by $(+,-,+)$ is compatible:
the two occurrences fit into $(+,+,-,+)$. The overlap is a constraint on
the words themselves, before the values of the multiplicative function
are considered.

\Needspace{25\baselineskip}
\begin{theorem}[Joint Poisson approximation for a word dictionary]
\label{thm:word-field}
Fix $0<\beta_-<\beta_+<\infty$ and $0<\varepsilon<1/3$. Uniformly for
$\beta_-\log N\le B\le\beta_+\log N$ and all dictionaries $\mathcal W$,
\begin{align}
 &\EE\,\dTV\left(\Law((I_{x,w})_{x\in I_N,w\in\mathcal W}
                       \mid\mathcal F_{Y_N^\sharp}),
            \bigotimes_{x\in I_N,w\in\mathcal W}\Pois(p_B)\right)
       \notag\\
 &\quad\ll_{\beta_-,\beta_+,\varepsilon}
 \Lambda\Omega(\mathcal W)
 +\Lambda(1+\Lambda)e^{-V_N+o(\nu_N)}\notag\\
 &\qquad\quad
 +N^{-1/3+\varepsilon}
       \left(\Lambda+\Lambda^{11/6}m^{1/6}
                       +\Lambda^{4/3}m^{2/3}\right).
       \label{eq:word-field-rate}
\end{align}
The implied constant and the little-oh term are independent of the words
and of $m$. In particular, fix $0<\eta<1/2$ and $C>0$. If
\begin{equation}\label{eq:dictionary-critical}
 m_N\le N^{1/2-\eta},\qquad
 |B_N-\log_2(Nm_N)|\le C,\qquad
 \Omega(\mathcal W_N)\longrightarrow0,
\end{equation}
then the entire site-and-word-labelled field converges in total variation
to the independent Poisson field. Its error is
\begin{equation}\label{eq:dictionary-critical-rate}
 O_{C,\eta}\!\left(\Omega(\mathcal W_N)
                  +e^{-V_N+o(\nu_N)}+N^{-\eta/2}\right).
\end{equation}
Every deterministic restriction of the site--word index set, and every
measurable function of the field, satisfies the same bound.
\end{theorem}
\begin{proof}
Use $Y=Y_N^\sharp$ and remove starts whose $B$-vertex support contains a
$Y$-defective integer. Let $D_Y^{\rm val}$ denote this set and put
$\mathcal M_B^{\rm val}=\sum_{x\in I_N}(2^{m_B(x)}-1)$ and $a=mp_B$.
Distinct words at a site are mutually exclusive, so $0<a\le1$ and
$\Lambda=Na$. By \cref{cor:absolute-marginal}, deleting the actual and
target coordinates costs at most
\[
 a\mathcal M_B^{\rm val}+2a|D_Y^{\rm val}|.
\]
All retained word events have conditional marginal $p_B$. Join all words
at the same site and all words at sites with intersecting prime supports
above $Y$, adding edges for overlapping supports if necessary. After
conditioning this is an exact dependency graph. Its first Poisson term is
$O(a^2(NB+E_Y))$, where $E_Y$ counts ordered pairs of distinct sites
sharing a prime above $Y$.

For $y=x+d$ with $1\le d<B$, two specified words are incompatible unless
$w\sim_d v$. In the compatible case they prescribe all $B+d$ values in
the union. Every vertex of a good union has a private coordinate above
$Y$, because its diameter is less than $2B<Y$. Hence
\begin{equation}\label{eq:absolute-overlap-law}
 \PP(I_{x,w}I_{x+d,v}=1\mid\mathcal F_Y)
          =2^{-(B+d)}\1_{\{w\sim_d v\}}.
\end{equation}
Summing ordered local pairs costs at most $2\Lambda\Omega(\mathcal W)$.
Different words at the same site contribute zero to the second term.

For separated good sites put $I_x^{\mathcal W}=\sum_{w\in\mathcal W}I_{x,w}$.
This is an indicator, not a count with possible multiplicities, and
$\PP(I_x^{\mathcal W}=1)=a$. Fourier inversion on the valuation rows also gives
$\PP(I_x^{\mathcal W}I_y^{\mathcal W}=1)\le a^2 2^{\rho_B^{\rm val}(x,y)}$. Combining the two
bounds \emph{before} summing over sites yields
\begin{align}
 \sum_{w,v\in\mathcal W}\PP(I_{x,w}I_{y,v}=1)
 &=\PP(I_x^{\mathcal W}I_y^{\mathcal W}=1)\notag\\
 &\le \min\{a,a^2 2^{\rho_B^{\rm val}(x,y)}\}\notag\\
 &\le a^2\left(1+
       \min\{a^{-1},2^{\rho_B^{\rm val}(x,y)}-1\}\right).
       \label{eq:dictionary-saturation}
\end{align}
Only the error summands have been grouped: the Poisson theorem is still
applied to the full family indexed by $(x,w)$. No word or position
label is discarded. Its averaged second term over separated graph
edges is therefore at most
$a^2(E_Y+R_2^{\mathrm{val},[1/a]}(N,B))$. We have proved the finite
transfer inequality
\begin{align}
 \EE\,\dTV
 &\ll a(\mathcal M_B^{\rm val}+|D_Y^{\rm val}|)
       +a^2\left(NB+E_Y+R_2^{\mathrm{val},[1/a]}(N,B)\right)
       +\Lambda\Omega(\mathcal W).
       \label{eq:dictionary-finite-transfer}
\end{align}

Here $\mathcal M_B^{\rm val}\le N^{1/2+o(1)}$,
$|D_Y^{\rm val}|/N\le e^{-V_N+o(\nu_N)}$, and
$E_Y/N^2\le e^{-V_N+o(\nu_N)}+N^{-1+o(1)}$.
Use \cref{prop:capped-profile} with $T=N/\Lambda$ and
$Q_B=2^B=Nm/\Lambda\ge T$. Its three contributions, after multiplication
by $a^2$, are bounded by
\[
 N^{-1/3+\varepsilon}
 \left(\Lambda^{11/6}m^{1/6}
                +\Lambda^{4/3}m^{2/3}+\Lambda\right).
\]
The smaller polynomial errors are absorbed with a smaller intermediate
$\varepsilon$: for example, $m\ge\Lambda/N$ implies
$\Lambda^2/N\le
N^{-1/3}\Lambda^{4/3}m^{2/3}$. This proves
\eqref{eq:word-field-rate}.
Under \eqref{eq:dictionary-critical}, $\Lambda$ is bounded above and
below and $B/\log N$ stays in one fixed compact interval. The polynomial
term is $O(N^{-\eta/2})$ on taking $\varepsilon=\eta/6$.
Restriction and measurable contraction cannot increase total variation.
\end{proof}

The exponent $1/2$ is a sufficient uniform bound, not an optimality claim. The
remaining restriction comes from the nonterminal term $NQ_B^{2/3}$:
after normalization it gives $N^{-1/3+\varepsilon}m^{2/3}$ at bounded
intensity. The terminal term has instead been capped by the marginal
probability. This distinction matters: a large raw relation count need
not produce a proportionally large probability of seeing a dictionary
word.

\begin{corollary}[Replacement by independent signs]\label{cor:iid-word-replacement}
Let $(\xi_n)$ be independent fair signs and let $I^{\rm iid}_{x,w}$ be the
same word indicators formed from them. Under
\eqref{eq:dictionary-critical},
\[
 \dTV\left(\Law((I_{x,w})_{x,w}),
                     \Law((I^{\rm iid}_{x,w})_{x,w})\right)\longrightarrow0.
\]
In particular the vector of counts of the individual words is jointly
close to independent Poisson variables of mean $N2^{-B}$, even when the
number of words tends to infinity.
\end{corollary}
\begin{proof}
For the independent sequence the graph needs only overlapping supports.
Equation \eqref{eq:absolute-overlap-law} is exact without conditioning,
and its process Poisson error is
$O(\Lambda\Omega(\mathcal W)+\Lambda^2B/N)$.
Compare both fields with the same independent Poisson field and apply
\cref{thm:word-field}. Summing sites is a measurable contraction.
\end{proof}

\subsection{Typical dictionaries: average before taking the worst case}
\label{sec:typical-dictionaries}
The uniform theorem protects against an adversarial choice of words. A
randomly chosen dictionary asks a different question: can its particular
selection avoid most of the arithmetic dependence? The useful average is
not the law obtained by forgetting the dictionary. We first fix the
selected words, compare the resulting labelled laws, and only then average
the distances.

For a dictionary $\mathcal W$ and a deterministic start mask $A\subset I_N$,
write
\begin{equation}\label{eq:typical-distance}
 \Delta_{N,A,Y}(\mathcal W)=
 \EE\,\dTV\left(\Law((I_{x,w})_{x\in A,w\in\mathcal W}\mid\mathcal F_Y),
           \bigotimes_{x\in A,w\in\mathcal W}\Pois(2^{-B})\right).
\end{equation}
The expectation in this definition is over the prime environment only.
At $A=I_N$ and $Y=Y_N^\sharp$, abbreviate it by $\Delta_N(\mathcal W)$.

\begin{theorem}[Typical word dictionaries]\label{thm:typical-dictionaries}
Fix $0<\beta_-<\beta_+<\infty$, $0<\lambda_-<\lambda_+<\infty$ and
$0<\varepsilon<1/3$. For all sufficiently large $N$, uniformly for integers $B,m$ with
\[
 \beta_-\log N\le B\le\beta_+\log N,\qquad
 1\le m\le2^B,\qquad \lambda_-\le\Lambda=Nm2^{-B}\le\lambda_+,
\]
choose $\mathcal W$ uniformly among the $m$-element subsets of
$\{\pm1\}^B$, independently of $f$. There is a deterministic bound
\begin{equation}\label{eq:typical-rate}
 \EE_{\mathcal W}\Delta_N(\mathcal W)\le r_N,\qquad
 r_N\ll_{\beta_-,\beta_+,\lambda_-,\lambda_+,\varepsilon}
             e^{-V_N+o(\nu_N)}+N^{-1/3+\varepsilon}.
\end{equation}
The little-oh is uniform in $B$ and $m$. For every $t>0$,
\begin{equation}\label{eq:typical-markov}
 \PP_{\mathcal W}\{\Delta_N(\mathcal W)>t\}\le \min\{1,r_N/t\}.
\end{equation}
Thus $t_N\to0$ and $r_N/t_N\to0$ give a class of dictionaries of
proportion tending to one, each with mean conditional distance at most
$t_N$. Every deterministic restriction of its site--word coordinates obeys the same
bound. Moreover the mean, over $\mathcal W$, of the unconditional
 distance between its multiplicative and independent-sign word fields
is at most $r_N+O(\Lambda^2B/N)$.
\end{theorem}
\begin{proof}
Put $p_B=Q^{-1}$, $Q=2^B$, $a=m/Q$, and
$b=m(m-1)/(Q(Q-1))$. For fixed words $u,v$,
\begin{equation}\label{eq:dictionary-selection}
 \PP_{\mathcal W}(u,v\in\mathcal W)
       =b+(a-b)\1_{\{u=v\}},\qquad 0\le b\le a^2\le a.
\end{equation}
In particular, if $\mathbf f_x=(f(x-1+j))_{0\le j<B}$, independence of the
selection from $f$ gives
\begin{equation}\label{eq:selected-pair}
 \EE_{\mathcal W}\PP_f(\mathbf f_x,\mathbf f_y\in\mathcal W)
       =b+(a-b)\PP_f(\mathbf f_x=\mathbf f_y)
       \le a^2+a\PP_f(\mathbf f_x=\mathbf f_y).
\end{equation}
For disjoint supports, let $\kappa(x,y)$ be the nullity of the $B$
valuation vectors for the ratios of corresponding values. Sending a
coefficient vector $t$ to $(t,t)$ injects its relation space into the
full $2B$-value relation space. Therefore
\begin{equation}\label{eq:word-collision}
 \PP_f(\mathbf f_x=\mathbf f_y)=2^{-B}2^{\kappa(x,y)}
                     \le p_B2^{\rho_B^{\rm val}(x,y)}.
\end{equation}
For good supports at distance $1\le d<B$ and $Y>2B$, all $B+d$ values
in their union are conditionally uniform. Their equality graph has $d$
chains and $B$ independent edges. Thus
$\PP_f(\mathbf f_x=\mathbf f_{x+d}\mid\mathcal F_Y)=p_B$.

Apply the process Chen--Stein bound to each fixed dictionary on each
prime fibre, using the same site--word graph as in
\cref{thm:word-field}. Average this \emph{nonnegative error bound} over
$\mathcal W$. The mean real deletion cost at each bad site is exactly
$a$, regardless of its arithmetic word law; the target cost is at most
$a$. Products of good marginals cost $O(a^2(NB+E_Y))$. Distinct words
at the same site cannot occur together. Equations
\eqref{eq:selected-pair}--\eqref{eq:word-collision} bound all remaining
pairs: their baseline is absorbed since $ap_B\le a^2$, and their
arithmetic excess costs $ap_BR_2^{\rm val}(N,B)$. More generally, for
every deterministic mask $A$ this proves
\begin{equation}\label{eq:typical-finite}
 \EE_{\mathcal W}\Delta_{N,A,Y}(\mathcal W)
 \ll a|D_Y^{\rm val}\cap A|+
 a^2\bigl(|A|B+E_Y(A)\bigr)+ap_BR_2^{\rm val}(A,B).
\end{equation}
Here $E_Y(A)$ counts ordered distinct-site pairs sharing a nonzero valuation-parity
coordinate above $Y$, and $R_2^{\rm val}(A,B)$ sums over ordered pairs
in $A$ at distance at least $B$. All sums may include deleted sites.
The companion, \cref{supp:sec:typical-details}, gives the full finite
bookkeeping. In particular no weighted defect term is removed from the
\emph{uniform} theorem: its removal here uses the dictionary average.

At $Y=Y_N^\sharp$, the deletion and edge bounds are those of
\cref{prop:cutoffs}. The coarse full-value profile gives
\[
 ap_BR_2^{\rm val}(N,B)
 \ll_\varepsilon \Lambda N^{-1/3+\varepsilon}
                           \left(1+\frac{\Lambda}{m}\right).
\]
Together with $a^2NB=\Lambda^2B/N$, this proves
\eqref{eq:typical-rate}. Markov's inequality proves
\eqref{eq:typical-markov}. Restrictions contract the distance for each
fixed dictionary, before either expectation. For independent signs there
are only local edges; the same selection calculation gives mean Poisson
error $O(\Lambda^2B/N)$. Comparing the two fields with their common
Poisson target proves the last assertion.
\end{proof}
For example, $m_N\asymp N$ and $B_N=2\log_2N+O(1)$ are allowed.
More generally $m_N\asymp N^\alpha$ for any fixed $\alpha\ge0$ is
allowed with $B_N=(1+\alpha)\log_2N+O(1)$ and a suitable fixed band.
There is no uniform assertion over all dictionaries of those sizes.
The dictionary is selected independently of the prime signs and then
held fixed; neither adaptation to the observed sequence nor an
almost-sure conclusion over all scales is part of the theorem.

\begin{corollary}[Random affine dictionaries]\label{cor:affine-dictionaries}
Identify sign words with $\F^B$. Choose an $r\times B$ matrix $A_0$
uniformly among the matrices of row rank $r$, where $0\le r\le B$,
and choose $b_0\in\F^r$ independently and uniformly. The random dictionary
\[
 \mathcal W=\{u\in\F^B:A_0u=b_0\},\qquad m=2^{B-r},
\]
is independent of $f$. Under the band and intensity hypotheses of
\cref{thm:typical-dictionaries}, all its conclusions hold for this
ensemble as well. In particular $r=\log_2N+O(1)$ and
$B=2\log_2N+O(1)$ give dictionaries of order $N$ specified by
$r(B+1)=O((\log N)^2)$ bits.
\end{corollary}
\begin{proof}
Each codimension-$r$ kernel has the same number of ordered row bases,
and uniform $b_0$ chooses uniformly among its cosets. A fixed word is
included with probability $a=2^{-r}$. For distinct $u,v$, invariance
under invertible linear maps gives
\[
 \PP(A_0(u-v)=0)=\frac{2^{B-r}-1}{2^B-1},\qquad
 \PP(u,v\in\mathcal W)=2^{-r}\frac{2^{B-r}-1}{2^B-1}
                      =\frac{m(m-1)}{Q(Q-1)}.
\]
These are precisely the one- and two-point probabilities used in
\eqref{eq:typical-finite}; no higher inclusion probability was used.
They transfer the nonnegative deletion, pair and overlap costs, not an
identity between the two ensembles' average total-variation distances.
The dictionary remains fixed while bounding each distance; only its
bound is then averaged over the actual matrix and offset.
The cases $r=0$ and $r=B$ obey the same formulas.
\end{proof}
The corollary supplies a short random description and a high-probability
statement, not a deterministic construction certified at every scale.
The proof uses the one- and two-point sampling identities of the affine
ensemble. A short matrix description by itself does not certify the
Poisson approximation for a chosen dictionary or control its overlap
weight.

The following exact identity controls local overlaps alone. It is useful
for the uniform theorem, but does not replace averaging the full
distance as in \cref{thm:typical-dictionaries}.

\begin{corollary}[Most dictionaries have small overlap]\label{cor:generic-dictionaries}
Let $B\ge1$ and $1\le m\le2^B$. Choose an $m$-element subset $\mathcal W$
uniformly among the subsets of $\{\pm1\}^B$.
This auxiliary choice is independent of $f$. Then
\begin{equation}\label{eq:generic-overlap}
 \EE_{\mathcal W}\Omega(\mathcal W)=\frac{m(B-1)}{2^B}.
\end{equation}
Under the first two conditions in \eqref{eq:dictionary-critical}, the
fraction of dictionaries with $\Omega(\mathcal W)>N^{-1/2}$ is
$O_{C,\eta}(\log N/\sqrt N)$. Every dictionary outside this exceptional
set has the field approximation \eqref{eq:dictionary-critical-rate}.
\end{corollary}
\begin{proof}
Put $Q=2^B$. At each proper displacement $1\le d<B$, there are
$2^d$ self-compatible words and $2^{B+d}$ ordered compatible pairs.
The number of distinct compatible pairs is therefore $2^d(Q-1)$.
A word belongs to the dictionary with probability $m/Q$; a distinct
ordered pair belongs with probability $m(m-1)/(Q(Q-1))$.
After multiplication by $2^{-d}/m$, the expected contribution of this
displacement is exactly
\[
 \frac{2^{-d}}m\left(2^d\frac mQ
       +2^d(Q-1)\frac{m(m-1)}{Q(Q-1)}\right)=\frac mQ.
\]
Sum over the $B-1$ displacements to get \eqref{eq:generic-overlap}.
This also covers $m=1$; for $B=1$ the sum is empty.
At criticality $m2^{-B}=\Lambda/N=O_C(1/N)$ and $B=O_\eta(\log N)$.
Markov's inequality proves the exceptional-fraction estimate. The result
is a statement about the size of a class of deterministic dictionaries;
no additional randomness is required in \cref{thm:word-field}.
\end{proof}

\begin{corollary}[Explicit dictionaries with no overlaps]
\label{cor:marker-dictionaries}
For all sufficiently large $B$, there is an explicitly described family
$\mathcal C_B\subset\{0,1\}^B$ of size at least $2^B/(32B)$ with no
proper suffix of any word equal to a prefix of any word, including itself.
Every subset of $\mathcal C_B$, after identifying $0,1$ with $-1,+1$,
has $\Omega=0$. In particular deterministic examples exist throughout
the range \eqref{eq:dictionary-critical}.
\end{corollary}
\begin{proof}
This is the binary marker construction of Chee, Kiah, Purkayastha and
Wang~\cite[Section III]{CheeKiahPurkayasthaWang2013}; we include the
short argument and the coarse size bound needed here.
Set $k=\lceil\log_2(2B)\rceil$ and take all words
\[
                0^k\,1\,u\,1,
       \qquad |u|=B-k-2,
       \qquad u\text{ contains no }0^k.
\]
The only occurrence of $0^k$ in such a word starts at its first position:
the first following letter and the last letter are both $1$.
An overlap of length at most $k$ is impossible because the prefix is all
zeros and the suffix ends in $1$. A longer proper overlap would put
$0^k$ at a later position, also impossible.
There are $2^{B-k-2}$ choices before the restriction on $u$. A union
bound over at most $B$ positions excludes at most the fraction
$B2^{-k}\le1/2$. Thus
$|\mathcal C_B|\ge2^{B-k-3}\ge2^B/(32B)$.
At the critical dictionary length, $m/2^B=O(1/N)$ and $B=O(\log N)$,
so any required $m$ words can be selected from this family for large $N$.
\end{proof}
For example, one may select $\lfloor N^{2/5}\rfloor$ such words at length
$B=(7/5)\log_2N+O(1)$. Their full labelled field has a Poisson
approximation. The coding construction is classical; the conclusion for
the multiplicative sequence is the application of the theorem above.

\subsection{Words up to a common sign}
For $a=(a_0,\ldots,a_L)\in\{\pm1\}^{L+1}$ put
$I_x(a)=I_{x,a}+I_{x,-a}$. Define
\begin{equation}\label{eq:overlap-definition}
 \mathcal O(a)=\{1\le d\le L:\exists\eta\in\{\pm1\},\
         a_{d+j}=\eta a_j\ (0\le j\le L-d)\},\qquad
 \Theta(a)=\sum_{d\in\mathcal O(a)}2^{-d}.
\end{equation}
For the two-word dictionary $\{a,-a\}$ one has
$\Omega(\{a,-a\})=\Theta(a)$: every compatible shift has two directed
sign choices.
\begin{corollary}[Poisson law for low-overlap patterns]\label{thm:patterns}
Fix $C>0$ and $\varepsilon>0$. Uniformly in $a=a_N$ and
$|L-\log_2N|\le C$,
\begin{equation}\label{eq:pattern-rate}
 \dTV\left(\Law\left(\sum_{x\in I_N}I_x(a)\right),\Pois(N2^{-L})\right)
 \ll_{C,\varepsilon}\Theta(a)+e^{-V_N+o_C(\nu_N)}+N^{-1/3+\varepsilon}.
\end{equation}
The same bound holds under any deterministic spatial mask, with target
mean $|A_N|2^{-L}$.
\end{corollary}
\begin{proof}
Apply \cref{thm:word-field} with $B=L+1$, $m=2$, and merge the two word
coordinates. Their events are disjoint and their target means add.
\end{proof}
If the least compatible shift $d_*(a_N)$ tends to infinity, then
$\Theta(a_N)\le2^{1-d_*(a_N)}\to0$. For the run-start word
$(-1,+1,\ldots,+1)$ only $d=L$ is compatible. Periodic words can instead
retain a short overlap and need declustering. Notice that the preceding
dictionary theorem also separates $a$ and $-a$: their asymptotic
independence is not assumed in this corollary.

The uniform and typical dictionary comparisons control the rare occurrences of each
prescribed word. We return to constant runs and keep their attributes
together: a start, its full length, its sign and all thresholds it exceeds.

\subsection{Exact marks without a growing-mark loss}
Almost surely every rightward constant run is finite. Indeed constancy on
$[x,\infty)$ would make all prime signs beyond $x$ equal, an event of
probability zero; take a countable union over $x$. Thus
\begin{equation}\label{eq:mark-partition}
 J_{x,L}=\sum_{e\ge0}K_{x,e},\qquad
 K_{x,e}=\1_{\{f(x-1)=-f(x),\ f(x)=\cdots=f(x+L+e-1),\
                              f(x+L+e)=-f(x)\}}.
\end{equation}
This system has $L+e+1$ independent homogeneous rows on a good support and
conditional marginal $p_e=2^{-L-e-1}$. Its homogeneous rows are nested in
those for a larger mark, although the terminal affine right-hand side
changes. Marks at the same start are mutually exclusive. We may retain the sign as
well:
\begin{equation}\label{eq:signed-mark}
 K^s_{x,e}=K_{x,e}\1_{\{f(x)=s\}},\qquad
 p_{e,s}=2^{-L-e-2},\qquad s\in\{\pm1\}.
\end{equation}
On a good support this event prescribes every value, so its conditional
probability is $p_{e,s}$. The exact identities are
$\sum_sK^s_{x,e}=K_{x,e}$ and $\sum_{e,s}K^s_{x,e}=J_{x,L}$.

\begin{theorem}[Finite exact-mark comparison]\label{thm:marked-field}
Let $E\ge0$, $Q=L+E+1$, and assume
$\beta_-\log N\le L+1\le Q+1\le\beta_+\log N$.
For $Y>2(Q+1)$ let $D_Y(Q)$ be the starts whose support
$\{x-1,\ldots,x+Q-1\}$ contains a $Y$-defective integer, and let $E_Y(Q)$
be the ordered graph-edge count for these maximal supports. Then
\begin{align}
 &\EE\,\dTV\left(\Law((K_{x,e})_{x\in I_N,\ 0\le e\le E}\mid\mathcal F_Y),
                 \bigotimes_{x\in I_N,\ 0\le e\le E}\Pois(p_e)\right)
 \notag\\
 &\quad\ll p\mathcal M_B+p|D_Y(Q)|
       +p^2\bigl(NQ+E_Y(Q)+R_2(N,L)\bigr),\qquad p=2^{-L}.
       \label{eq:marked-finite}
\end{align}
Exactly the same upper bound holds for the sign-resolved family
$(K^s_{x,e})_{x\in I_N,\,0\le e\le E,\,s\in\{\pm1\}}$, compared with
independent Poisson variables of means $p_{e,s}$. There is no polynomial
factor in the number of marks. At $Y=Y_N^\sharp$,
the right side is at most
\begin{equation}\label{eq:marked-hard}
 C_\varepsilon\lambda_N(1+\lambda_N)
       \left(e^{-V_N+o(\nu_N)}+N^{-1/3+\varepsilon}\right).
\end{equation}
\end{theorem}
\begin{proof}
On good maximal supports all marginals are $p_e$. Apply the process
Poisson theorem to the marked family, allowing all marks on neighbouring
supports to be neighbours, and all marks at the same site to be neighbours.
The two identities
\begin{equation}\label{eq:mark-folding}
 \sum_{e=0}^Ep_e\le p,\qquad
 \sum_{e,f=0}^E\PP(K_{x,e}K_{y,f}=1)\le\PP(J_{x,L}J_{y,L}=1)
\end{equation}
remove any double sum over marks before estimating the pair mass.
Local pairs cost $O(NQp^2)$ and separated pairs cost
$p^2(E_Y(Q)+R_2(N,L))$. Bad actual marks at one site have total mass at
most that of $J_{x,L}$, so their deletion is bounded by
$p\mathcal M_B+p|D_Y(Q)|$, regardless of which late vertex caused removal.
This proves \eqref{eq:marked-finite}.
The bounds $|D_Y(Q)|\le(Q+1)A(3N,Y)$ and \eqref{eq:edge-count} with
$B$ replaced by $Q+1$, together with \cref{thm:master-profile,prop:cutoffs},
give \eqref{eq:marked-hard}.

For the signed version, the same-site alternatives are still disjoint and
both sums in \eqref{eq:mark-folding} remain valid after summing signs.
It remains to check local pairs before that summation. Put $a=L+e$,
$b=L+f$, and $d=y-x>0$. If $d<a$ the pair is impossible. If $d=a$, the
two prescribed value strings share two vertices, require the second run
sign to be the opposite of the first, and have probability
$4p_{e,s}p_{f,t}$ when compatible. If $d=a+1$, they share one vertex,
require equal run signs, and have probability $2p_{e,s}p_{f,t}$.
For $d>a+1$ their actual supports are disjoint, and a good local union has
probability $p_{e,s}p_{f,t}$. These assertions follow by counting the
prescribed values in a union whose private coordinates make them uniform.
Thus the local sum is still $O(NQp^2)$ and all other terms are unchanged.
\end{proof}
The use of \eqref{eq:mark-folding} is essential. An exact mark $e<E$ does
not imply a threshold event at length $L+E+1$; a bad vertex can occur beyond
its support. We bound deletion by the base event, not the longest event.

The exact tail identity is
\begin{equation}\label{eq:mark-tail}
 \{\exists x\in I_N,e>E:K_{x,e}=1\}=
                      \{Z_{N,L+E+1}>0\}.
\end{equation}
The first-moment estimate bounds this probability by
$\lambda_N2^{-(E+1)}(1+N^{-1/2+o(1)})$ whenever the enlarged length is in
a fixed logarithmic band. The independent target tail has mass
$\lambda_N2^{-(E+1)}$. Couple both tails to zero.
At criticality choose $E=\lceil V_N/\log2\rceil$. This is $o(\log N)$;
\eqref{eq:marked-hard} and \eqref{eq:mark-tail} prove the full-field
estimate, including signs, in \cref{thm:poisson-main}. Riemann sums and
geometric tail tightness give its weak diffuse limit.

\begin{corollary}[An explicit compound-Poisson cluster law]
\label{cor:window-clusters}
Let $C_{N,L}$ count all constant length-$L$ windows whose left endpoint
lies in $I_N$, without requiring left maximality. At criticality its law
is at distance $o(1)$ from that of
\[
 \sum_{i=1}^{P_N}H_i,\qquad P_N\sim\Pois(\lambda_N),\qquad
 \PP(H_i=h)=2^{-h},\quad h\ge1,
\]
with independent $H_i$ and $P_N$. The target probability generating function
is $\exp[\lambda_N(z/(2-z)-1)]$ for $0\le z\le1$.
\end{corollary}
\begin{proof}
A maximal run of length $L+e$ contributes $e+1$ windows. Thus, apart from
runs crossing an endpoint of $I_N$,
$C_{N,L}=\sum_{x\in I_N,e\ge0}(e+1)K_{x,e}$.
A discrepancy implies a constant length-$L$ window starting at $N$ or
at $2N-1$. Each such event has $L-1$ relative constraints; private pivots
and the pointwise defect bound give probability at most
$2^{-(L-1)+O(\log N/\log\log N)}=N^{-1+o(1)}$.
The full marked-field comparison contracts under the measurable weighted
sum. Its target is the stated compound Poisson law, since $H=e+1$.
No moment convergence is inferred from this contraction.
\end{proof}

\subsection{Growing depth: distinguish positions from counts}
Put $b_N=\lfloor\log_2N\rfloor$, $\vartheta_N=\log_2N-b_N$. For
$d_N\in\NN$ with $d_N=o(\log N)$, set
\[
 L_N=b_N-d_N,\qquad \Lambda_N=N2^{-L_N}=2^{\vartheta_N+d_N}.
\]
Let $K^{\rm ex}_{x,q}$ denote the exact-length event for a left-maximal run
of length $q$ (zero for $q<1$). The moving field and its target are
\begin{equation}\label{eq:moving-field}
 \mathcal E_{N,d_N}=\sum_{x\in I_N}\sum_{r\ge-d_N}
             K^{\rm ex}_{x,b_N+r}\delta_{(x/N,r)},\qquad
 \EE\Pi_{N,d_N}\{(x/N,r)\}=2^{-b_N-r-1},
\end{equation}
where the target atoms are independent Poisson variables. Their total mean
is $\Lambda_N$. Define exact-level and threshold counts by
\[
 X_{N,r}=\sum_{x\in I_N}K^{\rm ex}_{x,b_N+r},\qquad
 T_{N,m}=\sum_{r\ge m}X_{N,r}=Z_{N,b_N+m}.
\]
For the target write $P_{N,r}=\Pi_{N,d_N}([1,2]\times\{r\})$ and
$S_{N,m}=\sum_{r\ge m}P_{N,r}$. Thus
$P_{N,r}\sim\Pois(2^{\vartheta_N-r-1})$ independently.

The preceding estimates bound the mean, over the small-prime environment,
of a conditional total-variation distance. Restricting this average to an
event $C_N\in\mathcal F_Y$ multiplies the error by at most
$1/\PP(C_N)$; this is the stable-kernel principle of
\cref{lem:stable-lift}. It is the first information cost below.

\begin{theorem}[Two levels of joint approximation]\label{thm:moving-comparison}
Fix $c>0$ and $0<\varepsilon<1/3$. Let $C_N$ be an arithmetic
conditioning event of positive probability and
put $I_N^{\rm info}=-\log\PP(C_N)$.

\textup{(i) Labelled field.} If $C_N\in\mathcal F_{Y_N^\sharp}$, a
sufficient condition for full-field total variation to tend to zero is
\begin{equation}\label{eq:labelled-hard-budget}
 I_N^{\rm info}+\log\Lambda_N+\log(1+\Lambda_N)\le V_N-c\nu_N.
\end{equation}
At a movable cutoff, if $C_N\in\mathcal F_{\widehat Y_N}$, the sufficient
condition improves to
\begin{equation}\label{eq:labelled-budget}
 I_N^{\rm info}+\log\Lambda_N\le\widehat V_N/2-c\widehat\nu_N.
\end{equation}
In the nontrivial range, the latter comparison has upper bound
\begin{align}
 &e^{I_N^{\rm info}+\log\Lambda_N-\widehat V_N/2+o(\widehat\nu_N)}
 +e^{I_N^{\rm info}+2\log\Lambda_N-\widehat V_N+o(\widehat\nu_N)}\notag\\
 &\hspace{20mm}+e^{I_N^{\rm info}}\Lambda_N(1+\Lambda_N)
                         N^{-1/3+\varepsilon+o(1)}.
 \label{eq:labelled-rate}
\end{align}
\textup{(ii) Aggregated exact levels.} If $C_N\in\mathcal F_{Y_N^\sharp}$
and
\begin{equation}\label{eq:staircase-budget}
 I_N^{\rm info}+\log\Lambda_N\le V_N-c\nu_N,
\end{equation}
then, for every $0<c'<c$ and $0<\varepsilon<1/3$,
\begin{equation}\label{eq:staircase-rate}
 \dTV\left(\Law((X_{N,r})_{r\ge-d_N}\mid C_N),
       \bigotimes_{r\ge-d_N}\Pois(2^{\vartheta_N-r-1})\right)
 \ll e^{-c'\nu_N}+N^{-1/3+\varepsilon}.
\end{equation}
Exactly the same distance holds for the full threshold paths
$(T_{N,m})_{m\ge-d_N}$ and $(S_{N,m})_{m\ge-d_N}$.
\end{theorem}
\begin{proof}
Conditioning multiplies any averaged kernel error by at most
$e^{I_N^{\rm info}}$. For (i), apply \cref{thm:marked-field} with base
$L_N$ and then remove the mark tail using \eqref{eq:mark-tail}.
For the hard bound one may take
$E=\lceil(I_N^{\rm info}+V_N+\log(2+\Lambda_N))/\log2\rceil$.
Under \eqref{eq:labelled-hard-budget} this is $o(\log N)$ and both tail
costs are absorbed by the displayed error.
At a free cutoff $Y=e^w$, the leading process errors are
\[
 e^{I_N^{\rm info}}\Lambda_N e^{-D(H_N/w)+o(H_N/w)},\qquad
 e^{I_N^{\rm info}}\Lambda_N^2e^{-w+o(H_N/w)}.
\]
Choose $w=\widehat V_N$, so $D(H_N/w)=w/2$, and take an equally large
$o(\log N)$ tail cutoff. This proves \eqref{eq:labelled-rate} and
\eqref{eq:labelled-budget}. The hard choice remains sharper at bounded
intensity; the movable choice improves the admissible growing intensity.

For (ii), aggregate sites \emph{before} solving the multivariate Poisson
Stein equation. If $q_e=2^{-(e+1)}$ are the normalized category masses,
Barbour's Lemmas~2--3 \cite[p.~179]{Barbour1988SteinProcess}, as
applied in the companion, bound the directional second-difference factor by
\[
 C\frac{1+\log^+(2\Lambda_N)}{\Lambda_N\sqrt{q_eq_f}}.
\]
The mark sum remains uniform because
$\sum_{e\ge0}\sqrt{q_e}=1+\sqrt2$.
The full argument and the precise finite-cutoff bound are given in
\cref{supp:thm:aggregated}. It yields a leading error
$e^{I_N^{\rm info}}\Lambda_N(1+\log^+(2\Lambda_N))e^{-V_N+o(\nu_N)}$
and a polynomial remainder after the enlarged-length arithmetic estimate.
All subpolynomial factors in that remainder are absorbed by choosing a
smaller initial $\varepsilon$. Choose a constant strictly between $c'$ and
$c$ for the tail cutoff. The scalar relative estimate at the shifted length
bounds both tails by $O(e^{-c'\nu_N})$; see \cref{supp:thm:aggregated}.
Finally, on finite-mass sequences, the map $X_r\mapsto\sum_{s\ge r}X_s$
is a measurable bijection with inverse $X_r=T_r-T_{r+1}$. Total variation
is preserved, not merely contracted.
\end{proof}
Part (ii) retains every statistic of the multiset of lengths, but not the
positions. A maximal coupling identifies all those retained statistics at
once, with no union bound over thresholds. A field coupling from (i) also
retains positions. The ranges are sufficient bounds for these methods, not
phase transitions. In particular the full one-factor labelled range is not
proved by these comparisons.


The fixed enlarged-cutoff bound remains useful for events exposed up to
$\widehat Y_N$. When their information can be exposed at a smaller
admissible cutoff, \cref{prop:information-labelled} optimizes the two
labelled costs as a function of that information.

\begin{corollary}[Sign-resolved growing fields]\label{cor:signed-growing}
Both conclusions of \cref{thm:moving-comparison} remain valid after retaining
the sign of each run. The labelled target has atom means $2^{-b_N-r-2}$
for each $s\in\{\pm1\}$, and the aggregated target has independent means
$2^{\vartheta_N-r-2}$ at each pair $(r,s)$.
\end{corollary}
\begin{proof}
For the labelled result use the signed version of \cref{thm:marked-field}
and the unchanged tail identity. For aggregation, the normalized category
weights are $q_{e,s}=2^{-e-2}$ and
$\sum_{e,s}\sqrt{q_{e,s}}=\sqrt2(1+\sqrt2)$.
The directional Stein bound therefore has the same summability.
At the largest retained support, the full value-relation space dominates
each mixed signed kernel by extension of coefficient vectors by zero;
\cref{prop:absolute-profile} supplies the same two-window envelope.
The retained support has $L+E+2$ vertices, differing by only one from the
relative-mark convention. Since $E=O(V_N)=o(\log N)$, its polynomial
remainder is absorbed exactly as in part (ii). The signed local-pair calculation and the same tail bound finish
the proof.
\end{proof}

\subsection{Target kernels, extremes and the Gaussian lower edge}
The independent target levels give, exactly,
\begin{align}
 S_{N,m}&\sim\Pois(2^{\vartheta_N-m}),\label{eq:target-marginal}\\
 S_{N,m+j}\mid S_{N,m}=n&\sim\operatorname{Bin}(n,2^{-j}),\label{eq:target-forward}\\
 S_{N,m-j}\mid S_{N,m}=n&\ \stackrel d=\ n+
      \Pois(2^{\vartheta_N-m}(2^j-1)),\label{eq:target-reverse}
\end{align}
when the lower levels are retained. The whole forward path is the number
of survivors among $n$ independent geometric lifetimes, each with
$\PP(E=e)=2^{-(e+1)}$. These are identities for the Poisson target, not
exact finite-$N$ Markov assertions for the multiplicative sequence.

For any integer $m\ge0$, the dyadic excess maximum
$M_{N,L}=\max\{\ell_x-L:x\in I_N,J_{x,L}=1\}$, with empty value $-\infty$,
satisfies the exact identity
\begin{equation}\label{eq:excess-void}
 \{M_{N,L}\le m\}=\{Z_{N,L+m+1}=0\}.
\end{equation}
Thus the scalar estimate at $L'=L+m+1$, with $\mu=N2^{-L'}$, gives
\[
 \PP(M_{N,L}\le m)=e^{-\mu}
 +O\left(\min\{1,\mu(e^{-V_N+o(\nu_N)}+N^{-1/3+\varepsilon})\}\right)
\]
whenever $L'+1$ is in a fixed logarithmic band. In particular one-hit upper
tails are relative when $N2^{-L'}\to0$.
For the $k$th largest run length among starts in $I_N$, along a subsequence
$\vartheta_N\to\vartheta$,
\begin{equation}\label{eq:order-statistic}
 \PP(R_N^{[k]}-b_N<m)\longrightarrow
 e^{-2^{\vartheta-m}}\sum_{j=0}^{k-1}\frac{(2^{\vartheta-m})^j}{j!}.
\end{equation}
This is just the event that fewer than $k$ starts exceed the threshold.

\begin{theorem}[Joint Poisson--Gaussian limit]\label{thm:bridge}
Assume $d_N\to\infty$ and the hypotheses of part (ii) of
\cref{thm:moving-comparison}. Put
\[
 G_N(j)=\frac{Z_{N,b_N-d_N+j}-\Lambda_N2^{-j}}{\sqrt{\Lambda_N}}.
\]
Along $\vartheta_N\to\vartheta$, for every fixed $J$ and every finite
$R\subset\Z$, conditionally on $C_N$,
\[
 ((G_N(j))_{0\le j\le J},(X_{N,r})_{r\in R})
 \Longrightarrow ((G(j))_{0\le j\le J},(P_r^{(\vartheta)})_{r\in R}),
\]
where $\EE G(j)G(k)=2^{-\max(j,k)}$, the variables
$P_r^{(\vartheta)}\sim\Pois(2^{\vartheta-r-1})$ are independent, and the
Gaussian and Poisson vectors are independent. The normalized Gaussian
sequence $H(j)=2^{j/2}G(j)$ is the stationary AR(1) chain
$H(j+1)=2^{-1/2}H(j)+2^{-1/2}\zeta_{j+1}$,
where $(\zeta_j)_{j\ge1}$ are independent standard normal random variables,
independent of $H(0)\sim N(0,1)$.
\end{theorem}
\begin{proof}
Transfer to the independent target by \eqref{eq:staircase-rate}.
Choose a fixed $r_0<\min R$ (any fixed $r_0$ if $R$ is empty).
The lower-edge sums restricted to $r<r_0$ are independent of the displayed
critical levels and satisfy the multivariate Poisson central limit theorem.
Their covariance after normalization tends to $2^{-\max(j,k)}$.
The centered sum over $r\ge r_0$ has bounded variance and hence vanishes
after division by $\sqrt{\Lambda_N}$. This proves the joint limit and
independence. The covariance of $H$ is $2^{-|j-k|/2}$, giving the stated
Gaussian autoregression.
\end{proof}
For fixed compact sets of levels, the untruncated centered field has the
phase-indexed target
\[
 {\rm PPP}\left(dt\otimes\sum_{r\in\Z}2^{\vartheta_N-r-1}\delta_r\right).
\]
Riemann sums give the corresponding Laplace approximation without assuming
that $\vartheta_N$ converges. A phase-convergent subsequence turns it into
a fixed weak limit; its intensity is locally finite although it diverges
towards $r=-\infty$. Full formulas and conditional configuration statements
are in \cref{supp:sec:target-consequences}. No continuous run lengths or
canonical coupling simultaneously over all $N$ is asserted.

% END ORIGINAL SOURCE: sections/05_patterns_fields.tex

% BEGIN ORIGINAL SOURCE: sections/06_conditioning.tex
\section{Arithmetic information and resolved conditional laws}
\label{sec:conditional-laws}
There are two different conditioning operations. An arithmetic event fixes
information about prime signs before the Poisson comparison. Conditioning
on a count or another observable then divides by the probability of that
observable. Keeping the two costs separate avoids treating a rare state as
though an absolute error resolved it automatically.


The first table records the sufficient budgets obtained directly by the
scalar and dependency-graph methods; it is not a list of the strongest
available hard-field conclusions. Here $I=-\log\PP(C)$,
$c>0$ is fixed, $Y^\sharp=\lfloor e^V\rfloor$ and
$\widehat Y=\lfloor e^{\widehat V}\rfloor$; $\nu$ and $\widehat\nu$ belong
to those cutoffs. The following rows concern the dyadic domain
$I_N=[N,2N)\cap\Z$ and the scalar or dependency-graph comparisons of
Sections~4--6. Their moving-field intensities $\Lambda$ are at least one.
\begin{center}\small
\begin{tabularx}{\textwidth}{@{}L{.25\textwidth}Xl@{}}\toprule
Observable & Sufficient budget & Prime field\\\midrule
Scalar, hard & $I+\log^+\lambda\le V-c\nu$ & $\mathcal F_{Y^\sharp}$\\
Scalar, soft & $2I+\log^+\lambda\le\widehat V-c\widehat\nu$ & $\mathcal F_{\widehat Y}$\\
All exact-level counts & $I+\log\Lambda\le V-c\nu$ & $\mathcal F_{Y^\sharp}$\\
All sites, graph / hard & $I+\log\Lambda+\log(1+\Lambda)\le V-c\nu$ & $\mathcal F_{Y^\sharp}$\\
All sites, graph / enlarged & $I+\log\Lambda\le\widehat V/2-c\widehat\nu$ & $\mathcal F_{\widehat Y}$\\
All sites, graph / adapted & $\log\Lambda\le\ell_{\rm lab,N}(I)-c\nu_N(I)$ & $\mathcal F_{Y_N(I)}$\\\bottomrule
\end{tabularx}
\end{center}
The last row uses \eqref{eq:information-saddle}--\eqref{eq:information-budget}, with $0\le I\le V_N$. Exact marks and signs can be retained in all site-labelled rows. The inequalities
have fixed negative margins; equality at the boundary and optimality for
the model are not asserted. A further conditioning on an observed count
has the separate probability cost analysed below.

\paragraph{The one-factor hard-field conclusion.}
The selected-pivot theorem improves the available site-labelled range
both on prefixes and, by restriction, on dyadic blocks:
\begin{center}\small
\begin{tabularx}{\textwidth}{@{}L{.29\textwidth}Xl@{}}\toprule
Observable and domain & Sufficient budget & Prime field\\\midrule
All prefix sites, exact excesses and signs &
$I_M+\log\Lambda\le V_M-c\nu_M$ & $\mathcal F_{Y_M^\sharp}$\\
All dyadic sites, exact excesses and signs &
$I_N^{\rm info}+\log\Lambda_N\le V_N-c\nu_N$ & $\mathcal F_{Y_N^\sharp}$\\\bottomrule
\end{tabularx}
\end{center}
The first row is \cref{thm:micro-run-tv}, with
$\Lambda=(M-L)2^{-L}\to\infty$ and
$L=\lfloor\log_2M\rfloor-a_M$, $0\le a_M=o(\log M)$.
The second is \cref{cor:dyadic-micro}, with
$\Lambda_N=N2^{-L_N}\to\infty$ and
$L_N=\lfloor\log_2N\rfloor-d_N$, $0\le d_N=o(\log N)$.
Both use the exact means $2^{-L-e-2}$ on their respective integer
index sets. The dyadic conclusion is obtained by restricting a prefix
of size $4N$ without changing $L_N$ or the integer start labels.
Thus, in this growing-intensity corridor, keeping the sites does not
require the extra intensity factor of the direct hard graph bound.
This does not enlarge the prime field. The direct routes remain useful
for their critical or relative rates and for events measurable only at
a larger or adapted cutoff; no dictionary extension is implicit.

\subsection{Stable kernels and information cost}
\begin{lemma}[Stable product lift]\label{lem:stable-lift}
Let $W$ take values in a standard Borel space, let $\mathcal F$ be a
sigma-field, and let $\nu$ be a deterministic target law. If
\[
 \EE\,\dTV(\Law(W\mid\mathcal F),\nu)\le\varepsilon,
\]
then for every $\mathcal F$-measurable random element $V$,
\[
 \dTV(\Law(V,W),\Law(V)\otimes\nu)\le\varepsilon.
\]
For $C\in\mathcal F$ with $\PP(C)>0$, the conditional bound is at most
$\varepsilon/\PP(C)$, both for $W$ and for the joint product lift with $V$.
\end{lemma}
\begin{proof}
Disintegrate the actual and product laws over $\mathcal F$. For a test set,
the difference of its conditional sections is bounded by the conditional
total-variation distance. Integrate, or integrate over $C$ and divide by
$\PP(C)$. Countable configuration spaces used here have measurable kernels;
the standard-Borel formulation permits the additional recorded variables.
\end{proof}

For $C_N\in\mathcal F_{Y_N^\sharp}$ write
$I_N^{\rm info}=-\log\PP(C_N)$. The hard scalar estimate becomes
\begin{equation}\label{eq:hard-conditioned}
 \dTV(\Law(Z_{N,L}\mid C_N),\Pois(\lambda_N))
 \ll \min\{1,e^{I_N^{\rm info}}\lambda_N
           (e^{-V_N+o(\nu_N)}+N^{-1/3+\varepsilon})\}.
\end{equation}
For $C_N\in\mathcal F_{\widehat Y_N}$ the soft proof gives instead
\begin{equation}\label{eq:soft-conditioned}
 \dTV(\Law(Z_{N,L}\mid C_N),\Pois(\lambda_N))
 \ll\min\{1,e^{I_N^{\rm info}-(\widehat V_N-\log^+\lambda_N)/2
                         +o(\widehat\nu_N)}
              +e^{I_N^{\rm info}}N^{-1/3+\varepsilon+o(1)}\}.
\end{equation}
Thus the sufficient scalar budgets are
\[
 I_N^{\rm info}+\log^+\lambda_N\le V_N-c\nu_N,
 \qquad
 2I_N^{\rm info}+\log^+\lambda_N\le\widehat V_N-c\widehat\nu_N.
\]
The hard route can be preferable at a larger information cost, but it
requires measurability with respect to its smaller prime field. These
budgets measure probabilities of events, not the number of primes exposed.
For instance, specifying $k$ independent prime signs costs $k\log2$;
a rank-$k$ compatible affine cylinder has the same cost.

At small intensity the hard estimate is relative: under
$I_N^{\rm info}\le V_N-c\nu_N$ and $\lambda_N\to0$,
\begin{equation}\label{eq:conditional-one-hit}
 \PP(Z_{N,L}>0\mid C_N)=\lambda_N(1+o(1)),\qquad
 \PP(Z_{N,L}=1\mid C_N)=\lambda_N(1+o(1)).
\end{equation}
The polynomial terms are negligible since $I_N^{\rm info}=o(\log N)$.
At large intensity, either valid scalar budget gives a conditional Gaussian
limit after centering by $\lambda_N$ and dividing by $\sqrt{\lambda_N}$.
The target Poisson normal-approximation error must still be added to the
arithmetic error; the quantitative and relative forms are in
\cref{supp:sec:conditional-details}.

\subsection{Choosing the labelled cutoff after the information cost}
\label{sec:information-cutoff}
In the dependency-graph proof, retaining all sites makes two errors
compete. Increasing the prime cutoff
removes shared-prime edges, but deletes more supports. After conditioning
on an event of cost $I$, the leading admissible logarithmic intensity is
\[
 g_{N,I}(w)=\min\{D(H_N/w)-I,\ (w-I)/2\}.
\]
The information cost enters the two constraints with different slopes.
For $0\le I\le V_N$, define
\begin{equation}\label{eq:information-saddle}
 2D(H_N/w_N(I))=w_N(I)+I,\qquad V_N\le w_N(I)\le\widehat V_N,
\end{equation}
and put
\begin{equation}\label{eq:information-budget}
 \nu_N(I)=\frac{H_N}{w_N(I)},\qquad
 Y_N(I)=\lfloor e^{w_N(I)}\rfloor,\qquad
 \ell_{\rm lab,N}(I)=\frac{w_N(I)-I}{2}.
\end{equation}
The root is unique: $2D(H_N/w)-w-I$ is strictly decreasing, with
endpoint values $V_N-I$ and $-I$. It is the crossing of a decreasing
and an increasing constraint, so $\ell_{\rm lab,N}(I)$ is the maximum
of $g_{N,I}$ on this cutoff interval.

\Needspace{15\baselineskip}
\begin{proposition}[Information-adapted labelled comparison]
\label{prop:information-labelled}
Let $C_N$ be events of positive probability, put
$I_N^{\rm info}=-\log\PP(C_N)$, and assume $0\le I_N^{\rm info}\le V_N$. For integer
$L_N\ge1$ put $\Lambda_N=N2^{-L_N}\ge1$. Suppose, for fixed $c>0$,
\begin{equation}\label{eq:information-labelled-condition}
 C_N\in\mathcal F_{Y_N(I_N^{\rm info})},\qquad
 \log\Lambda_N\le\ell_{\rm lab,N}(I_N^{\rm info})-c\nu_N(I_N^{\rm info}).
\end{equation}
For every $0<c'<c$ and $0<\varepsilon<1/3$,
\begin{equation}\label{eq:information-labelled-rate}
 \dTV\left(\Law((K^s_{x,e})_{x\in I_N,e\ge0,s\in\{\pm1\}}\mid C_N),
       \bigotimes_{x,e,s}\Pois(2^{-L_N-e-2})\right)
 \ll_{c,c',\varepsilon} e^{-c'\nu_N(I_N^{\rm info})}+N^{-1/3+\varepsilon}.
\end{equation}
All site labels, signs and exact excesses are retained. The estimate
also holds after any deterministic restriction or measurable contraction
of this field.
\end{proposition}
\begin{proof}
Write $w=w_N(I_N^{\rm info})$, $\nu=H_N/w$ and $\ell=\log\Lambda_N$.
The finite signed comparison at this cutoff has leading errors
$\exp(I_N^{\rm info}+\ell-D(H_N/w)+o(\nu))$ and
$\exp(I_N^{\rm info}+2\ell-w+o(\nu))$.
By \eqref{eq:information-saddle} and
\eqref{eq:information-labelled-condition}, their exponents are at most
$-c\nu+o(\nu)$ and $-2c\nu+o(\nu)$.
Take $E=3\lceil\widehat V_N/\log2\rceil$. Since
$I_N^{\rm info}+\ell\le w\le\widehat V_N$, the enlarged support is
$\log_2N+O(V_N)$, uniformly in the parameters, and the real conditional
and target tails are $O(e^{-2\widehat V_N})$.
This larger auxiliary truncation disappears from the final field. The polynomial remainder is at most
$e^{I_N^{\rm info}}\Lambda_N(1+\Lambda_N)N^{-1/3+\varepsilon/2+o(1)}$.
Since $I_N^{\rm info}+2\ell\le w=O(V_N)=o(\log N)$, it is absorbed by the
stated remainder. The free-cutoff estimates are uniform on
$[V_N,\widehat V_N]$; the detailed calculation, including the support
and tail checks, is in \cref{supp:sec:information-cutoff}.
\end{proof}
At zero information cost, $w_N(0)=\widehat V_N$ and
$\ell_{\rm lab,N}(0)=\widehat V_N/2$, recovering the enlarged-cutoff
labelled range. For $0<I<V_N$, the allowed leading value is strictly
larger than $\widehat V_N/2-I$. This is a comparison of proof budgets
for events measurable at the required cutoff, not a transfer of events
from a larger sigma-field into a smaller one. If $I_N^{\rm info}/V_N\to\theta\in[0,1]$,
then
\begin{equation}\label{eq:information-leading}
 \frac{\ell_{\rm lab,N}(I_N^{\rm info})}{V_N}
       \longrightarrow\frac{\sqrt{\theta^2+8}-3\theta}{4}.
\end{equation}
The exact saddle, rather than this first-order limit, is used in
\eqref{eq:information-labelled-condition} to retain its $\nu_N(I_N^{\rm info})$
margin.

\paragraph{Admissible information matters.}
For a fixed event known only to belong to
$\mathcal F_{\lfloor e^{w_0}\rfloor}$, with
$w_0\in[V_N,\widehat V_N]$, optimize over $w\ge w_0$, not over all
cutoffs. The best leading budget on this interval is attained at
$w=\max\{w_0,w_N(I)\}$. Since $w\ge w_0$, the original event
belongs to $\mathcal F_{\lfloor e^w\rfloor}$ by upward inclusion of the
prime sigma-fields; no new measurability assumption is required.
The sufficient condition there is
$\log\Lambda_N\le g_{N,I}(w)-cH_N/w$ with that event measurable in
$\mathcal F_{\lfloor e^w\rfloor}$. The proof is the same finite
comparison. These optimized upper bounds do not identify an intrinsic
phase boundary of the multiplicative model, nor do they use the scalar
soft-exception estimate as a field theorem.

The exact prefix comparison of \cref{thm:micro-run-tv}, and its dyadic
restriction \cref{cor:dyadic-micro}, use the hard field. Deep starts and
signed tails are checked independently in
\cref{supp:pivot:lem:inputs}. Its partition readouts require no mesh
hypothesis. Process limits still require their own target, ordering and
censoring estimates. Neither that contraction
nor this information-adapted graph calculation silently changes the
admissible prime field of the other route.

\subsection{The exact resolution cost}
\begin{lemma}[Sharp conditioning inequality]\label{lem:conditioning-floor}
Let $\mu,\nu$ be probability laws on $E\times F$ with
$\dTV(\mu,\nu)\le\varepsilon$. Let $B\subset E$ have
$p=\nu_E(B)>\varepsilon$ and write $q=\mu_E(B)$. Then $q>0$ and
\begin{equation}\label{eq:sharp-conditioning}
 \dTV(\mu_F(\cdot\mid B),\nu_F(\cdot\mid B))
 \le\frac{\varepsilon}{\max(p,q)}\le\frac{\varepsilon}{p}.
\end{equation}
\end{lemma}
\begin{proof}
The bound $|p-q|\le\varepsilon$ gives positivity. First suppose $q\le p$.
For any measurable $D\subset F$ put $A=B\times D$. Since $\mu(A)\le q$,
\[
 \frac{\mu(A)}q-\frac{\nu(A)}p
 \le\frac{\mu(A)-\nu(A)+p-q}{p}
 =\frac{\nu(B\times D^c)-\mu(B\times D^c)}p
 \le\frac\varepsilon p.
\]
The supremum of the positive differences equals total variation for
probability laws. If $q\ge p$, interchange the two laws.
\end{proof}
The denominator expresses the correct scale: convergence after conditioning
requires $\varepsilon/p\to0$, not merely $p>\varepsilon$.
This lemma applies to an entire future path at once.

Let $\lambda_{N,m}=2^{\vartheta_N-m}$ and
$p_{N,m}(n)=e^{-\lambda_{N,m}}\lambda_{N,m}^n/n!$.
Write $\varepsilon_{N,m,C}^{\rm path}$ for the total-variation error between
the actual complete future threshold path, under $C_N$, and the target
staircase based at $m$. The aggregated proof works equally for an arbitrary
base threshold in a fixed logarithmic band (including $\lambda_{N,m}<1$).
In its nontrivial information range it gives
\begin{equation}\label{eq:path-error}
 \varepsilon_{N,m,C}^{\rm path}
 \ll e^{I_N^{\rm info}}\lambda_{N,m}
       (1+\log^+(2\lambda_{N,m}))e^{-V_N+o(\nu_N)}
       +N^{-1/3+\varepsilon},
\end{equation}
with an arbitrarily small fixed exponent slack absorbing the enlarged
length and the subpolynomial information factor.

\begin{theorem}[Resolved thinning kernels]\label{thm:resolved-kernel}
If $p_{N,m}(n)>\varepsilon_{N,m,C}^{\rm path}$, then
\begin{equation}\label{eq:resolved-kernel}
 \dTV\left(\Law((Z_{N,b_N+m+j})_{j\ge0}\mid C_N,Z_{N,b_N+m}=n),
                     \operatorname{Thin}_n\right)
 \le\frac{\varepsilon_{N,m,C}^{\rm path}}{p_{N,m}(n)},
\end{equation}
where $\operatorname{Thin}_n$ is the path of $n$ independent geometric
lifetimes from \eqref{eq:target-forward}. If a fixed lower segment is also
retained, the same inequality uses the comparison error based at its
lowest level and gives the reverse immigration kernel
\eqref{eq:target-reverse} jointly with the future.
\end{theorem}
\begin{proof}
Apply \cref{lem:conditioning-floor} to the present count and the entire
retained path. The conditional target is exactly the geometric-lifetime
path by independent Poisson splitting. For lower levels, the increments
below the present level are independent Poisson variables. A finite
intensity half-line comparison cannot control the whole infinite reverse
path, whose total target mass diverges.
\end{proof}
For every integer $n\ge1$ and every $\lambda>0$, the local formula is
\[
 p_\lambda(n)=\frac{e^{-\delta_n}}{\sqrt{2\pi n}}
                 e^{-\lambda h(n/\lambda)},\qquad
 0\le\delta_n\le\frac1{12n},\qquad h(v)=v\log v-v+1.
\]
The correction is independent of $\lambda$; its relative error is at most
$1/(12n)$. For $n=0$ use the exact mass $e^{-\lambda}$ instead.
The telescoping proof is in \cref{supp:sec:conditional-details}.
For $n_N\to\infty$, a sufficient condition for the right side of
\eqref{eq:resolved-kernel} to vanish is
\begin{equation}\label{eq:resolved-budget}
 I_N^{\rm info}+\log^+\lambda_{N,m}
 +\log(1+\log^+(2\lambda_{N,m}))
 +\tfrac12\log n_N+\lambda_{N,m}h(n_N/\lambda_{N,m})
 \le V_N-c\nu_N.
\end{equation}
In the central regime $n_N=\lambda_{N,m}+t_N\sqrt{\lambda_{N,m}}+O(1)$,
$|t_N|=o(\lambda_{N,m}^{1/6})$, this reduces, with a fixed margin, to
\[
 I_N^{\rm info}+\tfrac32\log\lambda_{N,m}+\tfrac12t_N^2
 \le V_N-c\nu_N.
\]
The coefficient $3/2$ is a sufficient error budget, not a threshold of the
model. Its derivation, including the common-band check, is in
\cref{supp:sec:conditional-details}.

\subsection{Quenched comparison on geometric scales}
\begin{corollary}[Almost-sure environmental control]\label{cor:quenched}
Let $N_k\asymp q^k$, $q>1$, and suppose
$\log\Lambda_{N_k}\le V_{N_k}-c\nu_{N_k}$.
For each $0<\beta<c$, the conditional total-variation distance, given
$\mathcal F_{Y_{N_k}^\sharp}$, between the complete aggregated staircase
and its deterministic target is at most $e^{-\beta\nu_{N_k}}$ for all
sufficiently large $k$, almost surely.
\end{corollary}
\begin{proof}
The fibrewise proof of \cref{thm:moving-comparison}(ii) gives mean distance
$e^{-c\nu_N+o(\nu_N)}+N^{-1/3+o(1)}$.
Markov's inequality at $e^{-\beta\nu_N}$ gives a summable series along
$N_k\asymp q^k$, since $\nu_{N_k}\asymp\sqrt{k/\log k}$.
Borel--Cantelli concludes; no independence between scales is used.
\end{proof}
The same argument applies to the dyadic graph-based labelled field in
the sufficient range of that estimate, and to scalar kernels uniformly over the $O(\log N)$
lengths of any fixed logarithmic band with a fixed error margin.
A maximal coupling exists separately in each qualified environment and
scale. This does not furnish a canonical coupling over all scales.

% END ORIGINAL SOURCE: sections/06_conditioning.tex

% BEGIN ORIGINAL SOURCE: sections/07_prefix_boundary.tex
\Needspace{10\baselineskip}
\section{Finite prefixes and the arithmetic boundary}\label{sec:global-prefix}
A finite prefix is not a union of independent dyadic experiments. Its
retained bulk must be treated in one Poisson comparison, using the
macroscopic relation estimate. The starts near the origin need a different
argument: there the initial run costs one sign per prime, rather than one
sign per integer. The two candidate masses are
$\alpha_L=2^{-\pi(L)}$ at the boundary and
$\lambda_M=M2^{-L}$ in the bulk. To compare them, we must bound the other
microscopic starts, discard the intervening region, and approximate the
bulk with error $o(\lambda_M)$. A critical-scale error of size $o(1)$ is
not sufficient for this last task.

The limiting source weights arise by normalizing these two masses.
The stronger conclusion of \cref{thm:two-clock} also distinguishes their
marks: both excess laws are geometric, but one counts new primes and the
other new integer positions. This distinction is arithmetic, not a
property of the common geometric distribution.

\subsection{The exact boundary event}
Set
\begin{align*}
 C_L&=\1_{\{f(1)=\cdots=f(L)\}},&
 \mathbf B_L&=(C_L,(J_{x,L})_{2\le x\le2L^2}),\\
 B_L^{\rm mic}&=C_L+\sum_{2\le x\le2L^2}J_{x,L},&
 q_L&=\PP(B_L^{\rm mic}>0).
\end{align*}
Because $f(1)=1$, the event $C_L=1$ prescribes precisely the prime signs
up to $L$. Hence
\begin{equation}\label{eq:exact-border}
 \PP(C_L=1)=2^{-\pi(L)}
\end{equation}
exactly. The quantity $q_L$ is different: it includes all other microscopic
starts. The next result shows that their combined mass is smaller by a
fixed exponential factor in $\pi(L)$.

\begin{theorem}[Microscopic non-vacancy and localization]\label{thm:boundary}
Let $\delta_*=28271/332640$. As $L\to\infty$,
\begin{equation}\label{eq:interior-micro}
 \sum_{x=2}^{2L^2}\PP(J_{x,L}=1)
       \le2^{-(1+\delta_*-o(1))\pi(L)}.
\end{equation}
Consequently, for every fixed $0<\delta_0<\delta_*$,
\begin{equation}\label{eq:boundary-asymptotic}
 q_L=2^{-\pi(L)}(1+O_{\delta_0}(2^{-\delta_0\pi(L)})).
\end{equation}
In particular $\delta_0=1/12$ is admissible. Conditional on microscopic
non-vacancy, the exact border is the unique recorded start with probability
$1-O_{\delta_0}(2^{-\delta_0\pi(L)})$.
\end{theorem}
\begin{proof}
Let $M_x$ be the matrix whose rows are $v(n)$, $n\in V_x$.
The start matrix is $A_x=D_xM_x$, where
\[
 D_xz=(z_{x-1}+z_x,z_x+z_{x+1},\ldots,z_x+z_{x+L-1}).
\]
The kernel of $D_x$ consists of constant vectors. Therefore
\begin{equation}\label{eq:boundary-rank-drop}
 \rank A_x\ge\rank M_x-1,\qquad
 \PP(J_{x,L}=1)\le2^{-\rank A_x}.
\end{equation}
If $2\le x\le L-1$, both $2x$ and $2x-2$ lie in the proposed constant
run. The first forces $f(2)=1$, and the second forces $f(2)=-1$.
Thus these starts are impossible. For the three transition starts
$x\in\{L,L+1,L+2\}$, primes between $2\sqrt B$ and $B$, together
with prime vertices above $B$, give an identity minor of size
$(2-o(1))\pi(B)$. The explicit choice of rows is in
\cref{supp:lem:transition}.

For $B+2\le x\le B^2+2$, the uniform Laishram--Shorey bound gives
\[
 \omega\left(\prod_{n\in V_x}n\right)\ge(2-o(1))\pi(B).
\]
After discarding the prime divisors at most $B$, at least
$t_x\ge(1-o(1))\pi(B)$ primes $q>B$ remain. All vertices are at most
$B^2+B<q^2$, so such a prime occurs to valuation one at one vertex only;
two such primes cannot occupy the same vertex. Their columns span a
$t_x$-dimensional coordinate subspace.

Use the medium primes $B/11<p\le B$. Their total odd-valuation incidence
count satisfies
\begin{equation}\label{eq:medium-incidences}
 E_x\ge(H_{11}-1-o(1))\pi(B),\qquad H_{11}=\sum_{j=1}^{11}j^{-1}.
\end{equation}
Here the loss from even valuations is only $O(1)$ \emph{over the entire
prime range}: if $p^2\mid n\le B^2+B$ then $n=ap^2$ with $a\le122$,
and for each $a$ there are only $O(1)$ possibilities in an interval of
length $B$. This global count avoids paying one lost incidence per prime.

Delete the private rows. Each deleted row removes at most one medium
incidence, each remaining row has weight at most two, each column has
weight at most eleven, and a pair of medium primes cannot occur together
in two rows. The associated incidence graph is therefore simple.
The elementary simple-incidence estimate \cref{supp:lem:simple-incidence}
gives rank at least $E/12$. Passing to the quotient by the private
coordinate subspace yields
\[
 \rank M_x\ge t_x+\frac{(E_x-t_x)_+}{12}
 \ge\left(1+\frac{H_{11}-2}{12}-o(1)\right)\pi(B).
\]
Since $(H_{11}-2)/12=28271/332640$, \eqref{eq:boundary-rank-drop} proves the desired
uniform surplus in the quadratic range.

For $x>B^2+2$, the pointwise Balasubramanian--Shorey consequence
\cref{supp:lem:pointwise-maximum} gives
\begin{equation}\label{eq:gL}
 m_B(x)\le B-g_L,\qquad
 g_L=\frac B{\log B}
   (\log\log B-\log\log\log B-\theta_0+o(1)),
 \qquad \frac{g_L}{\pi(B)}\longrightarrow\infty.
\end{equation}
The affine bound then gives $\PP(J_{x,L}=1)\le2^{-g_L}$.
Sum the four ranges; their polynomial number of starts is absorbed in
the $o(\pi(B))$ exponent. Since $\pi(B)=\pi(L)+O(1)$, this proves
\eqref{eq:interior-micro}. Equation \eqref{eq:exact-border} and a union
bound prove the remaining conclusions. The complete incidence count,
the uniform literature input and the choice $K=11$ are detailed in
\cref{supp:sec:boundary}, with local product thresholds distinguished
from the thresholds of the asymptotic inputs.
\end{proof}

\subsection{Discarding deep starts without assuming independence}
\begin{lemma}[Two-defect windows at intermediate heights]
\label{lem:meso-defects}
Assume $B\asymp\log M$. The number of starts
$x\in[2L^2,2M]\cap\Z$ with at least two $B$-defective vertices is at most
$\exp(O(\log M/\log\log M))$. The same bound holds for every subpopulation;
in particular it is uniform on each $[X,2X)$ with $2L^2\le X\le M$.
\end{lemma}
\begin{proof}
Two such vertices have forms $d_1a^2,d_2b^2$ differing by a nonzero
integer of modulus at most $L$, with squarefree $B$-smooth $d_i$.
There are at most $4^{\pi(B)}=\exp(O(\log M/\log\log M))$ coefficient
pairs and $O(B^2)$ offset pairs. For $d_1\ne d_2$, use the uniform
Pell bound at ambient scale $M$; the coefficients and variables are
polynomially bounded in $M$. For equal coefficients use
$(a-b)(a+b)=(i-j)/d_1$ and the divisor bound. A solution determines the
start once the offsets are fixed. This counts the entire population up
to height $2M$ in a single Pell box, not once per dyadic interval.
\end{proof}

\begin{proposition}[Deep-start and mesoscopic estimates]\label{prop:deep-starts}
For each fixed $0<\delta<1$ and fixed logarithmic band,
\begin{equation}\label{eq:deep-starts}
 \sum_{2\le x<M^\delta}\PP(J_{x,L}=1)
 \ll \lambda_M M^{\delta-1}+\exp(-c\log M/\log\log M),
 \qquad\lambda_M=M2^{-L},
\end{equation}
where $c>0$ depends only on that band and $\delta$.
More precisely,
\begin{equation}\label{eq:mesoscopic}
 \PP(\exists x\in(2L^2,M^\delta):J_{x,L}=1)
 \le M^\delta2^{-L}+C_1 e^{C_2L/\log L}2^{-g_L}.
\end{equation}
If $L\to\infty$, $L=O(\log M)$ and $\lambda_M\to0$, the latter bound
is $o(q_L+\lambda_M)$.
\end{proposition}
\begin{proof}
For any deterministic $A\subset[2L^2,2M]\cap\Z$, starts with at most
one defect have probability at most $2^{-L}$. The remaining starts are
counted together by \cref{lem:meso-defects}, and \eqref{eq:gL} bounds each
probability by $2^{-g_L}$. Uniformly in $A$,
\[
 \sum_{x\in A}\PP(J_{x,L}=1)
 \le |A|2^{-L}+C_1 e^{C_2L/\log L}2^{-g_L}.
\]
The constants depend only on the fixed logarithmic band. Adding the
microscopic range, controlled by \cref{thm:boundary}, proves
\eqref{eq:deep-starts}; restricting this mask proves \eqref{eq:mesoscopic}.
No exterior factor counting height slices is needed. For the last assertion,
$M^\delta2^{-L}=o(\lambda_M)$, while
$g_L-\pi(L)\gg L\log\log L/\log L$ dominates the sieve prefactor.
Thus the second term is $o(2^{-\pi(L)})=o(q_L)$.
\end{proof}

The same argument gives cutoff stability. For $2L^2\le K\le2^L$,
\begin{equation}\label{eq:cutoff-stability}
 \PP(\exists x\in(2L^2,K]:J_{x,L}=1)
 \le K2^{-L}+O(e^{CL/\log L}2^{-g_L}).
\end{equation}
One may use the artificial ambient scale $2^L$ in
\cref{lem:meso-defects}. If $K=o(2^{L-\pi(L)})$, the entire bound is
$o(q_L)$. Hence the boundary asymptotic and conditional uniqueness remain
valid with cutoff $K$, and the two conditional microscopic configurations,
padded by zeros, are at total-variation distance $o(1)$ by
\cref{lem:conditioning-floor}.

\subsection{Global laws and the longest run}
\begin{theorem}[Finite-prefix Poisson approximation]\label{thm:prefix-poisson}
In a fixed logarithmic band, for every $\varepsilon>0$,
\begin{align}
 &\dTV\left(\Law\left(\sum_{2\le x<M}J_{x,L}\right),\Pois(M2^{-L})\right)
 \notag\\
 &\quad\ll\min\{1,\lambda_M(e^{-V_M+o(\nu_M)}+M^{-1/3+\varepsilon})\}
               +e^{-c\log M/\log\log M}.
               \label{eq:global-rate}
\end{align}
For $W_{M,L}=C_L+\sum_{2\le x\le M-L+1}J_{x,L}$, enlarge the
constant multiplying the existing deep-start remainder and add
\begin{equation}\label{eq:prefix-correction}
 2^{-\pi(L)}+O(L2^{-L}).
\end{equation}
In particular $\PP(R_M<L)=e^{-M2^{-L}}+o_C(1)$ at critical lengths.
\end{theorem}
\begin{proof}
Retain the single interval $U_{M,1/2}$. The proof of
\cref{thm:transfer} uses only support heights $O(M)$, the one-window
mass, the edge count and the separated-pair profile. These are supplied
by \eqref{eq:macro-defects}, \eqref{eq:edge-count} at upper scale $M$,
and \cref{thm:master-profile}. Thus \eqref{eq:hard-rate} applies on the
retained interval with mean $|U_{M,1/2}|2^{-L}$. Restoring deep starts
costs \eqref{eq:deep-starts}; the deterministic target-mean discrepancy
is $O(\lambda_M M^{-1/2})$. This proves \eqref{eq:global-rate}.

The event $W_{M,L}=0$ is exactly $R_M<L$: every long run contained in
the prefix either begins at one or has a left-maximal start at most
$M-L+1$. The full interior count differs only if the border event occurs
or a start lies in the excluded mask $M-L+2\le x<M$.
This mask has at most $L$ sites and lies above $2L^2$ for large $M$.
The mask bound in the proof of \cref{prop:deep-starts} gives total mass
\[
 L2^{-L}+O(e^{CL/\log L}2^{-g_L})
 \le L2^{-L}+O(e^{-c\log M/\log\log M}).
\]
Absorb the second term into the remainder already present in
\eqref{eq:global-rate}. The exact border probability then proves
\eqref{eq:prefix-correction} and the void law.
\end{proof}

\begin{corollary}[Almost-sure asymmetric envelopes]\label{cor:as-longest}
For every fixed $\varepsilon>0$, with probability one, eventually
\begin{align}
 R_M-\log_2M&\ge-\log_2(\log\log M)-O(1),
                                          \label{eq:as-lower}\\
 R_M-\log_2M&\le\log_2(\log M)
        +(1+\varepsilon)\log_2(\log\log M)+O_\varepsilon(1).
                                          \label{eq:as-upper}
\end{align}
In particular $R_M=\log_2M+O(\log\log M)$ almost surely.
\end{corollary}
\begin{proof}
Set $M_k=2^k$ and, for sufficiently large $k$, choose
\[
 L_k^-=\lfloor k-\log_2(\log k)-1\rfloor,\qquad
 L_k^+=\lceil k+\log_2k+(1+\varepsilon)\log_2(\log k)\rceil.
\]
The lower intensity satisfies $2\log k\le M_k2^{-L_k^-}<4\log k$,
so the target void probability is at most $k^{-2}$. The upper intensity
is at most $1/[k(\log k)^{1+\varepsilon}]$, a summable sequence.
Every error in \eqref{eq:global-rate}--\eqref{eq:prefix-correction} is also
summable: the cutoff error is a polynomial in $k$ times
$e^{-c\sqrt{k\log k}}$, and the deep-start and border errors are at most
$e^{-c'k/\log k}$. Borel--Cantelli gives
$L_k^-\le R_{2^k}<L_k^+$ for all sufficiently large $k$, almost surely.
For $2^k\le M<2^{k+1}$ use $R_{2^k}\le R_M\le R_{2^{k+1}}$.
The change from $k$ to $\log M/\log2$ and from $k$ to $k+1$ costs only
a bounded additive term. This proves both envelopes. No independence
between different scales has been used, and no matching lower bound for
the upper fluctuations is asserted.
\end{proof}

\begin{theorem}[Macroscopic exact-mark transport]\label{thm:macro-field}
Let $a_M=o(\log M)$ be a nonnegative integer, $b_M=\lfloor\log_2M\rfloor$,
$L_M=b_M-a_M$, and $\lambda=M2^{-L_M}$. Define
\[
 \mathcal E^{\rm base}_{M,a_M}
 =\sum_{x=2}^{M-L_M+1}\sum_{r\ge-a_M}
             K^{\rm ex}_{x,b_M+r}\delta_{(x/M,r)}.
\]
Its target has independent Poisson masses $2^{-b_M-r-1}$ on the same
lattice, and total intensity $(M-L_M)2^{-L_M}$.
For $A_M\in\mathcal F_{Y_M^\sharp}$ with $\PP(A_M)>0$, put
$I_M^{\rm info}=-\log\PP(A_M)$;
the full-field distance is bounded by
\begin{equation}\label{eq:macro-field-rate}
 C_\varepsilon e^{I_M^{\rm info}}
 \left[\lambda(1+\lambda)(e^{-V_M+o(\nu_M)}+M^{-1/3+\varepsilon})
            +\lambda M^{-1/2}+e^{-c\log M/\log\log M}\right].
\end{equation}
It tends to zero under the hard labelled budget
$I_M^{\rm info}+\log\lambda+\log(1+\lambda)\le V_M-c'\nu_M$.
The movable-cutoff labelled budget and the aggregated one-factor budget
of \cref{thm:moving-comparison} also hold, with $N$ replaced by $M$ and
exact-level target means $(M-L_M)2^{-b_M-r-1}$.
\end{theorem}
\begin{proof}
On $[\lceil M^{1/2}\rceil,M-L_M+1]$ use the finite marked comparison
with the macroscopic master profile; the identities
\eqref{eq:mark-folding} still collapse the separated-pair term to the
base threshold event. Restore lower starts with \eqref{eq:deep-starts}
and condition using \cref{lem:stable-lift}. Remove the exact-mark tail
using \eqref{eq:mark-tail} and the global scalar bound. A cutoff
$E=\lceil(I_M^{\rm info}+V_M+\log(2+\lambda))/\log2\rceil$ is
$o(\log M)$ whenever the bound is nontrivial; the deep-start tail error
is absorbed by the last term of \eqref{eq:macro-field-rate}.
This proves the displayed bound.
For the movable cutoff repeat with $w=\widehat V_M$.
For aggregation, sum over positions before applying the directional Stein
factor; the deep-start error is negligible under its information
budget. \Cref{supp:sec:macro-marked} details the finite-cutoff calculation.
\end{proof}
The superscript ``base'' is important: only the base $L_M$-window is
required to be contained. The exact right endpoint of a larger marked run
may lie beyond $M$. This is not the list of right-censored maximal runs of
the finite sequence. All measurable functionals of this base-contained
field transfer under the displayed labelled comparison.
\Cref{thm:micro-run-tv} below improves the hard-cutoff budget for its
growing-intensity signed field. The proof just given is kept independent,
as are the scalar and deep-start estimates on which that theorem relies.
Aggregated couplings by themselves do not retain locations.

\begin{figure}[htbp]\centering
\begin{tikzpicture}[x=.85cm,y=1cm]
\draw[->] (0,0)--(13,0);
\draw[thick] (1,0)--(11,0);
\foreach \x/\t in {1/1,6/x,11/M}{\draw (\x,.09)--(\x,-.09);
 \node[below,font=\small] at (\x,-.1) {$\t$};}
\draw[draw=FigBlue,line width=2pt] (6,.45)--(9.5,.45);
\node[above,font=\small,text=FigBlue] at (7.75,.45) {contained base window};
\draw[draw=FigRust,dashed,line width=1.2pt] (6,1.25)--(12,1.25);
\node[above,font=\small,text=FigRust] at (9,1.25) {exact run may continue beyond $M$};
\draw[dotted] (11,-.1)--(11,1.6);
\end{tikzpicture}
\caption{Containment of a base window does not censor an exact mark.
The longest-run void event instead uses the threshold-specific upper
start $M-L+1$. The two conventions agree where stated, but are not
interchangeable.}\label{fig:containment}
\end{figure}

\paragraph{Two coordinate conventions.}
The site-indexed convention in the next subsection is related to the
prefix convention by the exact relabelling $j=x-1$:
\begin{center}\small
\begin{tabularx}{\textwidth}{@{}lXX@{}}\toprule
Quantity & Prefix convention & Site-indexed convention\\\midrule
Interior starts & $2\le x\le M-L+1$ & $1\le j\le n$, $n=M-L$\\
Total target intensity & $(M-L)2^{-L}$ & $n2^{-L}$\\
Spatial coordinate & $x/M$ & $j/n$\\
Integer centre & $b_M=\lfloor\log_2M\rfloor$ & $\widetilde b_M=\lfloor\log_2n\rfloor$\\\bottomrule
\end{tabularx}
\end{center}
The rescaled positions differ uniformly by $O((L+1)/M)$, but this does
not identify distinct grids in total variation. An exact-length level
$r$ centred at $b_M$ becomes $r+b_M-\widetilde b_M$ in the other
convention. The centres need not be equal: at $M=2^k$ and positive
$L=o(M)$, they are $k$ and $k-1$ for large $k$. The limiting phase $1$
therefore retains a level shift relative to phase $0$. Full marks are
unchanged by this relabelling; observing their censored growth is a
separate process question.

\subsection{The full same-grid signed prefix comparison}
\label{sec:micro-run-comparison}
The graph-based transport above remains useful for its stated ranges.
We now keep the entire signed prefix field at one factor of intensity by
constructing the exact conditional law after forcing one occurrence.
Only the primes chosen for that forcing enter its change set.

Use the hard scales of \eqref{eq:saddle-definitions} with $N=M$ and set
\begin{equation}\label{eq:micro-normalization}
 L=\lfloor\log_2M\rfloor-a_M,\qquad n=M-L,\qquad p=2^{-L},\qquad
 \Lambda=np,\qquad a_M\in\NN,\quad a_M=o(\log M).
\end{equation}
Here $\NN=\{0,1,\ldots\}$, as elsewhere in this article.
For $1\le j\le n$, $e\ge0$ and $\sigma\in\{-1,+1\}$, put
\begin{equation}\label{eq:micro-indicators}
 I_{j,e,\sigma}=K^\sigma_{j+1,e}
 =\1_{\{j+1\text{ is a run start},\ \ell_{j+1}=L+e,\ f(j+1)=\sigma\}},
 \qquad \gamma_{e,\sigma}=2^{-e-2}.
\end{equation}
Thus the raw exact word occupies the integers $j,\ldots,j+L+e+1$.
Its base is contained in $[1,M]$, but its full rightward length may
continue beyond $M$. The run starting at $1$ is not indexed.
Write $\mathbf I_M=(I_{j,e,\sigma})_{j,e,\sigma}$, and let
$\mathbf P_M$ have independent coordinates
$P_{j,e,\sigma}\sim\Pois(p\gamma_{e,\sigma})$ on the \emph{same} index set.
Both are finite-mass configurations on a countable lattice; the target
has total mean $\Lambda$, whereas the source has at most one atom at a site.

Let $\mathcal A_M\in\mathcal F_{Y_M^\sharp}$ have positive probability,
put $I_M=-\log\PP(\mathcal A_M)$, and assume, for fixed $c>0$,
\begin{equation}\label{eq:micro-budget}
 \Lambda\to\infty,\qquad I_M+\log\Lambda\le V_M-c\nu_M.
\end{equation}
In particular $\Lambda\ge1$ for all sufficiently large $M$.

\begin{theorem}[The full same-grid signed comparison]
\label{thm:micro-run-tv}\label{C-MICRO-RUN-TV}
Under \eqref{eq:micro-normalization}--\eqref{eq:micro-budget}, for every
$0<c'<c$ and $0<\varepsilon<1/3$,
\begin{equation}\label{eq:micro-tv}
 \Delta_M^{\rm mic}:=
 \dTV\bigl(\Law(\mathbf I_M\mid\mathcal A_M),\Law(\mathbf P_M)\bigr)
 \ll_{c,c',\varepsilon} e^{-c'\nu_M}+M^{-1/3+\varepsilon}.
\end{equation}
All spatial labels, full exact excesses and actual signs are retained
simultaneously. No partition or mesh condition is needed for this
comparison with the moving same-grid target.
\end{theorem}
The pivot notation in the proof agrees with
$P^*(m)=P^+(\operatorname{core}(m))$ in
\cite[eq.~(5)]{deLaBretecheWangXu2026}, with value $1$ on squares.

\begin{proof}[Proof mechanism]
For an integer $m\ge1$ define
\[
 \kappa_{\rm odd}(m)=\max\{q\text{ prime}:v_q(m)\text{ is odd}\},
 \qquad \kappa_{\rm odd}(m)=1\quad\text{if }m\text{ is a square}.
\]
The pivot is not the ordinary largest prime factor. Use the finite cutoff
\begin{equation}\label{eq:micro-cutoff}
 E_M^*=\left\lceil
 \frac{I_M+V_M+\log(2+M2^{-L})}{\log2}\right\rceil,\qquad
 Q=L+E_M^*+1.
\end{equation}
On retained maximal supports each vertex has a private pivot above
$Y_M^\sharp$. To force a type $(e,\sigma)$, modify only the pivots of
its $L+e+2$ raw vertices, including both boundaries; the unused pivots
of the maximal support remain unchanged. This produces the
\emph{entire} law conditioned on that prescribed signed word. It leaves the nonpivot law, including the small-prime environment,
unchanged. This is \cref{supp:pivot:prop:forcing}.

The deterministic set $\mathscr D_j$ of sites possibly changed by this
forcing need not be symmetric. The dimension-free Palm--Stein inequality
of \cref{supp:pivot:lem:palm} charges its size, not every edge of the
all-large-prime graph. The uniform reciprocal estimate
\begin{equation}\label{eq:micro-reciprocal-intro}
 \sum_{\substack{m\le M+O(\log M)\\\kappa_{\rm odd}(m)>Y_M^\sharp}}
       \frac1{\kappa_{\rm odd}(m)}
 \le M\exp\{-2V_M+o(\nu_M)\}
\end{equation}
gives $\sum_j|\mathscr D_j|\ll
n^2e^{-2V_M+o(\nu_M)}+n(Q+1)^2$.
\Cref{supp:pivot:prop:reciprocal} proves this estimate by a uniform
Rankin bound and a convex saddle calculation. Precision $o(\nu_M)$,
not merely $o(V_M)$, preserves the fixed information margin.

With the deletion and shifted-tail cost $d_{0,M}$ of
\eqref{supp:pivot:eq:deletion}, the finite ledger is
\begin{equation}\label{eq:micro-ledger-main}
 \Delta_M^{\rm mic}\ll d_{0,M}+np^2+
 e^{I_M}p^2\left\{nQ+\sum_j|\mathscr D_j|
                    +R^{\rm val}_{2,U}(M,Q+1)\right\}.
\end{equation}
The directed-footprint term costs
$e^{I_M+2\log\Lambda-2V_M+o(\nu_M)}
 \le e^{-I_M-2c\nu_M+o(\nu_M)}$.
Deletion costs $e^{I_M+\log\Lambda-V_M+o(\nu_M)}$;
the full-value relation profile contributes $M^{-1/3+\varepsilon+o(1)}$.
\Cref{supp:sec:microscopic-pivots} gives the complete coupling and
ledger, including the signed local bound and all deletions.
Its tail input is the independent scalar theorem
\cref{thm:prefix-poisson}, proved independently above.
No process limit or Palm-void conclusion is used.
\end{proof}

% BEGIN ORIGINAL SOURCE: sections/07b_dyadic_restriction.tex
\paragraph{Restriction to a dyadic block.}
The one-factor hard-field budget is not confined to prefix geometry.
Restricting a larger prefix retains the original integer start labels.

\begin{dyadiccorollary}[Dyadic signed field at the one-factor budget]
\label{cor:dyadic-micro}
Let $L_N=\lfloor\log_2N\rfloor-d_N$, where $d_N\in\NN$ and
$d_N=o(\log N)$, and put $\Lambda_N=N2^{-L_N}\to\infty$.
Suppose $C_N\in\mathcal F_{Y_N^\sharp}$ has positive probability and,
for a fixed $c>0$,
\[
 I_N^{\rm info}:=-\log\PP(C_N),\qquad
 I_N^{\rm info}+\log\Lambda_N\le V_N-c\nu_N.
\]
For $0<c'<c$ and $0<\varepsilon<1/3$, with
$I_N=[N,2N)\cap\Z$,
\[
 \dTV\!\left(
  \Law((K^s_{x,e})_{x\in I_N,e\ge0,s=\pm1}\mid C_N),
  \bigotimes_{x\in I_N,e\ge0,s=\pm1}\Pois(2^{-L_N-e-2})
 \right)
 \ll_{c,c',\varepsilon}e^{-c'\nu_N}+N^{-1/3+\varepsilon}.
\]
The coordinates retain the full exact excess and actual sign at every
start; the target uses the same integer labels.
\end{dyadiccorollary}
\begin{proof}
Take $M=4N$ and keep the same length $L_N$. Then
\[
 \lfloor\log_2M\rfloor-L_N=d_N+2=o(\log M),\qquad
 (M-L_N)2^{-L_N}=(4-L_N/N)\Lambda_N.
\]
Every base starting in $I_N$ is contained in $[1,M]$ for large $N$;
no restriction is imposed on its full rightward length.

The hard scales are stable by monotonicity alone. Write $v(H)=V_{e^H}$
and $\nu(H)=H/v(H)$. The saddle equation is
$H=\nu(H)D(\nu(H))$. On the relevant positive branch both $D$ and
$zD(z)$ are strictly increasing. For $H\le K$, first compare the
solutions $\nu(H),\nu(K)$ and then their $D$-values to obtain
\[
 v(H)\le v(K),\qquad
 \nu(H)\le\nu(K)\le\frac KH\nu(H).
\]
At $K=H+\log4$, this gives $\nu_{4N}/\nu_N\to1$ and
$Y_{4N}^\sharp\ge Y_N^\sharp$. In particular $C_N$ belongs to the
hard field at $4N$, although the hypothesis still requires it to be
measurable in the original field at $N$.

Choose $c'<a<c$. Eventually $L_N\le N$, so the prefix intensity is
between $\Lambda_N$ and $4\Lambda_N$. The preceding inequalities give
\[
 V_{4N}-a\nu_{4N}-(V_N-c\nu_N)
 \ge \left(c-a-a\frac{\log4}{\log N}\right)\nu_N\longrightarrow\infty.
\]
They therefore absorb the bounded increase $\log4$ in log intensity:
$I_N^{\rm info}+\log((4N-L_N)2^{-L_N})\le V_{4N}-a\nu_{4N}$.
Apply \cref{thm:micro-run-tv} at $4N$ with margin $a$ and rate $c'$,
then project onto $x\in I_N$. Total variation contracts under this
projection, $e^{-c'\nu_{4N}}\le e^{-c'\nu_N}$, and the polynomial
error has the stated order. The coordinate means are unchanged.
\end{proof}
The passage is a restriction of integer coordinates, not an approximate
identification of two rescaled grids. It improves the available dyadic
hard-field range at growing intensity. It neither enlarges the permitted
prime sigma-field nor proves a comparison for arbitrary dictionaries;
these require their own hypotheses and estimates.

% END ORIGINAL SOURCE: sections/07b_dyadic_restriction.tex


\begin{corollary}[Every measurable readout of the retained field]
\label{cor:micro-readout}
Under \cref{thm:micro-run-tv}, every measurable map $\Phi_M$ into a
standard Borel space satisfies
\begin{equation}\label{eq:micro-readout}
 \dTV\bigl(\Law(\Phi_M(\mathbf I_M)\mid\mathcal A_M),
                    \Law(\Phi_M(\mathbf P_M))\bigr)\le\Delta_M^{\rm mic}.
\end{equation}
The same holds for any common Markov kernel, including auxiliary marking
followed by a measurable readout. The map or kernel may depend on $M$.
\end{corollary}
\begin{proof}
Pull an output event back to the input space. For a kernel $K_M$, apply
the variational formula to $K_M(\cdot,B)\in[0,1]$. The formula follows
by integrating level sets; indicator tests give equality.
\end{proof}
There is no continuity or number-of-queries restriction on this
comparison: an adaptive interrogation determined by the same retained
data is one measurable readout. Every deterministic partition, including
the singleton partition, inherits \eqref{eq:micro-tv} without a smallest-cell
penalty. Multiple target atoms receive separate auxiliary marks; a rule
initially defined only for simple configurations is extended measurably
to the other configurations.

This universality concerns an observation of the specified field, not a
universal limit law. It supplies no new starts, the missing boundary run,
an arbitrary word field, unbounded moments or an independence statement.
The recorded full lengths can be read beyond $M$, but the rest of the
future sequence is not recorded. Grid-to-continuum limits, censoring and
rare post-selection retain their own estimates. In particular neither
bounded nor vanishing $\Lambda$ is added to \eqref{eq:micro-budget} by this
corollary; \cref{thm:poisson-main,thm:relative-bulk} retain their separate
critical and relative roles.

% BEGIN ORIGINAL SOURCE: sections/07a_empirical_poisson.tex
\Needspace{34\baselineskip}
\subsubsection*{Empirical Poisson laws for almost every realization}
\phantomsection\label{sec:empirical-poisson}
\pdfbookmark[2]{Empirical Poisson laws for almost every realization}{bookmark:empirical-poisson}
The comparison above averages over the prime signs. A summable error
also permits a different order of quantifiers: first sample the entire
function $f$, then fix it and randomize only the spatial origin. The
following consequence concerns a count, not a comparison of the whole
local field after $f$ has been fixed.

\begin{empiricalcorollary}[Spatial empirical law for a fixed realization]
\label{cor:empirical-poisson}
Fix $0<\alpha<1$ and $\tau>0$. With $V_M$ as in
\eqref{eq:saddle-definitions}, for all sufficiently large $k$ set
\[
 M_k=2^k,\qquad
 L_k=k-\left\lfloor\frac{\alpha V_{M_k}}{\log2}\right\rfloor,
 \qquad p_k=2^{-L_k},
\]
\[
 n_k=M_k-L_k,\qquad h_k=\lfloor\tau 2^{L_k}\rfloor,
 \qquad N_k=n_k-h_k+1.
\]
Let $U_k$ be uniform on $\{0,\ldots,n_k-h_k\}$, independently of $f$, and put
\[
 Z_k(f,u)=\sum_{j=u+1}^{u+h_k}J_{j+1,L_k}(f),\qquad
 \mu_k^f=\frac1{N_k}\sum_{u=0}^{n_k-h_k}\delta_{Z_k(f,u)}.
\]
Then, for almost every $f$ in the model \eqref{eq:model},
\[
 \dTV\!\left(\mu_k^f,\Pois(\tau)\right)
 =\dTV\!\left(\Law_{U_k}(Z_k(f,U_k)),\Pois(\tau)\right)
 \longrightarrow0.
\]
Here $h_k$ counts potential start sites, the two signs are aggregated,
and a run is counted only at its left-maximal start. In particular,
\[
 h_kp_k\longrightarrow\tau,\qquad
 h_k=M_k\exp\{-\alpha V_{M_k}+O(1)\}=M_k^{1-o(1)}.
\]
The full-probability set is asserted for the fixed pair $(\alpha,\tau)$;
the conclusion is along the specified dyadic scales.
\end{empiricalcorollary}

The starts read by $Z_k$ range from $u+2$ to $u+h_k+1$. Thus every
base of length $L_k$ is contained in the global prefix $[1,M_k]$, and
the initial run from $1$ is excluded. A full run is not censored at
$M_k$, although its excess is not used by this count. In
$\Law_{U_k}$ only the origin is random; no independence between the
origins at different scales is required.

\begin{proof}
We first check the arithmetic budget and then do the frequency
calculation solely in the independent target. The hard-scale
expansions in \cref{prop:cutoffs} give
\[
 V_{M_k}\asymp\sqrt{k\log k},\qquad
 \nu_{M_k}\asymp\sqrt{k/\log k},\qquad
 V_{M_k}/\nu_{M_k}\longrightarrow\infty.
\]
Writing $\Lambda_k=n_kp_k$, the definitions imply
\[
 \log\Lambda_k=\alpha V_{M_k}+O(1),\qquad
 h_k/n_k=\exp\{-\alpha V_{M_k}+O(1)\}.
\]
Hence $\Lambda_k\to\infty$, $1\le h_k\le n_k$ eventually, and
\eqref{eq:micro-budget} holds with $\mathcal A_{M_k}$ equal to the whole
probability space and any fixed $c>0$, for all sufficiently large $k$.
\Cref{thm:micro-run-tv} consequently gives a sequence $\Delta_k$ with
\[
 \dTV\bigl(\Law(\mathbf I_{M_k}),\Law(\mathbf P_{M_k})\bigr)
 \le\Delta_k,\qquad
 \sum_k\Delta_k<\infty,\qquad
 \sum_k h_k/n_k<\infty.
\]
Indeed, for fixed admissible $c'$ and $\varepsilon$, the bound is
$O(e^{-c'\nu_{M_k}}+2^{-k(1/3-\varepsilon)})$, and both terms are
summable.

Sum the target over the signs and excesses at a site. Since
$\sum_{e,\sigma}\gamma_{e,\sigma}=1$, the resulting variables
\[
 X_{k,j}=\sum_{e\ge0,\,\sigma=\pm1}P_{k,j,e,\sigma}
 \sim\Pois(p_k),\qquad 1\le j\le n_k,
\]
are independent. For $0\le u<N_k$ and $r\in\NN$, define
\[
 S_{k,u}=\sum_{j=u+1}^{u+h_k}X_{k,j},\qquad
 F_{k,r}=\frac1{N_k}\sum_{u=0}^{N_k-1}\1_{\{S_{k,u}=r\}},
 \qquad q_{k,r}=\Pois(h_kp_k)\{r\}.
\]
Every $S_{k,u}$ has exactly the displayed Poisson law, so
$\EE F_{k,r}=q_{k,r}$. The indicators at origins $u,v$ are independent
if $|u-v|\ge h_k$. For the other pairs the upper bound on their covariance is $1/4$:
$\operatorname{Var}(A-B)\ge0$ and
$\operatorname{Var}(A),\operatorname{Var}(B)\le1/4$ give this directly
for indicators $A,B$. Counting at most $N_k(2h_k-1)$ ordered pairs gives
\[
 \Var(F_{k,r})\le\frac{2h_k-1}{4N_k}\ll\frac{h_k}{n_k},
\]
uniformly in $r$. Indeed $h_k/n_k\to0$ gives
$N_k=n_k-h_k+1\ge n_k/2$ eventually, so $\sum_k h_k/N_k<\infty$.
In particular, overlapping windows have not been treated as independent.

Almost surely all runs are finite, so summing
$I_{j,e,\sigma}$ over $(e,\sigma)$ gives $J_{j+1,L_k}$.
The empirical frequency is therefore the same measurable function of
the source field as $F_{k,r}$ is of the target field. For each fixed
$r\in\NN$ and $\eta>0$, \cref{cor:micro-readout} and Chebyshev's
inequality in the target yield
\[
 \PP_f\!\left(\left|\mu_k^f\{r\}-q_{k,r}\right|>\eta\right)
 \le\Delta_k+\frac{2h_k-1}{4N_k\eta^2}.
\]
This is one application to the event defined by the entire empirical
frequency: the error $\Delta_k$ is not multiplied by the number of
origins. The bound is summable. Borel--Cantelli and a countable
intersection over $r\in\NN$ and $\eta=1/m$, $m\ge1$, show that, almost
surely, $\mu_k^f\{r\}-q_{k,r}\to0$ for every $r$.

Finally, $0\le\tau-h_kp_k<p_k\to0$, so
$q_{k,r}\to q_r:=\Pois(\tau)\{r\}$. For any probability $\mu$ on
$\NN$ and any $R\in\NN$, normalization of the masses gives
\[
 \dTV(\mu,q)
 \le\sum_{r=0}^{R}|\mu\{r\}-q_r|+\sum_{r>R}q_r.
\]
First let $k\to\infty$ with $R$ fixed, and then let $R\to\infty$.
This proves the asserted total-variation convergence. Neither an
unbounded source moment nor a joint coupling of the targets at all
scales is used, and Borel--Cantelli requires no independence between
the source events at different scales.
\end{proof}

\paragraph{Scope and comparison.}
The normalization $h_k2^{-L_k}\to\tau$ is essential: the window-size
estimate with an unspecified $O(1)$ alone does not fix the limiting
mean. The inequality $\alpha<1$ leaves room in the arithmetic budget,
whereas $\alpha>0$ makes the target variance summable. This statement
fixes the entire $f$, unlike \cref{cor:quenched}. Najnudel's empirical
block laws \cite{Najnudel2020} use fixed block length.
Pandey--Wang--Xu \cite[arXiv version 4, Theorem~1.2 and footnote~1]{PandeyWangXu2024}
already study a typical multiplicative function from a uniform spatial
origin: their normalized Steinhaus sums have a complex Gaussian limit,
on sets of functions of measure $1-o(1)$ at each size. The present
statement instead counts rare left-maximal binary runs, converges in
total variation to a finite-mean Poisson law, and has a single
full-probability set for each fixed $(\alpha,\tau)$ along the specified
dyadic scales. Poisson genericity samples long words in a fixed sequence
\cite{AlvarezBecherMereb2023,AlvarezEtAlPoissonGenericity}; it is a different empirical
observable. No all-scale or simultaneous all-parameter assertion, and no
assertion about Liouville, follows here. No result about records,
occupation or scale flows is an input. The next observation shows why
the restriction to a count is substantive.

\begin{empiricalobstruction}[A support obstruction for the fixed local field]
\label{rem:empirical-field-obstruction}
Keep the scales of \cref{cor:empirical-poisson}. For any fixed $f$, let
\[
 \mathbf Z_k(f,u)=(J_{u+i+1,L_k}(f))_{1\le i\le h_k},\qquad
 \widehat\mu_k^f=\frac1{N_k}\sum_{u=0}^{N_k-1}\delta_{\mathbf Z_k(f,u)},
 \qquad Q_k=\bigotimes_{i=1}^{h_k}\Pois(p_k).
\]
Then
\[
 \liminf_{k\to\infty}\dTV(\widehat\mu_k^f,Q_k)
       \ge 1-e^{-\tau}(1+\tau)>0.
\]
Indeed, the support $\mathscr S_k(f)$ of $\widehat\mu_k^f$ contains at
most $N_k$ configurations. For large $k$, $p_k\le1$ and every
$z\in\NN^{h_k}$ with $|z|=\sum_i z_i\ge2$ satisfies
\[
 Q_k\{z\}=e^{-h_kp_k}\frac{p_k^{|z|}}{\prod_i z_i!}\le p_k^2.
\]
Testing the complement of $\mathscr S_k(f)$ within $\{|z|\ge2\}$ gives
\[
 \dTV(\widehat\mu_k^f,Q_k)
 \ge 1-e^{-h_kp_k}(1+h_kp_k)-N_kp_k^2.
\]
Now $h_kp_k\to\tau$ and $N_kp_k^2=M_k^{-1+o(1)}\to0$.
This holds for every fixed realization, not merely almost every one.
Thus the count law can converge even though the full local vector,
expressed relative to its sampled origin, cannot have the analogous
product-Poisson approximation in total variation.
\end{empiricalobstruction}

% END ORIGINAL SOURCE: sections/07a_empirical_poisson.tex


\subsection{Relative bulk approximation and the crossover}
Assume from now on $L\to\infty$, $L=O(\log M)$ and
$\lambda_M=M2^{-L}\to0$. Fix $0<\delta<1$ and put
$H_M=\lceil M^\delta\rceil$,
$I^{\rm bulk}_{M,L}=[H_M,M-L+1]\cap\Z$ and
$\Xi^{\rm bulk}_{M,L}=\sum_{x\in I^{\rm bulk}_{M,L}}J_{x,L}\delta_{x/M}$.
Let $\Pi^{\rm bulk,lat}_{M,L}$ have independent Poisson masses $p=2^{-L}$
at those same sites; its intensity is
$\lambda_M^{\rm bulk}=|I^{\rm bulk}_{M,L}|p\sim\lambda_M$.
Fix an upper-band constant $A_0$ with $L\le A_0\log M$ for all large
$M$. The rare-intensity assumption supplies the positive lower band.
Constants and little-oh terms in this subsection may depend on
$\delta$, $A_0$ and the displayed error exponent, but not on $M$ or $L$.

\begin{theorem}[Relative signed marked bulk kernel]\label{thm:relative-bulk}
Let $\mathcal K^{\rm bulk}_{M,L}=(K^s_{x,e})_{x\in I^{\rm bulk}_{M,L},\,e\ge0,\,s\in\{\pm1\}}$
and let $\mathcal P^{\rm bulk}_{M,L}$ have independent Poisson coordinates
of means $2^{-L-e-2}$ on this same index set. At $Y=Y_M^\sharp$, for every
fixed $0<\varepsilon<1/3$,
\begin{equation}\label{eq:relative-bulk-rate}
 \EE\,\dTV\left(\Law(\mathcal K^{\rm bulk}_{M,L}\mid\mathcal F_Y),
                         \Law(\mathcal P^{\rm bulk}_{M,L})\right)
 \ll_{\delta,A_0,\varepsilon}\lambda_M
               \left(e^{-V_M+o(\nu_M)}+M^{-1/3+\varepsilon}\right).
\end{equation}
In particular, summing marks and signs gives
\begin{equation}\label{eq:relative-bulk}
 \EE\,\dTV(\Law(\Xi^{\rm bulk}_{M,L}\mid\mathcal F_Y),
                    \Law(\Pi^{\rm bulk,lat}_{M,L}))=o(\lambda_M).
\end{equation}
\end{theorem}
\begin{proof}
For a finite mark cutoff $E$, repeat the signed comparison of
\cref{thm:marked-field} on this single macroscopic interval. With
$Q=L+E+1$ and $p=2^{-L}$ its error is at most
\[
 Cp(\mathcal M_B+|D_Y(Q)|)
       +Cp^2(MQ+E_Y(Q)+R_{2,\delta}(M,L)).
\]
The signed marks are summed before the separated-pair bound, so the
relation sum is at the base length $L$. Divide by $Mp=\lambda_M$.
The defect and support terms become $M^{-1/2+o(1)}$ and
$e^{-V_M+o(\nu_M)}$, respectively. The edge term becomes
$\lambda_M(e^{-V_M+o(\nu_M)}+M^{-1+o(1)})$.
By \eqref{eq:master-envelope} and $2^B=2/p$, the relation term is
$O_\varepsilon(M^{-1/3+\varepsilon}(1+\lambda_M))$.
The local term $pQ$ is smaller because $\lambda_M\to0$.

Take $E=\lceil V_M/\log2\rceil$. Then $E=o(\log M)$ and the base and
maximal support lengths stay in a fixed logarithmic band. The target
tail has mass at most $\lambda_M2^{-E-1}$. The actual tail is the event
that some retained site starts a run at length $L+E+1$; the macroscopic
masked first-moment bound gives the same upper bound times
$1+M^{-1/2+o(1)}$. Couple both tails to zero and average. Since
$\lambda_M\le1$ for large $M$, the preceding bounds give
\eqref{eq:relative-bulk-rate}. Measurable contraction proves
\eqref{eq:relative-bulk}.
\end{proof}
Every coordinate of $\mathbf B_L$ depends only on prime signs up to
$2L^2+L<Y_M^\sharp$. The stable product lift therefore gives
\begin{equation}\label{eq:boundary-bulk-factorization}
 \dTV\left(\Law(\mathbf B_L,\Xi^{\rm bulk}_{M,L}),
       \Law(\mathbf B_L)\otimes\Law(\Pi^{\rm bulk,lat}_{M,L})\right)
       =o(\lambda_M)=o(q_L+\lambda_M).
\end{equation}
This is asymptotic factorization at the indicated error scale, not exact
finite-$M$ independence.

\begin{theorem}[Two-source rare tail and conditional location]
\label{thm:crossover}
Under the preceding rare-intensity assumptions,
\begin{equation}\label{eq:two-source-tail}
 \PP(R_M\ge L)=q_L+\lambda_M+o(q_L+\lambda_M).
\end{equation}
Let $X^*_{M,L}$ be the least start of a contained run of length at least
$L$, on the event that one exists. If
$L-\log_2M-\pi(L)\to s\in\mathbb R$, then
\begin{equation}\label{eq:crossover-location}
 \Law(X^*_{M,L}/M\mid R_M\ge L)
 \Longrightarrow \frac1{1+2^{-s}}\delta_0+
                         \frac{2^{-s}}{1+2^{-s}}\Leb_{[0,1]}.
\end{equation}
The source-resolved version assigns a microscopic start its location divided
by $L^2$, and a bulk start its location divided by $M$; its limit is the
same mixture with the two source labels retained. The weights also have
their evident degenerate limits when $s\to\pm\infty$.
\end{theorem}
\begin{proof}
The mesoscopic region has mass $o(q_L+\lambda_M)$ by
\cref{prop:deep-starts}. Under the product target in
\eqref{eq:boundary-bulk-factorization}, a simultaneous microscopic and
bulk hit has probability $O(q_L\lambda_M)$, and at least two bulk hits
have probability $O(\lambda_M^2)$. Both are negligible on the combined
scale. This proves \eqref{eq:two-source-tail}.
Given a microscopic hit, \cref{thm:boundary} puts the start at one with
probability $1-o(1)$. Given exactly one bulk target point, its location
is uniform on the common lattice. Finally
\[
 \frac{\lambda_M}{q_L}
   =2^{-(L-\log_2M-\pi(L))}(1+o(1))\longrightarrow2^{-s}.
\]
Divide the two one-hit masses by \eqref{eq:two-source-tail} and pass from
the uniform grid to Lebesgue measure weakly. All total-variation comparisons
precede this final diffuse limit.
\end{proof}

\subsection{The prime clock}\label{sec:prime-clock}
Let $p_1=2<p_2<\cdots$. Almost surely
$J_\partial=\min\{j:f(p_j)=-1\}$ is finite and the maximal initial
positive run has length $\ell_\partial=p_{J_\partial}-1$: every smaller
integer has only positive prime factors, whereas that negative prime
changes sign. On this full-measure event and $C_L=1$, put
$G_L=J_\partial-\pi(L)-1$; assign $G_L=0$ elsewhere whenever a globally
defined random variable is needed. Then
\begin{equation}\label{eq:prime-clock}
 \PP(G_L=k\mid C_L=1)=2^{-(k+1)},\qquad
 \PP(\ell_\partial-L\ge h\mid C_L=1)
                    =2^{-[\pi(L+h)-\pi(L)]}.
\end{equation}
Thus prime rank, not integer distance, is its geometric clock.
The boundary theorem makes the law of $G_L$ given $B_L^{\rm mic}>0$
$O(2^{-\delta_0\pi(L)})$-close to $\nu_{\rm geo}$.
The prime number theorem gives $\ell_\partial/L\to1$ conditionally, but
the unscaled integer overshoot is not tight: for $L=m!+1$ the following
$m-1$ integers contain no prime.

\subsection{A lattice crossover with two geometric clocks}\label{sec:two-clock}
The two branches have the same geometric law but different clocks:
additional integers in a bulk run, and additional positive prime signs at
the initial boundary. A moving discrete target states their joint law
without requiring a limiting ratio of the two rare masses.

Write $\alpha_L=2^{-\pi(L)}$ and $\lambda^{\rm bulk}=\lambda_M^{\rm bulk}$.
On $\{R_M\ge L\}$, let $X^*=X^*_{M,L}$ and define
\[
 \Gamma_{M,L}=
 \begin{cases}
  (\partial,1,+1,G_L),&X^*=1,\\
  (\mathrm{bulk},X^*,f(X^*),\ell_{X^*}-L),&X^*\in I^{\rm bulk}_{M,L},\\
  \dagger,&\text{otherwise}.
 \end{cases}
\]
Here $\dagger$ records an exceptional start in the intervening region.
Let $\nu_\pm=(\delta_{-1}+\delta_{+1})/2$, and let
$\operatorname{Unif}(I^{\rm bulk}_{M,L})$ be the uniform law on that
integer grid.

\begin{theorem}[Sign-resolved, two-clock crossover]\label{thm:two-clock}
Under the rare-intensity assumptions of \cref{thm:crossover},
\begin{equation}\label{eq:two-clock-tv}
 \dTV\left(\Law(\Gamma_{M,L}\mid R_M\ge L),Q_{M,L}\right)\longrightarrow0,
\end{equation}
where the deterministic probability law on the same discrete state space is
\begin{align}
 Q_{M,L}
 &=\frac{\alpha_L}{\alpha_L+\lambda^{\rm bulk}}
        \delta_{(\partial,1,+1)}\otimes\nu_{\rm geo}
       \notag\\
 &\quad+\frac{\lambda^{\rm bulk}}{\alpha_L+\lambda^{\rm bulk}}
        \delta_{\mathrm{bulk}}\otimes
        \operatorname{Unif}(I^{\rm bulk}_{M,L})\otimes\nu_\pm
                     \otimes\nu_{\rm geo}.
       \label{eq:two-clock-target}
\end{align}
No convergence of $\lambda^{\rm bulk}/\alpha_L$ is needed. In particular,
\begin{equation}\label{eq:crossover-positive-sign}
 \PP(f(X^*)=+1\mid R_M\ge L)
  =\frac{\alpha_L+\lambda^{\rm bulk}/2}
         {\alpha_L+\lambda^{\rm bulk}}+o(1).
\end{equation}
The bulk excess in \eqref{eq:two-clock-tv} may instead be measured using
the right-censored run length in the finite word $(f(1),\ldots,f(M))$.
\end{theorem}
\begin{proof}
Fix $K\ge1$. Count the consecutive positive signs among the first $K$
primes after $L$, capped at $K$, and set the result to zero off $C_L=1$.
This is a genuine finite-cylinder random variable, agreeing almost surely
with $G_L\wedge K$. For fixed $K$, Bertrand's postulate places these
primes below $2^K L<Y_M^\sharp$ for large $M$. The stable lift applied to
\cref{thm:relative-bulk} therefore retains the microscopic vector and
this clock jointly with \emph{all} signed bulk marks, at error
$o(\lambda_M)$. No bulk-mark truncation is needed here.
The interior microscopic and mesoscopic errors are
$o(\alpha_L+\lambda_M)$ by \cref{thm:boundary,prop:deep-starts}.

Write $\alpha=\alpha_L$ and $b=\lambda^{\rm bulk}$.
In the product target, keep either a boundary hit with no bulk point,
or no boundary hit with exactly one bulk point. This event has exact mass
\[
 e^{-b}\bigl[\alpha+(1-\alpha)b\bigr].
\]
Its conditional source weights are
$\alpha/[\alpha+(1-\alpha)b]$ and
$(1-\alpha)b/[\alpha+(1-\alpha)b]$.
Their mixture differs from the weights in \eqref{eq:two-clock-target}
by at most
\[
 \frac{\alpha^2b}{(\alpha+b)[\alpha+(1-\alpha)b]}\le b.
\]
The discarded simultaneous and multiple hits cost
$O(\alpha b+b^2)=o(\alpha+b)$.
Given the one bulk point, its site, sign and excess have the independent
laws in \eqref{eq:two-clock-target}. Normalize all comparison errors by
$\alpha+b$, not by the smaller source mass. This proves the comparison
with the boundary clock capped at $K$, uniformly in the moving weights.

Under the actual law, a recorded boundary tail $G_L>K$ has conditional
probability at most $2^{-K-1}$, since
$\PP(C_L=1,G_L>K)=\alpha2^{-K-1}$ and
$\PP(R_M\ge L)\ge\alpha$. The target boundary tail is also at most
$2^{-K-1}$. Let $M\to\infty$ first and then $K\to\infty$.
This proves \eqref{eq:two-clock-tv}; projecting the sign gives
\eqref{eq:crossover-positive-sign}.

For the last assertion, under the one-hit bulk target a mark is censored
at site $x$ only if $e>M-x+1-L$. The average of this geometric tail is
\[
 \frac1{|I^{\rm bulk}_{M,L}|}
 \sum_{x\in I^{\rm bulk}_{M,L}}2^{-(M-x+2-L)}
       \le\frac1{|I^{\rm bulk}_{M,L}|}=o(1).
\]
The total-variation comparison transfers this estimate, so replacing that
mark changes the conditional law by $o(1)$.
\end{proof}
After scaling sites by $M$, convergence of the ratio recovers the mixture
in \cref{thm:crossover}, now with signs and both overshoot clocks retained.
The spatial limit is weak; the stronger statement
\eqref{eq:two-clock-tv} is entirely on the prelimit lattice. Its $o(1)$
error is on the combined source scale. It does not supply a relative
estimate for the smaller source when its mixture weight tends to zero.

\subsection{Affine information at the boundary}
\begin{theorem}[Affine-conditioned crossover]\label{thm:affine-crossover}
Let $A_M=\{G_M X(Y_M^\sharp)=b_M^{\rm aff}\}$ be a compatible affine
cylinder of rank $r_M$, and assume compatibility also with $C_L=1$.
Set
\[
 U_L=\Span\{e_p:p\le L\},\qquad
 \kappa_M=\dim(\operatorname{row}G_M\cap U_L),\qquad
 d_M=\pi(L)-\kappa_M.
\]
Whenever $L\le Y_M^\sharp$, the finite identity
$\PP(C_L=1\mid A_M)=2^{-d_M}$ holds exactly.
For the asymptotic conclusions, let $M,L\to\infty$ with
$L=O(\log M)$ and $\lambda_M=M2^{-L}\to0$.
Fix $c>0$ and assume $r_M\log2\le V_M-c\nu_M$ eventually.
The affine equations, their number and their right-hand sides may all
vary along the sequence. Then the rare-tail formula is
\[
 \PP(R_M\ge L\mid A_M)=2^{-d_M}+\lambda_M+o(2^{-d_M}+\lambda_M).
\]
If $L-\log_2M-d_M\to s$, the conditional location law given
$A_M$ and $R_M\ge L$ is \eqref{eq:crossover-location} with the same
weights. Microscopic non-vacancy under $A_M$ is concentrated on the border.
\end{theorem}
\begin{proof}
The stacked cylinder and border system has rank
$r_M+\pi(L)-\kappa_M$, proving the exact mass ratio.
Write $I=r_M\log2$. Apply the factor $e^I$ to the quantitative bound
\eqref{eq:relative-bulk-rate} \emph{before} taking a limit. Its normalized
cutoff term is at most $e^{-c\nu_M+o(\nu_M)}$, and its polynomial term
vanishes because $I=O(V_M)=o(\log M)$. Hence the conditional bulk error
is $o(\lambda_M)$.
The microscopic interior mass, after conditioning and division by
$2^{-d_M}$, is at most
$O(e^I2^{-\delta_0\pi(L)-\kappa_M})=o(1)$, since
$I=O(V_M)=o(\pi(L))$. Similarly the two terms of
\eqref{eq:mesoscopic} become $o(\lambda_M)$ and $o(2^{-d_M})$.
The conditional stable lift and the exact one-hit normalization in the
proof of \cref{thm:two-clock} now apply, with total normalization
$2^{-d_M}+\lambda_M^{\rm bulk}$.
\end{proof}
The parameter $\kappa_M$ counts linear consequences supported wholly on
the border coordinates, not input equations merely mentioning them.
No condition on future primes is needed for the rare tail, microscopic
concentration or location laws. The geometric future mark requires a
separate condition. Let $q_1<q_2<\cdots$ be the primes after $L$.
If, for each fixed $K$, eventually in $M$ with a threshold that may depend
on $K$,
\[
 (\operatorname{row}G_M+U_L)\cap\Span\{e_{q_1},\ldots,e_{q_K}\}=\{0\},
\]
then the geometric boundary mark and the full mixture of
\cref{thm:two-clock} survive under $A_M$, with $\alpha_L$ replaced by
$2^{-d_M}$. First fix $K$ and pass to the size limit, then let $K\to\infty$;
no common threshold for all $K$ or neutrality for a growing $K(M)$ is
assumed. In general, if the extended affine stack is compatible,
$\PP(G_L\ge K\mid A_M,C_L=1)$ equals
$2^{-[K-\dim((\operatorname{row}G_M+U_L)\cap\Span\{e_{q_1},\ldots,e_{q_K}\})]}$;
if incompatible, the tail is zero. This separates the information that
changes the border mass from information that changes its future mark.
\Cref{supp:sec:boundary-conditioning} records the rank calculation.

% END ORIGINAL SOURCE: sections/07_prefix_boundary.tex

% BEGIN ORIGINAL SOURCE: sections/08_discussion.tex
\section{What the method isolates}\label{sec:discussion}
Prime signs are independent, but their products satisfy exact identities
at arbitrarily large distances. The proof uses both facts. It counts the
homogeneous relation spaces created by multiplication, then uses the
prime environment either for an exact dependency graph or for an exact
one-word conditional coupling. In the latter route, potential dependence
is separated from the changes actually imposed by the forcing. The
three-term profile \eqref{eq:master-profile}, and its full-value extension
\eqref{eq:absolute-profile}, are the arithmetic outputs. The independent
Poisson field is obtained only after the separate marginal, local-overlap
and prime-support estimates have been verified.

\subsection{A class of observables, rather than one count}
The dictionary theorem separates local word combinatorics from nonlocal
arithmetic. Changing the prescribed signs does not change a valuation
matrix. Prescribing their absolute orientation removes at most two parity
conditions in a pair of windows; the host count controls the additional
relations. Consequently the same profile handles a growing dictionary and
retains the identity and position of every rare word. Explicit marker families make its overlap condition zero. For a typical
dictionary, one can do more: averaging before the worst-case estimate
reduces pair dependence to equality of the two complete words. The
finite collision bound removes the uniform cardinality restriction while
retaining the same labels and bounded total intensity. Short suffix--prefix matches can instead create
local clusters independently of any multiplicative relation; such
dictionaries require a separate analysis.

Long runs provide two complementary examples. Counting left-maximal starts
gives a Poisson field with geometric excesses and fair signs. Counting all
constant windows retains the clusters, and the field contracts to the
explicit compound-Poisson law of \cref{cor:window-clusters}. Both statements
come from the same comparison, not from two unrelated moment arguments.
The asymmetric almost-sure bounds in \cref{cor:as-longest} use its summable
quantitative errors; they do not require independence across scales.

The boundary is a different arithmetic regime. Its initial positive
cylinder imposes $\pi(L)$ independent prime constraints, whereas other
microscopic starts have an additional rank cost. The competition between
$2^{-\pi(L)}$ and $M2^{-L}$ therefore produces a change in the location
and sign of a rare long run. The two-clock theorem retains more than the
limiting source weights: it gives a prelimit lattice law, with prime-rank
overshoot at the boundary and integer overshoot in the bulk. The logistic
formula is a consequence of normalizing these two masses. The substantive
inputs are the microscopic rank surplus, the relative marked bulk kernel
and the negligible region between them.

\subsection{Arithmetic and probabilistic limits of the estimates}
Three terms reach $N^{5/3}$ at criticality in the present upper bound.
Improving only the terminal kernel energy would therefore not improve the
total exponent. The rational families in
\eqref{eq:rational-lower-dyadic}--\eqref{eq:rational-lower-macro} provide
lower exponents $4/3$ and $3/2$. Sharpening this gap is an arithmetic
question. It is distinct from improving the leading subexponential error,
which is governed by exceptional supports and shared prime coordinates.
For growing dictionaries the relevant quantity is not the full relation
sum, but its capped version. Mutual exclusivity at one site gives a
probability ceiling, and \cref{prop:capped-profile} preserves it through
the terminal count. This removes the terminal restriction on dictionary
size without improving the uncapped $5/3$ exponent. The remaining term
$NQ_B^{2/3}$ gives the sufficient exponent $1/2$ in
\eqref{eq:dictionary-critical}. Extending this uniform result requires control of the nonterminal
contribution or additional information about the dictionary. The typical
theorem provides the latter through its sampling law, not through a
smaller universal relation profile. A short affine description alone is
not a guarantee: the high-probability result concerns a uniformly sampled
full-rank matrix and an independent uniform right-hand side.

The Diophantine step uses only polynomial-height boxes. Localizing the
squarefree parts to the coefficient and root differences reduces the
count to two factors and a classical uniform Pell estimate; a height-free
bound on all integral points is not needed. The self-contained proof
fixes the required uniformity, rather than claiming priority for that
counting principle.

Different proof routes give sufficient budgets, not intrinsic prices
for retaining a label. In the hard corridor with $\Lambda\to\infty$,
\cref{thm:micro-run-tv,cor:dyadic-micro} retain all sites at the
one-factor budget, on a prefix and by restriction on a dyadic block.
Selected-prime reciprocals replace the all-large-prime footprint. The
tables in \cref{sec:conditional-laws} separate these available ranges
from the direct scalar and graph-based estimates. Local resolution still
requires an error negligible relative to the event being resolved.
At information-dependent cutoffs, \cref{prop:information-labelled}
balances deletion and graph costs; the event's measurability must be
retained along with its information cost.

\paragraph{What the Palm complement does and does not prove.}
\Cref{supp:sec:palm-complements} separates regular target configurations,
exact signed finite identities and consequences of the microscopic
comparison. Prescribed presence does not by itself control the absence
of further runs. The configuration-mass identity and
\cref{thm:micro-run-tv} imply the averaged one-sided Palm deficit
\cref{supp:palm:cor:palm-completion}; a stronger squared Walsh norm is
not asserted to vanish. Conversely,
\cref{supp:palm:thm:cumulant-obstruction} exhibits diverging all-order
absolute activity while microscopic distance tends to zero. Rare
configurations may dominate an exponential moment without preventing
ordinary total-variation approximation. The proof by selected pivots
never requires that activity to be small.

\subsection{Further questions}
Dictionaries with substantial short overlaps need a cluster analysis
compatible with the nonlocal arithmetic dependence. The compound-Poisson
word laws for Markov sources in
\cite{ReinertSchbath1998,RoquainSchbath2007} provide context, not an
automatic transfer. Our constant-window law is one model case, not a
classification of periodic or interacting patterns. Extending the
selected-coordinate coupling to nested word families, other prime
alphabets or larger conditioning fields would likewise require its own
exact forcing law, arithmetic footprint and tail estimates.

Records, censoring and occupation times require additional conventions
about the order and recognition of occurrences. Their process-level
analysis is outside the scope of this article. A common measurable
readout of the same-grid comparison does not by itself supply a new
Gaussian limit, a topology of convergence or asymptotic independence.
Similarly, the binary run-field theorem is not a comparison for every
prime-value distribution or for arbitrary persistent cluster targets.
These are boundaries of the results proved here, not assertions that
the corresponding questions are unresolved in all other models.

% END ORIGINAL SOURCE: sections/08_discussion.tex

\Needspace{12\baselineskip}
% BEGIN ORIGINAL SOURCE: declarations.tex
% BEGIN ORIGINAL SOURCE: formalization_v3_en.tex
\section*{Formalization and Palomar registrations}
\phantomsection\label{sec:formalization}
\addcontentsline{toc}{section}{Formalization and Palomar registrations}
A companion formalization in Lean~4 covers the principal arithmetic constructions
and probabilistic conclusions of this article and its technical companion.
The development uses Lean~4.34.0 and Mathlib~v4.34.0 and is available in the
\href{https://github.com/Vulkin-prog/paper-c-lean}{\texttt{paper-c-lean} repository}.
It includes explicit definitions of the random model, conditioning events,
observables and comparison laws, together with proofs and reproducible verification
instructions. The repository's correspondence between the manuscripts and Lean
statements records the precise coverage, hypotheses and proof substitutions.

The formalization is relative to seven explicit literature inputs: scalar Poisson
Stein factors; finite Arratia--Goldstein--Gordon process comparison; the prime
number theorem remainder; multivariate Poisson Stein solution bounds; the uniform
prime-divisor estimate of Laishram--Shorey; Shorey's square-product exclusion; and
the Nicolas--Robin divisor bound. These propositions are hypotheses of the formal
theorems that use them; their proofs are not supplied by this development. In
particular, the analytic Stein solution used in \cref{supp:pivot:lem:palm} remains
an input to its Lean proof. The finite coupling argument is formally checked
under this premise, while the semigroup construction given in the companion is
outside that formal proof. This is a limitation of formal coverage, not an
additional assumption in the written argument.

The formal proofs contain no proof placeholders and use only Lean's standard
foundational axioms \texttt{propext}, \texttt{Classical.choice} and
\texttt{Quot.sound}. This foundational check does not discharge the literature
hypotheses above. The written proofs and the formal development provide
complementary accounts, whose precise correspondence is documented in the
repository.

\Needspace{17\baselineskip}
Seven Palomar registrations cover a selection of 89 declarations from the
development. All entries below are version~1; the counts refer to the registered
selection, not to the total number of results in the repository.

\begin{center}
\small
\setlength{\tabcolsep}{4pt}
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\textwidth}{@{}>{\raggedright\arraybackslash}X r l@{}}
\toprule
\textbf{Scope} & \textbf{Count} & \textbf{Palomar identifier} \\
\midrule
Critical Poisson fields and conditional transfers & 16 &
\href{https://palomar-registry.org/entry?id=PALOMAR-2026-09-20-000003\&version=1}{\texttt{PALOMAR-2026-09-20-000003}} \\
Word dictionaries, exact marks and compound clusters & 11 &
\href{https://palomar-registry.org/entry?id=PALOMAR-2026-09-20-000004\&version=1}{\texttt{PALOMAR-2026-09-20-000004}} \\
Threshold staircases and the Poisson--Gaussian bridge & 29 &
\href{https://palomar-registry.org/entry?id=PALOMAR-2026-09-20-000007\&version=1}{\texttt{PALOMAR-2026-09-20-000007}} \\
Microscopic boundary, prefixes and longest runs & 7 &
\href{https://palomar-registry.org/entry?id=PALOMAR-2026-09-20-000011\&version=1}{\texttt{PALOMAR-2026-09-20-000011}} \\
Microscopic--bulk crossover and conditional affine laws & 11 &
\href{https://palomar-registry.org/entry?id=PALOMAR-2026-09-20-000012\&version=1}{\texttt{PALOMAR-2026-09-20-000012}} \\
Microscopic signed fields and empirical Poisson laws & 6 &
\href{https://palomar-registry.org/entry?id=PALOMAR-2026-09-21-000002\&version=1}{\texttt{PALOMAR-2026-09-21-000002}} \\
Regular configurations, Palm deficits and cumulant obstructions & 9 &
\href{https://palomar-registry.org/entry?id=PALOMAR-2026-09-21-000003\&version=1}{\texttt{PALOMAR-2026-09-21-000003}} \\
\bottomrule
\end{tabularx}
\end{center}

Each registration identifies a precise set of formal statements and a fixed
repository revision. Comparator checks agreement between the selected statements
and their proved counterparts; the proofs are checked by Lean and independently
replayed through NanoDa. Palomar's separate editorial review assesses the
correspondence with the informal account and the mathematical interest of the
submission. That review is automated and does not constitute independent human
peer review. The registrations document the verification of the selected formal
results under their stated hypotheses; they do not certify every argument or
claim in the article and companion. The verification and review protocol is
specified in
\href{https://github.com/PalomarRegistry/PalomarPolicy/blob/main/docs/specification.md}{\texttt{PalomarPolicy}}.

% END ORIGINAL SOURCE: formalization_v3_en.tex


\section*{Declaration on artificial-intelligence assistance}
\phantomsection\label{sec:ai-assistance}
The research was organized and directed by the author with substantial
assistance from language-model systems, including Claude Fable 5 and
Claude Opus 4.8 (Anthropic), and GPT-5.6 Sol and GPT-6 Astra (OpenAI).
These systems were used to explore candidate lemmas and proof routes,
draft and reorganize manuscripts, assist with the separate Lean development,
write diagnostic programs, check references and build artifacts, and
conduct adversarial reviews through separate instances.

GPT-6 Astra assisted in particular with the selected-pivot and Palm
arguments, the empirical corollary, the dyadic restriction and local-field
obstruction, and the exposition, bibliography and preparation of this
edition. This attribution preserves the other models' historical
contributions. The author determines the claims and scope and accepts
responsibility for the arguments, references and any remaining errors.
AI-assisted checks are not independent human peer review. The
mathematical claims rest on the printed arguments and stated external
results, not on model output or the success of finite diagnostic tests.

\section*{Competing interests}
The author declares no competing interests.

% END ORIGINAL SOURCE: declarations.tex

\printbibliography[heading=bibintoc,title={References}]

\end{refsection}
\clearpage
\renewcommand{\LRtocfile}{companioncontents}
\pagenumbering{arabic}
\setcounter{section}{0}
\setcounter{subsection}{0}
\setcounter{subsubsection}{0}
\setcounter{equation}{0}
\setcounter{theorem}{0}
\setcounter{figure}{0}
\setcounter{table}{0}
\setlength{\cftbeforesecskip}{3pt}
\fancyhead[L]{Technical companion: long runs and rare patterns}
\title{\textbf{Technical companion to}\\[2mm]
\large\emph{Long runs and rare patterns of a random\\completely multiplicative function}}
\author{Brice Pouly}
\date{Version 3}
\makeatletter
\patchcmd{\@maketitle}{\vskip 2em}{\vskip 0pt}{}{}
\makeatother

\begin{refsection}

\maketitle
\vspace{-0.6em}
\noindent\begin{minipage}{\textwidth}\footnotesize
This research received substantial assistance from AI systems; see the
\hyperref[sec:ai-assistance]{declaration at the end of the article}.
\end{minipage}\medskip
\begin{abstract}
This companion completes the proof package for the article, Version 3.
Appendices A--E give the growing-degree Runge and bounded-height Pell
calculations, prime cutoffs, scalar smoothing, marked and dictionary
comparisons, conditional kernels, and the boundary and relative-bulk
estimates. The common macroscopic square-product argument remains in
the article.

Appendix F proves the full same-grid signed prefix comparison. It
assembles the existing arithmetic inputs and an independently proved
scalar tail, then gives the finite Palm--Stein inequality, the uniform
reciprocal-pivot estimate and the exact one-word conditional coupling.
Appendix G is a separate methodological complement: regular target
configurations, signed Palm-void identities and an arithmetic obstruction
to global absolute-cumulant smallness. Its vanishing Palm deficit is a
consequence of Appendix F, never an input to its proof.
\end{abstract}
\medskip
\noindent\textbf{Reading convention.} The two documents form one proof package.
Appendix letters and bibliographic numbers are local; unqualified
numbered statements refer to the article. All unlabelled logarithms are
natural. The scope, explicit literature inputs and limitations of the Lean
development, together with its seven Palomar registrations, are described in the
article's section
\hyperref[sec:formalization]{``Formalization and Palomar registrations''}.

\tableofcontents
\clearpage
% BEGIN ORIGINAL SOURCE: companion/00_external_inputs.tex
\section*{External inputs and internal deductions}
\addcontentsline{toc}{section}{External inputs and internal deductions}
This register identifies the non-elementary inputs, their precise role,
and the calculations proved in this package. References are local to this
companion; statement links lead to the relevant document. Classical
finite-dimensional linear algebra, conditioning and elementary inequalities
are not separate imported hypotheses.
The equation numbers (2.14)--(2.15) cited for Krokowski refer to arXiv version 3;
the cited Lemma~4 of Cilleruelo--Garaev refers to their arXiv text.

\begingroup
\small
\renewcommand{\arraystretch}{1.16}
\begin{longtable}{@{}L{.205\textwidth}L{.30\textwidth}L{.435\textwidth}@{}}
\toprule
Input and source & Form consumed & Use and internal deduction\\\midrule
\endfirsthead
\toprule Input and source & Form consumed & Use and internal deduction\\\midrule
\endhead
\bottomrule\endfoot

Prime number theorem; standard prime sums \cite{Tenenbaum2015}
& $\pi(t)\sim t/\log t$; elementary weighted consequences
& \Cref{supp:prop:cutoff-calculation} proves both cutoff balances; \cref{supp:pivot:lem:rankin,supp:pivot:prop:reciprocal} prove the uniform reciprocal-pivot estimate.\\
Finite-dependency Poisson approximation \cite{ArratiaGoldsteinGordon1989,ArratiaGoldsteinGordon1990}
& Scalar and process theorems; $b_3=0$ for an exact graph
& \Cref{thm:transfer,thm:word-field,thm:marked-field}; the supremum convention for variation is half the sources' norm.\\
Scalar Stein factors \cite[(2.14)--(2.15)]{Krokowski2017}
& Zeroth and first differences, of orders $\lambda^{-1/2}$ and $\lambda^{-1}$
& \Cref{supp:lem:soft-retention} proves the exceptional-set decomposition; no Malliavin input is used.\\
Directional Poisson factor \cite[p.~179, Lemmas~2--3]{Barbour1988SteinProcess}
& Weighted quadratic Hessian bound and $|\Delta_{ij}g|\le1$, for the same solution
& \Cref{supp:eq:directional-factor,supp:thm:aggregated}; Hessian conjugation, independent Poisson filling and geometric mark summation are explicit.\\
Palm--Stein method \cite{Barbour1988SteinProcess,ChenXia2004}
& Immigration--death generator and conditional-law identity
& \Cref{supp:pivot:lem:palm} proves the finite total-variation bound with no graph assumption; the exact arithmetic forcing is proved in \cref{supp:pivot:prop:forcing}.\\
Quadratic ideals and units \cite{HalterKoch2013}; divisors \cite{NicolasRobin1983}
& Ideal factorization, unit orbits and $\tau(n)=\exp(O(\log n/\log\log n))$
& \Cref{supp:lem:pell} gives a self-contained proof of the classical uniform count; split-product localization is internal.\\
Uniform polynomial-height Pell count \cite[arXiv text, Lemma~4]{CillerueloGaraev2011}; \cite[Lemma~2.10, Remark~2.12]{Tao2026}
& $\exp(O_K(\log M/\log\log M))$ solutions in polynomial-height boxes
& The estimate itself has these antecedents; Appendix A.2 fixes the normalization and height dependence needed here.\\
Largest-prime reciprocal asymptotics \cite[eq.~(1.3)]{ErdosIvicPomerance1986}
& $S_+(x)=x\exp\{-2V_x+o(\nu_x)\}$ at the precision used here
& Appendix F.3 deduces the odd-pivot bound by $n=a^2v$. The direct Rankin proof is retained; no separate analytic priority is claimed.\\
Laishram--Shorey \cite[eq.~(14), p.~331]{LaishramShorey2004}
& $\omega(\prod_{j<B}(n+j))\ge(2-\varepsilon)\pi(B)$, uniformly for $n>B$
& Appendix E.2 records the quantifiers; the private-coordinate and medium-prime rank arguments are internal.\\
Balasubramanian--Shorey \cite{BalasubramanianShorey1993}, as recorded in \cite[pp.~55--56]{Shorey1994BakerApplications}
& Square-product exclusion with $P^+(b)\le k$ and the stated height/density conditions
& \Cref{supp:lem:pointwise-maximum} gives the substitutions and contrapositive. No size bound on the smooth coefficient is added.\\
Berry's normal approximation bound \cite{Berry1941}
& Third-absolute-moment normal approximation
& Appendix D uses the explicit Poisson specialization; the marked comparison and joint limiting independence are proved here.\\
Binary marker construction \cite{CheeKiahPurkayasthaWang2013}
& A classical cross-bifix-free family
& \Cref{cor:marker-dictionaries} proves the particular binary construction and its size bound. This illustrative application is not an input to the arithmetic theorem.\\
\end{longtable}
\endgroup

\medskip
Runge's split-product estimate, the Hamming-ball argument, the raw and
capped two-window profiles, the simple-incidence rank lemma and the
relative bulk normalization are proved in the article--companion pair.
Evertse--Silverman is contextual only. The selected-pivot comparison and
Palm complements are proved in Appendices F--G, with the latter using,
not establishing, the microscopic comparison. The table concerns the
mathematical inputs to the written arguments. In particular the directional estimate is obtained
directly from Barbour's two lemmas, while the arithmetic comparison and
Poisson filling are proved here.


\paragraph{Written inputs and formal coverage.}
This table describes the written proof. The scope and limitations of the
Lean development, including its seven explicit literature inputs, are recorded in
the article's section
\hyperref[sec:formalization]{``Formalization and Palomar registrations''}.
In particular, the directional input supplies the entrywise bound and the
weighted quadratic bound for the same Stein solution. An entrywise bound by one
is not an unweighted Euclidean operator-norm bound. The derivation in
\eqref{supp:eq:directional-factor} uses the weighted and entrywise estimates.

% END ORIGINAL SOURCE: companion/00_external_inputs.tex

\Needspace{16\baselineskip}
% BEGIN ORIGINAL SOURCE: companion/00_reading_routes.tex
\section*{How to use this companion}
\addcontentsline{toc}{section}{How to use this companion}
This companion and the article form a single proof package. The article gives
the objects, main arguments and conclusions; the entries below identify the
calculation behind each excursion and the point at which the main argument
resumes. Unqualified numbers refer to the article; lettered numbers refer
to this companion.

\begin{center}\small
\renewcommand{\arraystretch}{1.3}
\begin{tabularx}{\textwidth}{@{}>{\raggedright\arraybackslash}p{.24\textwidth}>{\raggedright\arraybackslash}X>{\raggedright\arraybackslash}p{.25\textwidth}@{}}
\toprule
Entry from the article & Calculation here & Return to the article\\\midrule
\Cref{lem:runge} & Coefficients, integrality and the growing-degree remainder in \cref{supp:lem:runge-calculation}. & Weighted defects, \cref{prop:defects}.\\
\Cref{macro:lem:pell,lem:split-bounded} & Uniform ideal and unit counts, then localization to two square classes in A.2--A.3. & Bounded components and the deep-core estimates.\\
\Cref{prop:cutoffs,thm:scalar-rates} & Weighted prime sums, degree bounds and scalar retention in B.1--B.3. & Scalar laws or the fields of Section~5.\\
\Cref{thm:word-field,thm:typical-dictionaries} & Marginal cap, selection probabilities and collision bounds in C.1. & Uniform, typical and affine dictionaries.\\
\Cref{thm:moving-comparison}(ii) & The two Stein bounds, Poisson filling and the single tail event in C.3--C.4. & The staircase; information-adapted labels in C.5 and \cref{prop:information-labelled}.\\
\Cref{thm:resolved-kernel} & Poisson atom estimates and the resolution cost in D.1--D.2. & Conditional laws in Section~6.\\
\Cref{thm:boundary} & Prime incidences, quotient rank and the post-quadratic bound in E.1--E.5. & The microscopic mass and the prefix law.\\
\Cref{thm:relative-bulk,thm:two-clock} & The normalized error table, finite-cylinder clock and affine conditioning in E.6--E.7. & The two-source comparison and its affine variant.\\\bottomrule
\end{tabularx}
\end{center}

\paragraph{The prefix coupling and its optional complement.}
\Cref{thm:micro-run-tv} enters Appendix F. The signed input register
\cref{supp:pivot:lem:inputs} points to the one-window and two-window
estimates, local signed ranks, deep-start bound and independently proved
scalar tail. The argument then proceeds through the finite Palm--Stein
inequality, uniform Rankin estimate, reciprocal pivots and exact forcing.
It returns to the full same-grid field and
\cref{cor:micro-readout}. The restriction to the dyadic field in
\cref{cor:dyadic-micro} needs only the stability of the hard scales
under $M=4N$ and projection of integer labels. No result of Appendix G
is used along this route.
Appendix G separately proves regular-cloud statements, finite signed
Palm identities and the absolute-cumulant obstruction. Its vanishing
Palm deficit is downstream of Appendix F.
The empirical consequence \cref{cor:empirical-poisson} is proved in the
article, immediately after \cref{cor:micro-readout}: it uses concentration
of overlapping-window frequencies in the independent target and the
summable microscopic error, with no additional arithmetic input.
\Cref{rem:empirical-field-obstruction} then gives a finite-support
obstruction to the analogous local-vector comparison after fixing $f$.

\paragraph{Support and target conventions.}
For interior starts, the following probabilities hold on good supports
conditionally on the exposed prime signs.
\begin{center}\small
\begin{tabularx}{\textwidth}{@{}Xccc@{}}\toprule
Event & Values & Independent equations & Target probability\\\midrule
Run of length at least $L$ & $L+1$ & $L$ & $2^{-L}$\\
Absolute word of length $B$ & $B$ & $B$ & $2^{-B}$\\
Exact run, excess $e$ & $L+e+2$ & $L+e+1$ & $2^{-L-e-1}$\\
Exact run, excess $e$, sign $s$ & $L+e+2$ & $L+e+2$ & $2^{-L-e-2}$\\\bottomrule
\end{tabularx}
\end{center}
Summing the last row over signs gives $2^{-L-e-1}$, and then summing
over $e\ge0$ gives $2^{-L}$. With $e\le E$, the maximal support has
$Q+1$ values, $Q=L+E+1$. This is distinct from the base support
$B=L+1$ and from the exponential profile parameter $Q_B=2^B$.

\paragraph{Two different summations.}
For the graph-based spatially labelled comparison of Section~5, marks
are summed without weights before the pair estimate, leaving the
relation profile at the base length.
For an aggregated vector, the directional Stein weights are inserted
first, and the maximal-length profile is required. Section~C.3 keeps
this distinction explicit before any asymptotic absorption. Its signed
local factor can be four, not two. Full marks of base-contained starts
are not right-censored run lengths.

In the selected-pivot comparison of Appendix F the signed low types are
instead controlled on maximal supports and summed with their exact
weights. Its dimension-free bound has no inverse minimum type
probability. This is a third ledger, not a use of an aggregated
Stein factor on an unrecorded spatial field.

% END ORIGINAL SOURCE: companion/00_reading_routes.tex

\medskip
\appendix
% BEGIN ORIGINAL SOURCE: companion/A_runge_diophantine.tex
\section{Runge coefficients and uniform Diophantine inputs}\label{supp:sec:diophantine}
This appendix supplies the growing-degree Runge calculation and the
bounded-height Pell argument used in the article. The two types of
uniformity are different: the Runge degree is allowed to grow, whereas a
component's degree and its polynomial-height exponent are fixed before
the ambient scale. No height-free integral-point estimate is needed.

\subsection{The split-product calculation}
\begin{lemma}[Coefficient-level Runge estimate]\label{supp:lem:runge-calculation}
Under the hypotheses of \cref{lem:runge}, including $U\in\Z$, the bound
$U\le(C_0dR)^{2d}$ holds with an absolute constant $C_0$.
\end{lemma}
\begin{proof}
The $\gamma_\nu$ being distinct and $d\ge2$, we have $R\ge1$ and $G$ is not the square of a polynomial. Set
\[
 F(z)=\prod_{\nu=1}^{d}(1+\gamma_\nu z)^{1/2}
      =\sum_{m\ge0}c_mz^m,
\]
where each square root is the analytic branch equal to $1$ at $0$, and
\[
 P(T)=\sum_{m=0}^{k}c_mT^{k-m}.
\]

For $m\le k$, the identity
\[
 \binom{1/2}{j}
 =(-1)^{j-1}\frac{\operatorname{Cat}_{j-1}}{2^{2j-1}}
 \quad(j\ge1)
\]
implies $2^{2j}\binom{1/2}{j}\in\Z$ and
$|\binom{1/2}{j}|\le1$. More explicitly,
\[
 c_m=
 \sum_{j_1+\cdots+j_d=m}
 \prod_{\nu=1}^d
 \binom{1/2}{j_\nu}\gamma_\nu^{j_\nu}.
\]
The number of weak compositions of $m$ into $d$ parts is
$\binom{m+d-1}{d-1}\le2^{m+d}$. Since $m\le k=d/2$ and
$|\gamma_\nu|\le R$, we obtain simultaneously
\begin{equation}\label{supp:eq:runge-coefficients}
 2^dc_m\in\Z,
 \qquad
 |c_m|\le2^{m+d}R^m\le(8R)^d,
 \qquad 0\le m\le k.
\end{equation}
In particular $2^dP\in\Z[T]$, uniformly as $d$ grows with the ambient scale.

If $2R\le U\le4R$, the conclusion is immediate after enlarging $C_0$. Assume therefore $U>4R$. On the circle $|z|=(2R)^{-1}$,
\[
 |F(z)|\le(3/2)^{d/2}=:M_d.
\]
Cauchy's formula gives $|c_m|\le M_d(2R)^m$. Since $2R/U<1/2$,
\begin{align}
 |\sqrt{G(U)}-P(U)|
 &=U^k\left|\sum_{m\ge k+1}c_mU^{-m}\right|\notag\\
 &\le
 U^kM_d\sum_{m\ge k+1}(2R/U)^m\notag\\
 &\le
 \frac{2M_d(2R)^{k+1}}{U}
 \le\frac{(C_1dR)^d}{U}.
 \label{supp:eq:runge-tail}
\end{align}

If the difference is nonzero, $\sqrt{G(U)}$ is an integer and, by \eqref{supp:eq:runge-coefficients}, $P(U)\in2^{-d}\Z$. It is therefore at least $2^{-d}$ in absolute value. Inequality \eqref{supp:eq:runge-tail} then implies
\[
 U\le2^d(C_1dR)^d\le(C_0dR)^{2d}.
\]

Assume finally $\sqrt{G(U)}=P(U)$. The polynomial
\[
 Q(T)=2^{2d}G(T)-(2^dP(T))^2
\]
has integer coefficients. It is nonzero, since $G$ is not a square. The expansion at infinity gives $\sqrt{G(T)}-P(T)=O(T^{-1})$, whence
\[
 \deg Q\le k-1.
\]
The coefficient of $T^r$ in $G$ is a symmetric polynomial containing at most
$\binom dr\le2^d$ monomials, each of modulus at most $R^d$ after the
uniform upper bound $R\ge1$. By \eqref{supp:eq:runge-coefficients}, every
coefficient of $2^dP$ is at most $(16R)^d$ in modulus. A coefficient
of $(2^dP)^2$ is the sum of at most $d+1$ products of two such
coefficients. Consequently
\[
 H(2^{2d}G)\le(8R)^{2d},
 \qquad
 H((2^dP)^2)\le(d+1)(16R)^{2d},
\]
and therefore, for an absolute constant $C_2$,
\[
 H(Q)\le(C_2dR)^{2d}.
\]
This estimate is uniform in $d$; it hides no step where
$d$ would be fixed. An integer zero of a nonzero integer polynomial has
modulus at most $1+H(Q)$, by the elementary Cauchy bound. After
enlarging $C_0$, we again obtain \eqref{eq:runge-bound}.
\end{proof}


The integrality of $U$ is used precisely in the separation by $2^{-d}$.
Without it the statement is false: for $G(T)=T(T+1)$ the real numbers
$U_m=(\sqrt{1+4m^2}-1)/2$ tend to infinity and satisfy $G(U_m)=m^2$.

\subsection{Pell equations with varying coefficients}
The uniform polynomial-height count below is a classical Pell estimate;
see \cite[arXiv text, Lemma~4]{CillerueloGaraev2011} and
\cite[Lemma~2.10 and Remark~2.12]{Tao2026}.
For comparison with $x^2-Dy^2=E$, multiply $Az^2-Cw^2=e$ by $A$
and set $x=Az$, $D=AC$, $E=Ae$. The nonsquare and polynomial-height
conditions are preserved. The proof below is retained to expose its
uniformity, not as a claim that this counting result is new.
\begin{lemma}[Uniform Pell count in a polynomial range]\label{supp:lem:pell}
Fix $K_0>0$. Let $A,C$ be positive squarefree integers such that $A/C$ is not a rational square, and let $e\ne0$. Assume
\[
 A,C,|e|\le M^{K_0}.
\]
The number of integer solutions $(z,w)$ of
\[
 Az^2-Cw^2=e,
 \qquad
 |z|,|w|\le M^{K_0},
\]
is
\begin{equation}\label{supp:eq:pell-uniform}
 \exp\!\left(O_{K_0}\!\left(\frac{\log M}{\log\log M}\right)\right).
\end{equation}
\end{lemma}

\begin{proof}
Set $g=(A,C)$, $A=gA_0$, $C=gC_0$. Since $A,C$ are squarefree, $A_0,C_0$ are squarefree, coprime, and their prime supports are disjoint. If a solution exists, $g\mid e$; write $e=ge_0$. After multiplying by $A_0$ and substituting $X=A_0z$, the equation becomes
\begin{equation}\label{supp:eq:pell-norm}
 X^2-Dw^2=m_0,
 \qquad
 D=A_0C_0,
 \qquad
 m_0=A_0e_0.
\end{equation}
The product $D$ is squarefree. The hypothesis on $A/C$ gives $D>1$.

In the real quadratic field $K_D=\mathbb Q(\sqrt D)$, using the standard
ideal-and-unit description of quadratic orders \cite{HalterKoch2013}, the element
\[
 \alpha=X+w\sqrt D
\]
belongs to the ring of integers $\mathcal O_{K_D}$, since $\sqrt D$ is an algebraic integer. Its norm is $m_0$. Consequently
\[
 (\alpha)(\bar\alpha)=(m_0),
\]
and the integral ideal $(\alpha)$ divides the ideal $(m_0)$. For each rational prime $p^a\Vert m_0$, the number of choices for the part above $p$ is at most $(a+1)^2$: it equals $(a+1)^2$ if $p$ splits, $a+1$ if it is inert, and $2a+1\le(a+1)^2$ if it ramifies, since then $(p)=\mathfrak p^2$. The number of ideal divisors of $(m_0)$ is therefore at most
\[
 \prod_{p^a\Vert m_0}(a+1)^2=\tau(|m_0|)^2.
\]

Only the principal ideals $\mathfrak a=(\alpha)$ of absolute norm
$|m_0|$ that admit a generator of norm exactly $m_0$ can come from a
solution of \eqref{supp:eq:pell-norm}. Fix such an ideal and a generator
$\alpha_0$ of norm $m_0$. Then
\[
 |\sigma_1(\alpha_0)\sigma_2(\alpha_0)|=|m_0|.
\]
All relevant generators are among
\[
 \pm\alpha_0\varepsilon_D^n,
 \qquad n\in\mathbb Z,
\]
where $\varepsilon_D>1$ generates the free part of the unit group. If a unit of norm $-1$ exists, the norm-$m_0$ condition selects only one parity class of $n$, which cannot increase the number of solutions.

The bounds on $A,C,z,w$ give, for the two real embeddings $\sigma_1,\sigma_2$,
\[
 |\sigma_j(\alpha)|\le H:=M^{O_{K_0}(1)}.
\]
The inequalities
\[
 |\sigma_1(\alpha_0)|\varepsilon_D^n\le H,
 \qquad
 |\sigma_2(\alpha_0)|\varepsilon_D^{-n}\le H
\]
define an interval of integers $n$ of length at most
\[
 \frac{2\log H-\log|m_0|}{\log\varepsilon_D}+O(1)
 =O_{K_0}(\log M),
\]
since every real quadratic unit $\varepsilon>1$ satisfies
$\varepsilon\ge(1+\sqrt5)/2$. Indeed its conjugate is $\pm\varepsilon^{-1}$
and its trace is an integer. For norm $-1$, this gives
$\varepsilon-\varepsilon^{-1}\ge1$; for norm $+1$,
$\varepsilon+\varepsilon^{-1}\ge3$. Both give the asserted lower bound.
Thus the number of solutions is at most
\[
 O_{K_0}(\tau(|m_0|)^2\log M).
\]
Since $|m_0|\le M^{O_{K_0}(1)}$, the standard explicit divisor bound
\cite{NicolasRobin1983} gives
\[
 \tau(|m_0|)^2\log M
 =
 \exp\!\left(
 O_{K_0}\!\left(\frac{\log M}{\log\log M}\right)
 \right),
\]
which proves \eqref{supp:eq:pell-uniform}; the sign of $m_0$ does not affect the
argument.
\end{proof}
The exponent $K_0$ is fixed in every use. There is no appeal to a uniform
lower bound for a regulator that deteriorates with the discriminant: the
elementary absolute lower bound on a real quadratic fundamental unit
already controls the number of generators in each ideal class.

\subsection{Square-class localization in bounded height}
\label{supp:sec:split-localization}
In \cref{lem:split-bounded}, the decisive set is
$S=\{p:p\mid e\prod_{i<j}(h_i-h_j)\}$.
For a solution, a prime outside $S$ can divide only one shifted factor;
its valuation there is therefore even. We enumerate the squarefree parts
of only the first two factors, not of all $d$ factors. There are at most
$4^{|S|}$ pairs, and the two-factor equation has a nonzero right-hand side
because $h_1\ne h_2$.

Here is an explicit check on the order of parameters. Fix $d,K$ first.
For $M\ge2$,
\[
 \log|e\Delta|\le K\log M+
         \binom d2(K\log M+\log2)=O_{d,K}(\log M).
\]
Thus $4^{|S|}=\exp(O_{d,K}(\log M/\log\log M))$.
Independently, every retained squarefree coefficient is at most $2M^K$,
not an unrestricted product of primes in $S$. The coefficients, right-hand
side and square variables of its Pell equation are therefore all covered
by the fixed exponent $K+1$ (increasing the ambient threshold if needed).
The Pell constant does not depend on $S$, the offsets or the chosen pair.
When the squarefree coefficients agree, divisor factorization treats the
case that the nonsquare Pell lemma deliberately excludes.

For a component, $d=|J|\le K_0$ and the normalizing coefficient is the
squarefree part of $d_C\prod_{i\in I}(x+i)$. The smooth coefficient
$d_C\le\exp(O(B))$ and $|I|\le K_0$ put this integer in one fixed
polynomial range. This supplies exactly the uniformity consumed by
\cref{lem:one-sided-component}, even when the two starts have different
scales.

\subsection{Why a single rational unit is not contracted}
Two units force the reduced channel height to be at most $L$. One unit
does not. For example, take $N=2^{100}$, $L=100$, $a=103$, $b=102$ and
$t=12427947061061072563693168687$, with $x=bt$, $y=at$.
Both starts lie in $[N,2N)$, and $103i=102j$ has only $(i,j)=(0,0)$
in $\{-1,\ldots,99\}^2$. Yet $103>B=101$, and $t\equiv100\pmod{103}$
gives $v_{103}(x)=0$, $v_{103}(y)=1$.
No vertex of the first window is divisible by $103$, so the second
occurrence is pinned rather than isolated with the first. This is why the
article defines $\sigma=0$, $D^\#=D$ and $c^\#=c$ when $m\le1$.
The aligned deep-core proof removes the at most two components meeting
a single unit directly; it never assigns that unit a false private-kernel
identity.

\subsection{Uniformity across the master interval}
All vertices are $O(M)$ and at least $M^\delta-1$. Squarefree
$B$-smooth coefficients have size $M^{O(1)}$, while a fixed number of
offsets and smooth choices contributes $M^{o(1)}$. On a larger-start
terminal slice $X\le\max(x,y)<2X$, $B/\log X$ remains in a fixed
compact interval depending only on $(\delta,\beta_-,\beta_+)$. Thus the
Pell estimates can be applied at scale $X$ there, even when the other
start is smaller. The proof never identifies a macroscopic cross-block
pair with an intra-block pair, and never applies a fixed-ratio theorem
with a growing ratio.

% END ORIGINAL SOURCE: companion/A_runge_diophantine.tex

% BEGIN ORIGINAL SOURCE: companion/B_cutoffs.tex
\section{Prime cutoffs and scalar smoothing}\label{supp:sec:cutoffs}
This appendix verifies the free cutoff before specializing it to either
saddle. The distinction matters: the hard route deletes all exceptional
starts, whereas the soft scalar route leaves them in the Stein equation.

\subsection{A uniform weighted prime sum}
\begin{proposition}[Free Rankin calculation and both saddles]
\label{supp:prop:cutoff-calculation}
The bounds and expansions in \cref{prop:cutoffs} hold uniformly for $w$
in a fixed multiplicative neighbourhood of $\sqrt{H\log H}$,
where $H=\log N$.
\end{proposition}
\begin{proof}
Every $Y$-defective integer has the unique form $n=a^2s$ with $s$
squarefree and $Y$-smooth. Therefore, for $0<\zeta<1/2$,
\begin{equation}\label{supp:eq:rankin}
 \frac{A(X,Y)}X\le X^{-\zeta}
                         \prod_{p\le Y}(1+p^{-1+\zeta}).
\end{equation}
Indeed sum $\sqrt{X/s}$ over $s\le X$ and use
$s^{-1/2}\le X^{1/2-\zeta}s^{-1+\zeta}$.
Put $\nu=H/w$, choose the larger solution $u\ge1$ of $e^u/u=\nu$,
and set $\zeta=u/w$. Uniformly in the stated range,
\[
 u\sim\tfrac12\log H,\quad
 \nu\asymp\sqrt{H/\log H},\quad
 \zeta\asymp\sqrt{\log H/H},\quad
 \log H=o(\nu).
\]
For $X/N$ in a fixed compact subset of $(0,\infty)$,
$\zeta\log X=u\nu+o(\nu)$.

We require the prime number theorem inside the weighted Euler product,
not merely an upper bound with an unspecified constant. Let
$R(t)=\pi(t)-\operatorname{Li}(t)$. Given $\epsilon>0$, choose a fixed
lower split point beyond which $|R(t)|\le\epsilon t/\log t$.
Stieltjes integration by parts gives, uniformly in $w$,
\begin{align*}
 \sum_{p\le e^w}p^{-1+\zeta}
 &=\int_2^{e^w}\frac{t^{-1+\zeta}}{\log t}\,dt\nobreak
       +O_\epsilon(1)
       +O\left(\epsilon\frac{e^u}{w}
           +\epsilon\int_2^{e^w}\frac{t^{-1+\zeta}}{\log t}\,dt\right)\\
 &=\operatorname{Ei}(u)+o(\nu).
\end{align*}
For the second equality first let $N\to\infty$ and then
$\epsilon\downarrow0$. The substitution $z=\zeta\log t$ costs only
$O(\log(1/\zeta))=o(\nu)$ at the lower endpoint, and
$\operatorname{Ei}(u)\asymp e^u/u=\nu$. Also
$\sum_p p^{-2+2\zeta}=O(1)$, so logarithms of the Euler factors give
\begin{equation}\label{supp:eq:weighted-primes}
 \sum_{p\le e^w}\log(1+p^{-1+\zeta})
       =\operatorname{Ei}(u)+o(\nu).
\end{equation}
Floors in $Y=\lfloor e^w\rfloor$ alter these estimates by a negligible
quantity. Equations \eqref{supp:eq:rankin}--\eqref{supp:eq:weighted-primes}
yield
\[
 \log(A(X,Y)/X)\le-\nu u+\operatorname{Ei}(u)+o(\nu)
                      =-D(\nu)+o(\nu).
\]
A bad integer contaminates at most $B$ windows; $\log B=o(\nu)$ absorbs
this factor. The graph-edge estimate in the article gives
$E_Y/N^2\le e^{-w+o(\nu)}+N^{-1+o(1)}$.
A window uses at most $CBH/w$ large primes, and each prime meets at most
$CB(N/Y+1)$ starts. Hence
\begin{equation}\label{supp:eq:max-degree}
 \Delta_Y\ll B^2(H/w)(N/e^w+1),\qquad
 \Delta_Y/N\le e^{-w+o(\nu)}+N^{-1+o(1)}.
\end{equation}
This maximum-degree bound applies to the graph on all starts, including
exceptional ones.

For a unified expansion write $V_a=aD(H/V_a)$ for $a=1$ or $2$,
$\nu_a=H/V_a$, and $u_a=u(\nu_a)$. Since $D'=u>0$, the defining
function decreases strictly in $V_a$. At zero its sign is positive;
at $H/e$ its sign is negative for sufficiently large $H$. Thus the root
is unique. The parametric identities are
\[
 \nu_a=e^{u_a}/u_a,\qquad
 V_a=a(e^{u_a}-\operatorname{Ei}(u_a)).
\]
Using $\operatorname{Ei}(u)=e^u u^{-1}(1+u^{-1}+O(u^{-2}))$ gives
\[
 \log H=2u_a-\log u_a+\log a+O(u_a^{-1}),\qquad
 u_a=\tfrac12(\log H+\log\log H-\log(2a))+o(1),
\]
and $V_a^2/H=a(u_a-1+O(u_a^{-1}))$. Substitution gives
\eqref{eq:hard-expansion} and \eqref{eq:soft-expansion}.
\end{proof}
The second-order constants belong to this PNT main-term optimization.
They are not matching lower bounds for the probability distance.

\subsection{A finite soft-exception Stein lemma}
\begin{lemma}[Retention of exceptional indicators]\label{supp:lem:soft-retention}
Let a finite indicator family $(I_\alpha)_{\alpha\in\mathcal I}$ have an
exact dependency graph conditionally on $\mathcal F$. Split
$\mathcal I=\mathcal G\sqcup\mathcal X$ deterministically, and assume
$\EE(I_\alpha\mid\mathcal F)=p$ for every $\alpha\in\mathcal G$.
Put $W=\sum I_\alpha$, $\lambda=|\mathcal I|p$,
$\lambda_{\mathcal X}=|\mathcal X|p$ and
$W_{\mathcal X}=\sum_{\alpha\in\mathcal X}I_\alpha$.
Let $p_\beta(\omega)=\EE(I_\beta\mid\mathcal F)(\omega)$. The mean
conditional Poisson distance is at most a constant times
\begin{align}
 &(1\wedge\lambda^{-1})\left[|\mathcal G|p^2+
     p\sum_{\alpha\in\mathcal G}\sum_{\beta\in B_\alpha\setminus\{\alpha\}}
                                     \EE I_\beta
     +\sum_{\alpha\in\mathcal G}\sum_{\beta\in B_\alpha\setminus\{\alpha\}}
                                     \EE(I_\alpha I_\beta)\right]\notag\\
 &\hspace{10mm}+(1\wedge\lambda^{-1/2})
                          (\lambda_{\mathcal X}+\EE W_{\mathcal X}).
 \label{supp:eq:soft-finite}
\end{align}
The two minimum factors are interpreted as one if $\lambda=0$.
No marginal assumption is made for exceptional neighbours.
\end{lemma}
\begin{proof}
If $\lambda=0$, all good indicators vanish almost surely and
$\dTV(\Law(W\mid\mathcal F),\delta_0)\le
\EE(W_{\mathcal X}\mid\mathcal F)$; averaging proves the claim.
For $\lambda>0$ and a target test set $A\subset\NN$, let $f=f_{\lambda,A}$ solve
$\lambda f(k+1)-kf(k)=\1_A(k)-\Pois(\lambda)(A)$.
The standard factors are
\[
 \|f\|_\infty\le1\wedge\sqrt{2/(e\lambda)},\qquad
 \|\Delta f\|_\infty\le(1-e^{-\lambda})/\lambda
                         \le1\wedge\lambda^{-1}
\]
\cite[(2.15)]{Krokowski2017}; see also
\cite[(2.4)]{Rollin2012SteinFactors}.
For $\alpha\in\mathcal G$ set
$U_\alpha=\sum_{\beta\in B_\alpha\setminus\{\alpha\}}I_\beta$ and
$V_\alpha=W-I_\alpha-U_\alpha$. On each fibre $I_\alpha$ is independent
of $V_\alpha$. Add and subtract
$p\EE[f(V_\alpha+1)\mid\mathcal F]$. Telescoping gives
\begin{align*}
 &|p\EE[f(W+1)\mid\mathcal F]-\EE[I_\alpha f(W)\mid\mathcal F]|\\
 &\quad\le\|\Delta f\|_\infty
 \left[p^2+p\sum_{\beta\in B_\alpha\setminus\{\alpha\}}p_\beta
             +\sum_{\beta\in B_\alpha\setminus\{\alpha\}}
                            \EE(I_\alpha I_\beta\mid\mathcal F)\right].
\end{align*}
The remaining term is bounded directly by
\[
 |\EE[\lambda_{\mathcal X}f(W+1)-W_{\mathcal X}f(W)\mid\mathcal F]|
 \le\|f\|_\infty
                  (\lambda_{\mathcal X}+\EE[W_{\mathcal X}\mid\mathcal F]).
\]
The bound is uniform in the test set, so taking the supremum on each
fibre and then averaging proves \eqref{supp:eq:soft-finite}.
\end{proof}

\subsection{Applying the lemma at a movable cutoff}\label{supp:sec:soft-application}
Take $\mathcal X=D_Y$, $p=2^{-L}$ and $\lambda=Np>0$.
Set $\ell=\log^+\lambda$ and $K=\max(1,\lambda)$.
The elementary inequalities
\[
 (1\wedge\lambda^{-1})\lambda^2\le K,\qquad
 (1\wedge\lambda^{-1})(\lambda^2+\lambda)\le2K,\qquad
 (1\wedge\lambda^{-1/2})\lambda\le\sqrt K
\]
hold on both sides of intensity one.
For neighbour marginals use the maximum degree of the graph on all sites:
\[
 p\sum_{x\in\mathcal G}\sum_{y\in B_x\setminus\{x\}}\EE J_{y,L}
 \le p\Delta_Y\EE Z_{N,L}
 \le p\Delta_Y(\lambda+p\mathcal M_B).
\]
The first-difference factor makes this at most
$K(\Delta_Y/N)(1+N^{-1/2+o(1)})$.
The true marginals of exceptional neighbours have not been replaced by $p$.
For the joint sum, overlapping starts contribute zero and the local bound
treats touching pairs; the separated pairs give
\[
 \sum_{x\in\mathcal G}\sum_{y\in B_x\setminus\{x\}}
       \PP(J_{x,L}J_{y,L}=1)
 \ll p^2\bigl(N^{1+o(1)}+N\Delta_Y+R_2(N,L)\bigr).
\]
Since $R_2/N^2\ll N^{-1/3+\varepsilon/4}(1+\lambda^{-1})$,
its contribution, after the Stein factor, is at most
$K N^{-1/3+\varepsilon/2}$ after enlarging the threshold.
The diagonal and local polynomial costs are smaller.
Finally the exceptional zeroth-order term is at most
\[
 C\sqrt K\left(\frac{\mathcal M_B}{N}+\frac{|D_Y|}{N}\right)
 \le C e^{\ell/2-D(H/w)+o(H/w)}+K N^{-1/3+\varepsilon/2}.
\]
Together with \eqref{supp:eq:max-degree}, these estimates give
\eqref{eq:soft-free-rate} for all intensities.

At $w=\widehat V_N$, let $\nu=H/w$ and choose a common $\eta_N\ge0$,
$\eta_N\to0$, dominating the displayed remainder terms. Put
\[
 q=e^{-(w-\ell)/2+\eta_N\nu},\qquad
 r=e^{\ell-w+\eta_N\nu},\qquad P=N^{-1/3+\varepsilon/2}.
\]
Up to an absolute constant, the error is at most $q+r+KP$.
If $q+N^{-1/3+\varepsilon}\ge1$, the trivial total-variation bound suffices.
Otherwise $q<1$ implies $\ell\le w$ and $r\le q$.
Since $w=o(\log N)$, eventually $K\le e^w\le N^{\varepsilon/2}$, so
$KP\le N^{-1/3+\varepsilon}$. This proves the soft minimum bound,
including $\lambda<1$, at the same cutoff and on the full $\mathcal F_Y$.
All thresholds depend only on the fixed band and the error margin, not on
its length parameter. The numerical constants in a coarser ledger do not
affect the claimed asymptotic bound.

All terms admit their conditional versions: integrate the fibrewise
bound over $C\in\mathcal F_Y$ and divide by $\PP(C)$. A single factor
$e^I$ multiplies the ledger, where $I=-\log\PP(C)$; the leading soft
exponent consequently has sufficient budget $2I+\log^+\lambda<\widehat V$.
This argument applies only to events measurable in the chosen prime
field. It does not turn arbitrary post-selection on a run statistic into
an arithmetic conditioning event.

% END ORIGINAL SOURCE: companion/B_cutoffs.tex

% BEGIN ORIGINAL SOURCE: companion/C_fields.tex
\section{Marked fields and spatial aggregation}\label{supp:sec:fields}
The distinction between a labelled field and its aggregate is already visible
in a finite family. This appendix gives the finite inequalities first; all
passages to countably many marks are then controlled by a single tail event.

\subsection{Uniform and averaged dictionary comparisons}
The marginal-ceiling calculation in the proof of
\cref{thm:word-field} gives the finite inequality
\[
 \EE\Delta_{\mathcal W}\ll
 a\bigl(\mathcal M_B^{\rm val}+|D_Y^{\rm val}|\bigr)
 +a^2\bigl(NB+E_Y+R_2^{\mathrm{val},[1/a]}(N,B)\bigr)
 +\Lambda\Omega(\mathcal W),
\]
where $a=m2^{-B}$ and $\Lambda=Na$.
The comparison retains the full site--word index set; only the error
terms have been summed over words. We verify here the substitutions
and the uniform absorption of secondary terms.
With $T=N/\Lambda$ and $Q_B=Nm/\Lambda\ge T$, the three capped-profile
terms become, respectively,
\[
 N^{-1/3+\varepsilon}\Lambda^{11/6}m^{1/6},\qquad
 N^{-1/3+\varepsilon}\Lambda^{4/3}m^{2/3},\qquad
 N^{-1/3+\varepsilon}\Lambda.
\]
These substitutions are exact because $a=\Lambda/N$ and
$Q_B=Nm/\Lambda$. The restrictions $1\le m\le2^B$ also give
$0<a\le1$, so the arithmetic cap $T=1/a$ is always at least one.
The secondary edge contribution is absorbed in every admissible
intensity regime, not only at criticality: $m\ge\Lambda/N$ implies
\[
 \frac{\Lambda^2}{N}
 \le N^{-1/3}\Lambda^{4/3}m^{2/3}.
\]
Factors $N^{o(1)}$ in these smaller terms are absorbed by fixing a
smaller arithmetic error exponent first.
At bounded positive $\Lambda$ and $m\le N^{1/2-\eta}$, the powers of
$N$ in the three terms, with $\varepsilon=\eta/6$, are at most
\[
 -\frac14,\qquad -\frac\eta2,\qquad
 -\frac13+\frac\eta6\le-\frac\eta2\quad(0<\eta<1/2).
\]
Thus all three are $O(N^{-\eta/2})$, uniformly in the dictionary. No independent random marks
are added to the multiplicative sequence, and no word identity is lost
when the error terms are summed.

\subsubsection*{Averaging the finite dictionary comparison}
\label{supp:sec:typical-details}
We verify \eqref{eq:typical-finite} before using any asymptotic profile.
Fix $N\ge2$, $B\ge1$, $Y>2B$, a deterministic start mask
$A\subset I_N$, and a random size-$m$ dictionary, $1\le m\le2^B$,
independent of the prime signs. Only the inclusion probabilities
\eqref{eq:dictionary-selection} are needed; both ensembles in the article
have them. For each realization of the dictionary, apply the process
inequality on the finite coordinate set $A\times\mathcal W$.

\paragraph{Deletion and the exact graph.}
Let $D=D_Y^{\rm val}\cap A$. Removing real indicators has mean cost,
after averaging the dictionary and the prime environment,
\[
 \sum_{x\in D}\EE_f\PP_{\mathcal W}(\mathbf f_x\in\mathcal W)=a|D|.
\]
The Poisson target has total deleted mean $a|D|$. This uses no uniformity
of a bad window's law. At a retained site every coordinate has conditional
mean $p_B=2^{-B}$. Its neighbours include all words at its own site,
at each overlapping support and at each support sharing an odd prime
coordinate above $Y$. Families with no edge depend on disjoint remaining
prime signs, so their conditional independence is exact. The site graph
and its good set do not depend on $\mathcal W$.
The first Chen--Stein sum, including each diagonal coordinate, is bounded
by $Ca^2(|A|B+E_Y(A))$.

\paragraph{The second sum.}
At the same site its off-diagonal terms vanish by mutual exclusivity;
a coordinate's own diagonal belongs to the first sum only. For distinct
sites the sum over word labels is exactly
$\PP_f(\mathbf f_x,\mathbf f_y\in\mathcal W\mid\mathcal F_Y)$.
After both averages it is $b+(a-b)\PP_f(\mathbf f_x=\mathbf f_y)$, as in
\eqref{eq:selected-pair}. At a good local pair with displacement $d<B$,
write the union values as $z_0,\ldots,z_{B+d-1}$. The equations
$z_j=z_{j+d}$, $0\le j<B$, form the $d$ paths with residue classes
modulo $d$. There are $B+d$ vertices and $B$ edges. A prime above
$Y>2B$ can occur at only one union vertex, and each vertex has such
an odd coordinate, so the conditional vector is uniform. Its collision
probability is exactly $2^{-B}$ on every fibre.

For disjoint supports, let $v(n)$ denote the complete binary valuation
vector at the finite set of primes dividing the union. The equality
matrix has rows $v(x-1+j)+v(y-1+j)$. Its coefficient kernel maps
injectively by
\[
 t\longmapsto(t,t),\qquad
 \sum_jt_j\bigl(v(x-1+j)+v(y-1+j)\bigr)=0,
\]
into the coefficient kernel of the stacked value matrix. The equality
right-hand side is zero, so its probability is exactly
$2^{-B+\kappa}$, regardless of whether either value window is individually
uniform. This proves \eqref{eq:word-collision} with the full, not the
large-prime-truncated, nullity. It is used only after averaging the prime
fibres. Consequently a separated edge costs at most
\[
 a^2+ap_B+ap_B(2^{\rho_B^{\rm val}}-1)
 \le 2a^2+ap_B(2^{\rho_B^{\rm val}}-1).
\]
The local sum is $O(a^2|A|B)$ and the separated sum is
$O(a^2E_Y(A)+ap_BR_2^{\rm val}(A,B))$. All site pairs here are ordered;
the two orientations are included. Adding the two deletion costs proves
\eqref{eq:typical-finite} with an absolute implied constant. This finite
inequality also covers $m=1$, $B=1$ and the empty mask.

\paragraph{Normalization and the order of averaging.}
With $a=\Lambda/N$ and $p_B=\Lambda/(Nm)$, the two terms of the coarse
profile give, separately,
\[
 ap_BN^{5/3}=\frac{\Lambda^2}{m}N^{-1/3},\qquad
 ap_BN^{2/3}2^B=\Lambda N^{-1/3}.
\]
The factor $N^\varepsilon$ is inserted only after these exact identities.
For an independent-letter word field, distinct disjoint supports need no
edge, and every local collision again has probability $p_B$. Its averaged
Poisson error is therefore $O(a^2|A|B)$, without a restriction on $m$.
This is a fresh finite calculation, not an application of the uniform
dictionary theorem outside its range.

The supremum over target tests is taken for each fixed dictionary and
prime fibre \emph{before} averaging their distances. Finite sums over the
auxiliary dictionary law then give the claimed mean bound. The class of
good dictionaries supplied by Markov controls all coordinate restrictions
simultaneously by contraction. No mask may use the observed prime signs.
A dictionary-dependent deterministic restriction is harmless once that
dictionary has been selected. No coupling over different scales, or
almost-sure choice of a good dictionary at every scale, is inferred.

\subsection{Local pairs and the role of the maximal support}\label{supp:sec:local-marks}
From here on, $p=2^{-L}$, $q_e=2^{-e-1}$, $p_e=pq_e$,
$\lambda=Np$ and $0\le e\le E$. The largest row length is
$Q=L+E+1$; its support has $Q+1$ vertices. An unsigned mark of excess $e$
imposes $L+e+1$ relative equations; prescribing its sign fixes all
$L+e+2$ values on the same support.

A good maximal support contains no $Y$-defective vertex and has diameter
$Q<Y$. Every exact-mark system on that support consequently has full rank
above $Y$, with conditional marginal $p_e$. If two actual supports intersect,
their union has diameter at most $2Q<Y$; every vertex of the union then has a
prime coordinate private to that union. For two compatible exact runs the
only possible intersecting cases are a common terminal/border vertex or a
common terminal edge. Equivalently, for $x<y$, they satisfy
$y-x=L+e+1$ or $y-x=L+e$. The union of the two constraint trees has cycle
rank at most one, so its probability is at most $2p_ep_f$.

The actual supports can also be disjoint while the \emph{maximal} supports
intersect. In that case all their private coordinates are still distinct,
and the probability is $p_ep_f$. Thus the bound
\begin{equation}\label{supp:eq:local-marked}
 \PP(K_{x,e}K_{y,f}=1\mid\mathcal F_Y)\le 2p_ep_f
 \qquad(0<|x-y|\le Q)
\end{equation}
holds for every good pair; it does not assert that only two separations are
possible among pairs of intersecting maximal supports. At the same site,
different exact marks are mutually exclusive.

For $|x-y|>Q$, zero-extension of the coefficient vectors embeds the mixed
relation space into the two-window space at row length $Q$. Therefore
\begin{equation}\label{supp:eq:mixed-domination}
 \PP(K_{x,e}K_{y,f}=1)\le p_ep_f\,2^{\rho_Q(x,y)}.
\end{equation}
This is an averaged, not necessarily fibrewise, estimate: the small-prime
phase in the affine formula can vary with the environment.


For signed marks, put $p_{e,s}=p_e/2$ and let $a=L+e$, $b=L+f$.
On a good local union the raw value vector is uniform. For $x<y$,
$d=y-x$, the exact joint probabilities are
\[
 \PP(K^s_{x,e}K^t_{y,f}=1\mid\mathcal F_Y)=
 \begin{cases}
  0,&d<a,\\
  4p_{e,s}p_{f,t}\1_{\{t=-s\}},&d=a,\\
  2p_{e,s}p_{f,t}\1_{\{t=s\}},&d=a+1,\\
  p_{e,s}p_{f,t},&a+1<d\le Q.
 \end{cases}
\]
The final case concerns disjoint actual supports inside intersecting
maximal supports. The factor four, rather than two, is needed when signs
are prescribed. After summing signs one recovers
\eqref{supp:eq:local-marked}. For separated signed systems, zero-extension
instead gives the bound
$p_{e,s}p_{f,t}2^{\rho^{\rm val}_{Q+1}(x,y)}$.
\Cref{prop:absolute-profile} controls its weighted sum.

\subsection{The directional multivariate Stein factor}\label{supp:sec:stein-factor}
Let $\pi=\bigotimes_{i=1}^d\Pois(\lambda_i)$, with $d\ge2$,
$\lambda_i>0$, $\lambda'=\sum_i\lambda_i$ and
$\mu_i=\lambda_i/\lambda'$. For a test set $A\subseteq\NN^d$, use
Barbour's immigration--death solution, with its sign chosen so that
\[
 \mathcal A g(z)=\1_A(z)-\pi(A),\qquad
 \mathcal A g(z)=\sum_i\{\lambda_i\Delta_i g(z)
                           +z_i[g(z-\boldsymbol e_i)-g(z)]\}.
\]
At $z_i=0$ the corresponding death term is zero.
Write $H(z)=(\Delta_{ij}g(z))_{i,j}$ and
$c(\lambda')=(1+2\log^+(2\lambda'))/2$.
Barbour's Lemmas~2 and~3 \cite[p.~179]{Barbour1988SteinProcess}
give, for this same solution,
\[
 |\Delta_{ij}g(z)|\le1,\qquad
 |\alpha^T H(z)\alpha|
 \le c(\lambda')\sum_i\frac{\alpha_i^2}{\lambda_i}.
\]
The weighted bound, stated there for integer directions, extends to
rational directions by homogeneity and to real directions by continuity.
The literature input consists of the solution and these analytic bounds,
not an arithmetic or aggregated total-variation estimate.

For clarity, the constant entrywise bound also follows directly from the
solution's semigroup representation. Couple the processes started at
$z$, $z+\boldsymbol e_i$, $z+\boldsymbol e_j$ and
$z+\boldsymbol e_i+\boldsymbol e_j$ by adding two independent particles
of rate-one exponential lifetimes. The second difference vanishes once
either added particle dies; before that time the second difference of an
indicator is at most two in absolute value. Its time integral is thus
bounded by $\int_0^\infty 2e^{-2t}\,dt=1$. The argument includes $i=j$.

To obtain the weighted entrywise bound, conjugate the symmetric matrix
$H(z)$ by $D=\operatorname{diag}(\sqrt{\lambda_i})$.
The quadratic inequality gives $\|D H(z)D\|_{2\to2}\le c(\lambda')$.
Bounding each entry by this norm, and using the constant bound above,
yields
\begin{equation}\label{supp:eq:directional-factor}
 |\Delta_{ij}g(z)|\le
 \min\left\{1,\frac{1+2\log^+(2\lambda')}
                    {2\lambda'\sqrt{\mu_i\mu_j}}\right\}
 \le\min\left\{1,\frac{C(1+\log^+(2\lambda'))}
                    {\lambda'\sqrt{\mu_i\mu_j}}\right\}.
\end{equation}
The logarithmic term in multivariate Stein factors is discussed in
\cite{Rollin2012SteinFactors}; the two bounds used above are taken directly
from Barbour.
For the truncated geometric marks,
$S_E=\sum_{e=0}^Eq_e\in[1/2,1]$, $\lambda'=\lambda S_E$ and
$\mu_e=q_e/S_E$. The second expression is thus bounded by
$C(1+\log^+(2\lambda))/(\lambda\sqrt{q_eq_f})$.
The summability that removes dependence on the number of marks is
\begin{equation}\label{supp:eq:square-root-sum}
 \sum_{e\ge0}\sqrt{q_e}=1+\sqrt2.
\end{equation}

\begin{samepage}
\begin{theorem}[Finite aggregated comparison]\label{supp:thm:aggregated}
Let $X_e=\sum_{x\in I_N}K_{x,e}$, and let $D_Y(Q)$ be the bad maximal
supports. Put
\[
 s(\lambda)=1\wedge
          \frac{C(1+\log^+(2\lambda))}{\lambda}.
\]
Uniformly while $L+1$ and $Q+1$ remain in a fixed positive logarithmic
band and $2Q<Y$, one has
\begin{align}\label{supp:eq:aggregate-finite}
 &\EE\,\dTV\left(\Law((X_e)_{0\le e\le E}\mid\mathcal F_Y),
                  \bigotimes_{e=0}^E\Pois(\lambda q_e)\right)\notag\\
 &\hspace{1cm}\ll p\mathcal M_B+p|D_Y(Q)|
    +s(\lambda)p^2\bigl(NQ+E_Y(Q)+R_2(N,Q)\bigr).
\end{align}
Here $\mathcal M_B$ is the one-window defect mass at the base length,
and $E_Y(Q)$ is the ordered large-prime edge count for maximal supports.
\end{theorem}
\end{samepage}
\begin{proof}
Write $D=D_Y(Q)$ and retain the good-site counts
$G_e=\sum_{x\notin D}K_{x,e}$. Independently of the arithmetic variables,
introduce independent $Z_e\sim\Pois(\zeta_e)$ with
$\zeta_e=|D|pq_e$, and put $W=G+Z$. Removing the actual bad-site marks
and adding this Poisson filling changes the law by at most
\[
 \sum_{x\in D}\PP(J_{x,L}=1)+\sum_{e=0}^E\zeta_e
 \ll p\mathcal M_B+p|D|.
\]
This is an averaged conditional coupling bound. It has no factor $E$,
since the actual marks are disjoint refinements of $J_{x,L}$ and
$\sum_{e\le E}q_e\le1$.

Use the Stein solution for the \emph{full} target means $\lambda q_e$.
For the coordinate vector $\boldsymbol e_e$ and
$\Delta_eg(w)=g(w+\boldsymbol e_e)-g(w)$, Poisson summation gives,
conditionally on the good indicators and $\mathcal F_Y$,
\[
 \EE\bigl[Z_e\Delta_eg(W-\boldsymbol e_e)\mid G,\mathcal F_Y\bigr]
 =\zeta_e\EE\bigl[\Delta_eg(W)\mid G,\mathcal F_Y\bigr].
\]
Indeed $z\PP(Z_e=z)=\zeta_e\PP(Z_e=z-1)$ for $z\ge1$; the term with
$Z_e=0$ is zero. Thus the filling terms cancel the $\zeta_e$ part of the
birth rates in
\[
 \sum_{e=0}^E\{\lambda q_e\Delta_eg(W)
              -W_e\Delta_eg(W-\boldsymbol e_e)\}.
\]
Only the good indicators remain to be telescoped. The target parameters
are still $\lambda'=\lambda S_E$ and $\mu_e=q_e/S_E$, so
$\lambda'\sqrt{\mu_e\mu_f}=\lambda\sqrt{q_eq_f}$ exactly.
No comparison of the good-site intensity with $\lambda$ is needed,
even if every site was removed. The independent vector $Z$ stays in the
outside count during each telescoping step, so the good-site dependency
graph is unchanged. For good sites, apply the multivariate dependency-graph Stein argument
\cite{Barbour1988SteinProcess,Rollin2012SteinFactors} to the \emph{counts by mark}, not to
counts by site and mark. On the first-difference telescoping terms each
pair of types contributes its joint or product mass multiplied by
$|\Delta_{ef}g|$. The product masses are $p_ep_f$.
For local pairs use \eqref{supp:eq:local-marked}; for separated pairs use
\eqref{supp:eq:mixed-domination} after averaging. Decompose
$2^{\rho_Q}=1+(2^{\rho_Q}-1)$ and sum the second part by $R_2(N,Q)$.
The diagonal and all local pairs are bounded by $O(NQp^2)$.

Using the first bound in \eqref{supp:eq:directional-factor} sums the mark
weights to at most one. Using the second sums them to
$\sum_{e,f}\sqrt{q_eq_f}\le(1+\sqrt2)^2$. Take the better of these two estimates. This proves \eqref{supp:eq:aggregate-finite}
when $E\ge1$. For $E=0$, retain both signs during the same comparison,
so there are two positive target coordinates. At fixed excesses the
factor does not depend on the chosen signs; summing their probabilities
before the relative-pair bound again gives $R_2(N,Q)$.
The signed local factor is at most four, an absolute constant.
Thus the result also covers $E=0$ without extending the literature
statement to dimension one.
Notice that the arithmetic argument is used at $Q$, rather than merely
at $L$: after insertion of the directional weights, the unweighted
folding identity for marks is not the inequality being summed.
\end{proof}

At $Y=Y_N^\sharp$, the first two arithmetic error sources in
\eqref{supp:eq:aggregate-finite} are bounded by
\[
 \lambda e^{-V_N+o(\nu_N)}+N^{-1/2+o(1)}\lambda,
 \qquad
 \lambda(1+\log^+(2\lambda))e^{-V_N+o(\nu_N)}.
\]
For clarity, before absorbing subpolynomial factors the profile term is
\[
 p^2R_2(N,Q)
 \ll N^{-1/3+\varepsilon}
              \bigl(\lambda^2+\lambda 2^{E+2}\bigr).
\]
If $I+\log^+\lambda=O(V_N)$ and $E=O(V_N)$, multiplication by
$e^Is(\lambda)$ leaves only an $N^{o(1)}$ factor. Start with a smaller
arithmetic $\varepsilon$ to obtain the final remainder
$N^{-1/3+\varepsilon}$ at any prescribed positive $\varepsilon$.
This is the step that prevents an unrecorded exponential mark-cutoff loss.


For aggregation with signs retained, replace $q_e$ by
$q_{e,s}=q_e/2$ and $R_2(N,Q)$ by $R^{\rm val}_2(N,Q+1)$ in
\eqref{supp:eq:aggregate-finite}. The local factor is at most four and
$\sum_{e,s}\sqrt{q_{e,s}}=\sqrt2(1+\sqrt2)$, both fixed constants.
The full-value profile at $Q+1$ vertices has the same envelope as the
relative profile at $Q$ rows. It therefore leaves the same polynomial
remainder and the same one-factor budget. By contrast, the unweighted
labelled argument sums signs and marks first and continues to use
$R_2(N,L)$ at the base length.

\subsection{Removing the mark cutoff}
Assume $I+\log^+\lambda\le V_N-c\nu_N$, and choose
$0<c'<c''<c$. Set
\[
 E_N=\left\lceil
        \frac{\log^+\lambda+c''\nu_N}{\log2}\right\rceil .
\]
Then $E_N=O(V_N)=o(\log N)$, and the target tail has mass
$\mu_N=\lambda2^{-E_N-1}\le e^{-c''\nu_N}$.
The actual tail event is exactly a hit at length $L+E_N+1$.
The scalar hard-conditioning inequality at that shifted length gives
\[
 \PP(\exists x,\ e>E_N:K_{x,e}=1\mid C)
 \le \mu_N+
 O\left(e^I\mu_N
      (e^{-V_N+o(\nu_N)}+N^{-1/3+\varepsilon})\right).
\]
It is $O(\mu_N)$ under the displayed budget. Couple each tail to zero,
apply the finite theorem, and absorb
$\log(1+\log^+\lambda)=o(\nu_N)$. This gives
\[
 \dTV\left(\Law((X_e)_{e\ge0}\mid C),
                  \bigotimes_{e\ge0}\Pois(\lambda q_e)\right)
 \ll e^{-c'\nu_N}+N^{-1/3+\varepsilon}.
\]
For finite-mass sequences, tail summation is a measurable bijection,
with inverse $X_e=T_e-T_{e+1}$. The entire threshold path has exactly the
same total-variation distance.

\subsection{Keeping the spatial labels}
\label{supp:sec:information-cutoff}
For the labelled comparison no directional Stein factor is inserted.
The identities
\[
 \sum_e p_e\le p,\qquad
 \sum_{e,f}\PP(K_{x,e}K_{y,f}=1)\le\PP(J_{x,L}J_{y,L}=1)
\]
fold the separated-pair contribution at the base length. Their signed
versions hold after summing both signs. Thus the relation profile is
$R_2(N,L)$, whereas the graph and deleted set use the maximal support.

\paragraph{Free-cutoff bound and tails.}
Let $H=\log N$, $w\in[V_N,\widehat V_N]$, $Y=\lfloor e^w\rfloor$
and $\nu_w=H/w$. Suppose $C\in\mathcal F_Y$ has
$I=-\log\PP(C)\ge0$, and $\lambda=N2^{-L}\ge1$ with
$\ell=\log\lambda$ and $I+\ell=O(V_N)$. Choose
\[
 E=\left\lceil\frac{I+\ell+w}{\log2}\right\rceil,
 \qquad Q=L+E+1.
\]
Since $L=(H-\ell)/\log2$, the maximal number of values is
\[
 Q+1=\frac{H+I+w}{\log2}+O(1)=\log_2N+O(V_N).
\]
Both the base and maximal lengths remain in a fixed positive logarithmic
band, and $Y>2(Q+1)$ eventually, uniformly in these parameters.
By \cref{thm:marked-field} and the uniform free-cutoff estimates, the
finite field distance, multiplied by the information cost, is at most
\begin{equation}\label{supp:eq:information-free}
 C\left\{
 e^{I+\ell-D(H/w)+o(\nu_w)}+
 e^{I+2\ell-w+o(\nu_w)}+
 e^I\lambda(1+\lambda)N^{-1/3+\varepsilon/2+o(1)}
 \right\}.
\end{equation}
The weighted one-window term and the $N^{-1+o(1)}$ part of the edge
bound fit the polynomial term. Constants in the little-oh terms are
uniform for $w$ in the displayed interval and for the enlarged lengths.
A fixed multiplicative enlargement of this interval is also permitted by
\cref{supp:prop:cutoff-calculation}.

An excess greater than $E$ is exactly a threshold hit at $L+E+1$.
The unconditional first moment gives actual conditional tail cost at most
\[
 e^I\lambda2^{-E-1}(1+N^{-1/2+o(1)})=O(e^{-w}),
\]
and the target tail has no larger cost. This step uses conditioning only
as the bound $\PP(H\mid C)\le e^I\PP(H)$; no conditioning theorem at
a different cutoff is invoked. Coupling both tails to zero adds
$O(e^{-w})$ to \eqref{supp:eq:information-free}. The full site--sign--excess
space is countable, with finite expected total mass, so these tail
couplings exhaust its coordinates.

In the applications below, $I+\ell\le w\le\widehat V_N$. One may instead
use the common truncation $E=3\lceil\widehat V_N/\log2\rceil$, for which
$e^I\lambda2^{-E-1}\le \tfrac12 e^{-2\widehat V_N}$.
It still has $Q+1=\log_2N+O(V_N)$, so the finite comparison is unchanged
and both tails are $O(e^{-2\widehat V_N})$. This is the truncation used
in the proof of \cref{prop:information-labelled}; the sharper choice
above is not needed for that conclusion.

\paragraph{The information-dependent crossing.}
For $0\le I\le V_N$, set $F_I(w)=2D(H/w)-w-I$.
The function $D$ is continuous and strictly increasing on the larger
saddle branch. Since $H/w$ strictly decreases with $w$, $F_I$ is
continuous and strictly decreasing. Its endpoint values on
$[V_N,\widehat V_N]$ are $V_N-I$ and $-I$.
There is therefore exactly one root $w_N(I)$, including the endpoints
for $I=V_N$ and $I=0$. The first branch of
$g_{N,I}(w)=\min\{D(H/w)-I,(w-I)/2\}$ decreases strictly; the second
increases strictly. The root is their unique maximum on the interval.
It obeys
\[
 D(H/w_N(I))=\frac{w_N(I)+I}{2},\qquad
 \ell_{\rm lab,N}(I)=D(H/w_N(I))-I.
\]
If $\ell\le\ell_{\rm lab,N}(I)-c\nu_w$, the first two terms in
\eqref{supp:eq:information-free} are respectively
$e^{-c\nu_w+o(\nu_w)}$ and $e^{-2c\nu_w+o(\nu_w)}$.
The polynomial multiplier is $N^{o(1)}$ because
$I+2\ell\le w_N(I)=O(V_N)$. Also $w/\nu_w=w^2/H\asymp\log H\to\infty$,
so $e^{-w}$ is smaller than any fixed $e^{-c'\nu_w}$.
This proves \cref{prop:information-labelled}. The hypotheses are
vacuous at $I=V_N$ for $\lambda\ge1$, since then the allowed leading
logarithmic intensity is zero and the margin is strictly negative.

For $I>0$, the root satisfies $w_N(I)<\widehat V_N$. Hence
$2D(H/w_N(I))>2D(H/\widehat V_N)=\widehat V_N$, and its defining
equation gives $w_N(I)+I>\widehat V_N$.
Thus $w_N(I)>\widehat V_N-I$, giving the claimed strict improvement
of the leading budget over $\widehat V_N/2-I$ without differentiating
the implicitly defined root.
Uniformly for $w=tV_N$, $1\le t\le\widehat V_N/V_N$, the expansion
$D(\nu)=\nu u(\nu)(1+O(1/u(\nu)))$ gives
$D(H/w)/V_N=t^{-1}+o(1)$. The saddle equation consequently yields
$t^2+\theta t-2=0$ when $I/V_N\to\theta$.
The positive root is $(\sqrt{\theta^2+8}-\theta)/2$; subtracting
$\theta$ and dividing by two proves \eqref{eq:information-leading}.

\paragraph{A fixed event and a constrained cutoff.}
Suppose $C\in\mathcal F_{\lfloor e^{w_0}\rfloor}$ for a given
$w_0\in[V_N,\widehat V_N]$. The prime sigma-fields increase with the
cutoff. On the certified admissible interval $[w_0,\widehat V_N]$,
$g_{N,I}$ increases up to $w_N(I)$ and decreases afterwards; hence its
maximum is at $\max\{w_0,w_N(I)\}$. For any admissible $w$ the two
inequalities
\[
 I+\ell\le D(H/w)-cH/w,\qquad
 I+2\ell\le w-2cH/w
\]
are sufficient. In particular no event measurable only above the proposed
cutoff is retained by an algebraic optimization. All cutoffs here are
deterministic functions of the event and its probability, not of the
realized prime environment. This is an optimization of the proved
labelled bounds, not an optimality assertion about the law.

\subsection{Macroscopic transport and containment}\label{supp:sec:macro-marked}
Let $L_M=\lfloor\log_2M\rfloor-a_M$, $a_M=o(\log M)$, and
$\lambda=M2^{-L_M}$. The base-contained interior starts are
$2\le x\le M-L_M+1$; their number is $M-L_M$.
First retain $\lceil M^{1/2}\rceil\le x\le M-L_M+1$.
The conditional ranks and graph construction are local and unchanged.
The one-window defect masses, summed on dyadic slices, are
$M^{1/2+o(1)}$; the edge estimate uses the upper height $M$; and the
separated-pair mass is bounded by the macroscopic profile
\eqref{eq:master-envelope}. Thus the labelled finite comparison is
\[
 \ll\lambda(1+\lambda)
             \bigl(e^{-V_M+o(\nu_M)}+M^{-1/3+\varepsilon}\bigr).
\]
Restoring the deep starts costs the bound \eqref{eq:deep-starts},
\[
 O\left(\lambda M^{-1/2}
             +\exp(-c_0\log M/\log\log M)\right).
\]
The target deletion costs only $O(\lambda M^{-1/2})$.
Conditioning on a compatible event in the chosen prime field multiplies
the averaged error by at most $e^I$.

Choose $E=\lceil(I+V_M+\log(2+\lambda))/\log2\rceil$ in the hard
labelled regime. The exact mark-tail identity and the global scalar
bound control the real tail. Besides the intensity-linear tail term,
the scalar bound has the separate deep-start error just displayed;
that error is already present in the full-field inequality. In
particular it must not be silently dropped from a finite-prefix tail
calculation. Both tails are absorbed by
\eqref{eq:macro-field-rate}. The free-cutoff proof is identical with
$V_M$ replaced by the optimized cutoff in its two leading terms.

For aggregated counts apply \cref{supp:thm:aggregated} on the retained
macroscopic set, with $R_{2,\delta}(M,Q)$ in place of $R_2(N,Q)$.
The same geometric square-root sum applies. Under
$I+\log^+\lambda\le V_M-c\nu_M$, the deep-start correction times
$e^I$ is smaller than every fixed $e^{-c'\nu_M}$, and the aggregate
mark cutoff above removes both tails. This proves the macroscopic
one-factor staircase without an additional marked arithmetic input.

At exact centered level $r$ the target mean is precisely
$(M-L_M)2^{-b_M-r-1}$, not $M2^{-b_M-r-1}$.
All these fields retain the full rightward mark of a base-contained
start. They are not the list of blocks censored at the right end of the
finite sequence. The containment diagram in the article distinguishes
the two objects.

\subsection{Deterministic masks and diffuse limits}
A deterministic restriction of a lattice field contracts total
variation. At bounded intensity, if
$N^{-1}\sum_{x\in A_N}\delta_{x/N}\Rightarrow\mu$, the restricted marked
field therefore converges weakly to
$\operatorname{PPP}(\lambda\mu\otimes\nu_{\rm geo})$, along a subsequence
$N2^{-L}\to\lambda$. The same statement holds for the macroscopic field,
with scaling by $M$. It follows directly from the target Laplace
functional and weak convergence of the deterministic empirical measures.
There is no universal numerical rate for this last deterministic step
without a quantitative assumption on the mask or the test function.

Nor is the diffuse limit a total-variation target. The actual field is
supported on its lattice, while the diffuse target has positive
probability $1-e^{-\lambda\mu(\mathbb R)}$ of a nonempty configuration
outside that lattice. The approximation in total variation always takes
place \emph{before} passing from the grid to the continuum.

% END ORIGINAL SOURCE: companion/C_fields.tex

% BEGIN ORIGINAL SOURCE: companion/D_conditioning.tex
\section{Conditional laws and the Poisson target}\label{supp:sec:conditional-details}
The arithmetic theorems provide total-variation errors. This appendix
records exactly when those errors are small enough to transfer a local
probability, a conditional state, or a Gaussian limit. Properties of the
independent Poisson target are not additional arithmetic results.

\subsection{Scalar Gaussian and local estimates}
Let $P_\lambda\sim\Pois(\lambda)$ and let $\epsilon_N$ denote any proved
conditional scalar total-variation bound in the article. If $\lambda\to\infty$,
\[
 \sup_t\left|\PP\left(\frac{Z-\lambda}{\sqrt\lambda}\le t\mid C\right)
                       -\Phi(t)\right|
 \le\epsilon_N+C\lambda^{-1/2}.
\]
For example, write $P_\lambda$ as a sum of $\lceil\lambda\rceil$
independent Poisson variables of mean at most one and use the classical
Berry bound \cite{Berry1941}. Their third absolute centered moments
are $O(\lambda/\lceil\lambda\rceil)$, which gives the displayed scale.
The normalization uses the deterministic comparison mean $\lambda$;
replacing it by an actual conditional mean requires a separate moment
estimate and is not inferred from total variation alone.

For every integer $n\ge1$ and every $\lambda>0$, one has the exact form
\begin{equation}\label{supp:eq:poisson-atom}
 \PP(P_\lambda=n)=\frac{e^{-\delta_n}}{\sqrt{2\pi n}}
                    \exp[-\lambda h(n/\lambda)],\qquad
 0\le\delta_n\le\frac1{12n}.
\end{equation}
Here $h(u)=u\log u-u+1$, and $\delta_n$ is independent of $\lambda$. To check the remainder, set
\[
 \delta_n=\log(n!)-(n+\tfrac12)\log n+n-\tfrac12\log(2\pi).
\]
Stirling's limit gives $\delta_n\to0$. With $t=(2n+1)^{-1}$,
\[
 \begin{aligned}
 \delta_n-\delta_{n+1}
 &=(n+\tfrac12)\log(1+1/n)-1
   =\sum_{j\ge1}\frac{t^{2j}}{2j+1},\\
 0\le\delta_n-\delta_{n+1}
 &\le\frac{t^2}{3(1-t^2)}=\frac1{12n(n+1)}.
 \end{aligned}
\]
Telescoping to infinity proves the stated bounds. Substitution of the
factorial identity into $e^{-\lambda}\lambda^n/n!$ proves
\eqref{supp:eq:poisson-atom}; moreover
$0\le1-e^{-\delta_n}\le1/(12n)$, uniformly in the intensity.
For $n=0$ the mass is exactly $e^{-\lambda}$ and the square-root formula
is not used.

Thus relative transfer at $n$ requires
$\epsilon_N/\PP(P_\lambda=n)\to0$. For the hard scalar rate a sufficient
condition is
\[
 I+\log^+\lambda+\tfrac12\log n+\lambda h(n/\lambda)
                  \le V_N-c\nu_N,
\]
and for the soft rate a sufficient condition is
\[
 2I+\log^+\lambda+\log n+2\lambda h(n/\lambda)
                  \le\widehat V_N-c\widehat\nu_N.
\]
In both cases the event must be measurable at the corresponding prime
cutoff. The logarithms of the reciprocal target masses are then
$O(V_N)$ or $O(\widehat V_N)$, so division by that mass does not destroy
the polynomial remainder.

In the central range
$n=\lambda+t\sqrt\lambda+O(1)$, $|t|=o(\lambda^{1/6})$,
Taylor expansion gives
$\lambda h(n/\lambda)=t^2/2+o(1)$ and $\log n=\log\lambda+o(1)$.
The respective central local budgets are therefore
\[
 I+\tfrac32\log\lambda+\tfrac12t^2\le V_N-c\nu_N,
 \qquad
 2I+2\log\lambda+t^2\le\widehat V_N-c\widehat\nu_N.
\]
A global scalar approximation at its upper intensity endpoint does not
by itself imply a relative local approximation there.

For completeness, the analogous relative upper-tail transfer has the
additional denominator $\PP(P_\lambda\ge\lceil\lambda+t\sqrt\lambda\rceil)$.
Stirling's formula and summation of successive Poisson masses yield the
usual Gaussian moderate-tail comparison for
$0\le t=o(\lambda^{1/6})$. Equivalently, expanding the exponent uniformly
on the contributing $O(\sqrt\lambda/(1+t))$ scale gives relative error
$O((1+t^3)/\sqrt\lambda)$; the exponentially smaller outer part of the
sum is bounded by the ratio of successive masses. Since
$1-\Phi(t)\asymp e^{-t^2/2}/(1+t)$ for $t\ge0$, a sufficient hard
arithmetic condition is
\[
 I+\log^+\lambda+\tfrac12t^2+\log(1+t)\le V_N-c\nu_N.
\]
This condition must be combined with the intrinsic Poisson restriction
$t=o(\lambda^{1/6})$. It is not a claim about larger deviation scales.

\subsection{Resolving an entire conditional path}
Let $\epsilon^{\rm path}$ be the aggregate staircase error based at the
present threshold. The sharpened conditioning inequality
\eqref{eq:sharp-conditioning} gives a full future-path error at most
$\epsilon^{\rm path}/p_\lambda(n)$, where
$p_\lambda(n)=\PP(P_\lambda=n)>\epsilon^{\rm path}$.
The same assertion holds jointly with any fixed available lower segment.
Forward transitions are binomial thinning; backward transitions add
independent Poisson increments.

Before division, the aggregate proof gives
\[
 \epsilon^{\rm path}\ll
 e^I\lambda(1+\log^+(2\lambda))e^{-V_N+o(\nu_N)}
                      +N^{-1/3+\varepsilon},
\]
provided the information, log-intensity and mark cutoff are $O(V_N)$.
Combining this with \eqref{supp:eq:poisson-atom} proves the budget
\eqref{eq:resolved-budget}. Its hypotheses also validate that use of the
common logarithmic band. Indeed, for $n\ge2$, if $\lambda\ge1$ the budget
bounds $\log\lambda$ by $O(V_N)$. If $\lambda<1$, use
\[
 \lambda h(n/\lambda)=n\log n-n+\lambda-n\log\lambda
\]
to conclude $|\log\lambda|=O(V_N)$. Thus the base offset, information
cost and chosen mark cutoff are all $O(V_N)=o(\log N)$.
The atom's reciprocal is $\exp(O(V_N))=N^{o(1)}$, which can be absorbed
by first choosing a smaller arithmetic $\varepsilon$.
For a fixed present state at a fixed positive limiting intensity the
atom stays bounded away from zero, and no extra logarithmic cost is needed.

The condition $p_\lambda(n)>\epsilon^{\rm path}$ only makes the
conditional law well-defined and gives a bound. Convergence requires
$\epsilon^{\rm path}/p_\lambda(n)\to0$. Nothing here supplies a single
finite-intensity comparison for the entire reverse path: its target
mass diverges toward thresholds $m\to-\infty$.

\subsection{Quenched transfer along geometric scales}
The proof of \cref{cor:quenched} gives the single-length
Markov--Borel--Cantelli argument. Its uniform extension uses the bound
\[
 e^{-(c-\beta)\nu_{N_k}+o(\nu_{N_k})}
       +e^{\beta\nu_{N_k}}N_k^{-1/3+\varepsilon},\qquad 0<\beta<c,
\]
for the exceptional probability on $N_k\asymp q^k$, $q>1$.
Since $\nu_{N_k}\asymp\sqrt{k/\log k}$, multiplication by
$O(\log N_k)=O(k)$ preserves summability. A single union bound therefore
covers every integer length in a fixed logarithmic band, provided the
error margin is uniform. The resulting almost-sure statement concerns
these conditional laws at individual scales; it requires no independence
between scales and asserts no canonical coupling over all scales.

\subsection{Exact target consequences}\label{supp:sec:target-consequences}
For a phase $\vartheta\in[0,1]$, let
$P_r\sim\Pois(2^{\vartheta-r-1})$ independently for $r\in\Z$, and
$S_m=\sum_{r\ge m}P_r$. Each tail sum is finite almost surely. Then
\[
 S_m\sim\Pois(2^{\vartheta-m}),\qquad
 \operatorname{Cov}(S_m,S_n)=2^{\vartheta-\max(m,n)}.
\]
Splitting $S_m=P_m+S_{m+1}$ gives the exact forward thinning and
backward immigration kernels in the article. For $s\ge0$ the joint
probability generating function is
\[
 \EE\prod_{j=0}^s z_j^{S_{m+j}}
 =\exp\left[
 \sum_{r=m}^{m+s-1}2^{\vartheta-r-1}
       \left(\prod_{j=0}^{r-m}z_j-1\right)
 +2^{\vartheta-m-s}\left(\prod_{j=0}^sz_j-1\right)\right].
\]
Conditionally on $S_m=n$, the $n$ points have independent uniform
positions and independent geometric overshoots
$\PP(E=e)=2^{-e-1}$. For lattice targets the positions are uniform on
the corresponding finite grid. Conditioning the critical prelimit field
on a fixed count transfers this assertion by the sharp conditioning
lemma; only the final grid-to-uniform passage is weak convergence.

The complete centered target is locally finite on $[1,2]\times\Z$,
although it has infinite total mass toward $r=-\infty$. For every
nonnegative continuous compactly supported $g$ put
\[
 \Psi_\vartheta(g)=
 \exp\left[-\int_1^2\sum_{r\in\Z}2^{\vartheta-r-1}
                                (1-e^{-g(t,r)})\,dt\right].
\]
Compact support meets finitely many levels. At those levels the lattice
comparison and a Riemann sum give
\[
 \EE(e^{-\langle g,\mathcal E_N\rangle}\mid C_N)
                         =\Psi_{\vartheta_N}(g)+o_g(1)
\]
whenever $I_N\le V_N-c\nu_N$. A convergent phase subsequence is needed
only to replace this moving deterministic target by a single limit law.

For a base intensity $\lambda_N\to\infty$, the exact maximum identity
in the article, applied at
$m_N=\lfloor\log_2\lambda_N\rfloor+j$, gives, along
$\{\log_2\lambda_N\}\to\vartheta$,
\[
 \PP(M_{N,L}-\lfloor\log_2\lambda_N\rfloor\le j)
                         \longrightarrow e^{-2^{\vartheta-j-1}}.
\]
This is a lattice extreme law. The identity uses the scalar comparison
at the shifted critical length, not an unproved uniform law for every
large marked field.

Finally, the Poisson--Gaussian bridge follows by splitting the independent
levels at a fixed critical level $r_0$. Below it, the normalized Poisson
sums satisfy a multivariate central limit theorem with covariance
$2^{-\max(j,k)}$. The centered contribution of the levels at or above
$r_0$ has variance $O(\Lambda_N^{-1})$ after normalization, whereas
those levels retain their Poisson law and are independent of the lower
part. This proves joint independence in the limit. No continuous-time
embedding or phase-removing auxiliary randomization is used in this
arithmetic argument.

% END ORIGINAL SOURCE: companion/D_conditioning.tex

% BEGIN ORIGINAL SOURCE: companion/E_boundary.tex
\section{The microscopic boundary and relative factorization}\label{supp:sec:boundary}
The exact border cylinder and microscopic non-vacancy are different
objects. The first has probability $2^{-\pi(L)}$ without an error term;
the second also allows an interior start. This appendix proves that the
entire interior first moment is exponentially smaller than the border
probability. A union bound is then sufficient; no microscopic overlap
expansion is needed.

\subsection{Valuation matrices and the transition range}
Put $B=L+1$, $V_x=\{x-1,\ldots,x+L-1\}$, and let $M_x$ have rows
$v(n)=(v_p(n)\bmod2)_p$ for $n\in V_x$.
The start matrix is $A_x=D_xM_x$, where $D_x$ is the edge-difference
matrix of a tree on $V_x$. Its kernel consists exactly of the constant
vectors. Consequently
\begin{equation}\label{supp:eq:boundary-rank}
 \rank A_x\ge\rank M_x-1,\qquad
 \PP(J_{x,L}=1)\le2^{-\rank A_x}.
\end{equation}
For $2\le x\le L-1$ a start is impossible: both $2x$ and $2x-2$
lie inside the putative constant run, and complete multiplicativity
would force $f(2)=1$ and $f(2)=-1$ simultaneously.

\begin{lemma}[Transition identity minor]\label{supp:lem:transition}
Uniformly for $x\in\{B-1,B,B+1\}$,
$\rank A_x\ge(2-o(1))\pi(B)$.
\end{lemma}
\begin{proof}
Write $a=x-1\in\{B-2,B-1,B\}$ and $Y_0=2\sqrt B$.
For each prime $Y_0<p\le B$ choose
$m_p=p\lceil a/p\rceil\in[a,a+B-1]$.
The cofactor satisfies
$\lceil a/p\rceil\le B/p+1<\sqrt B/2+1<Y_0<p$ for large $B$.
Thus $m_p$ has valuation one at $p$ and no other prime factor above
$Y_0$. Each prime $B<q\le a+B-1$ occurs as its own vertex.
On these rows and columns the valuation matrix has an identity minor
of size $\pi(a+B-1)-\pi(Y_0)=(2-o(1))\pi(B)$.
Apply \eqref{supp:eq:boundary-rank}.
\end{proof}

\subsection{The uniform prime-divisor input}
The precise literature input is
\begin{equation}\label{supp:eq:LS-uniform}
 \forall\varepsilon>0\ \exists B_0(\varepsilon)\ \forall B\ge B_0(\varepsilon)
 \ \forall n>B:\qquad
 \omega\left(\prod_{j=0}^{B-1}(n+j)\right)\ge(2-\varepsilon)\pi(B).
\end{equation}
It is equation~(14) of Laishram--Shorey \cite{LaishramShorey2004}.
Its uniformity is important. For $n>17B/12$, their Theorem~2 gives
at least $\pi(2B)$ prime divisors, apart from a fixed exceptional pair.
For $B<n\le17B/12$, every prime at most $B$ divides the block product,
and each prime in $[n,n+B-1]$ is an additional divisor. Hence the number
is at least
\[
 \pi(B)+\pi(n+B-1)-\pi(n-1)=(2-o(1))\pi(B),
\]
uniformly in that bounded ratio, by the prime number theorem.
This verifies the quantifier order in \eqref{supp:eq:LS-uniform}.

Suppose now that $B+2\le x\le B^2+2$. Every vertex is at most
$B^2+B$. Equation~\eqref{supp:eq:LS-uniform} supplies
$t_x\ge(1-o(1))\pi(B)$ primes $q>B$ in the block.
Each has valuation one, since $q^2\ge(B+1)^2>B^2+B$; it occurs at
one vertex only, since $q>B-1$; and two such primes cannot occupy the
same vertex, since their product exceeds $B^2+B$.
Their columns span a coordinate subspace $U_x$ of dimension $t_x$.
Call its rows the private rows.

\subsection{A global count of even valuations}
Fix an integer $K\ge2$ and use the medium primes $B/K<p\le B$.
Let $E_x$ be their total odd-valuation incidence mass in $V_x$.
We claim
\begin{equation}\label{supp:eq:medium-mass}
 E_x\ge(H_K-1-o(1))\pi(B),\qquad H_K=\sum_{j=1}^K\frac1j.
\end{equation}
The block contains at least $\lfloor B/p\rfloor$ multiples of $p$.
An incidence of positive even valuation is among those with
$n=ap^2\le B^2+B$. Thus $a\le K^2+1$ for sufficiently large $B$.
For fixed $a$, successive distinct primes $p,q>B/K$ give values
$ap^2,aq^2$ separated by more than $2aB/K$; only
$1+K/(2a)$ such values can lie in an interval of length $B$.
Summing over the fixed number of possible $a$ shows that the total
number of square divisibilities is $O_K(1)$, uniformly in $x$.
Subtracting all of them is a permissible overcount of even valuations.

The main incidence count can be evaluated without rounding every summand:
\begin{align*}
 \sum_{B/K<p\le B}\left\lfloor\frac Bp\right\rfloor
 &=\sum_{j=1}^{K-1}\bigl(\pi(B/j)-\pi(B/K)\bigr)\\
 &=(H_{K-1}-(K-1)/K+o(1))\pi(B)\\
 &=(H_K-1+o(1))\pi(B).
\end{align*}
This proves \eqref{supp:eq:medium-mass}. The improvement over a
prime-by-prime loss is essential: subtracting one for each prime would
lose a positive proportion of $\pi(B)$.

\subsection{Rank of a simple incidence matrix}
\begin{lemma}[Simple incidence rank]\label{supp:lem:simple-incidence}
Let $H$ be a finite binary matrix. Suppose every row has weight at most
two, every column has weight at most $K$, and no pair of columns occurs
as the support of two weight-two rows. If $E$ is the number of entries
equal to one, then
\[
 \rank H\ge\frac{E}{K+1}.
\]
\end{lemma}
\begin{proof}
Remove zero columns and view a weight-two row as an edge between its
columns, and a weight-one row as a pin. In a connected component with
$r$ active columns, an unpinned component has rank $r-1$ and a pinned
component has rank $r$. The first assertion follows because the edge
vectors of a connected graph span the even-parity subspace; one pin
then spans its missing direction.

A pinned component has incidence mass at most $Kr$, which is at most
$(K+1)$ times its rank. An unpinned component has $r\ge2$; simplicity
and the degree bound give incidence mass at most
$\min\{r(r-1),Kr\}$. If $r\le K+1$ use the first bound, and if
$r\ge K+1$ use the second. In either case the mass is at most
$(K+1)(r-1)$. Sum over the components.
\end{proof}

Delete the private rows from the medium-prime columns of $M_x$, and
call the resulting matrix $H_x$. For fixed $K$ and sufficiently large
$B$, a private row carries at most one medium prime: otherwise a vertex
would exceed $B(B/K)^2>B^2+B$. Thus deletion removes at most $t_x$
incidences. A remaining row carries at most two medium primes, since
$(B/K)^3>B^2+B$. A column has weight at most $K$.
Finally two weight-two rows with the same support $p,q$ would be two
multiples of $pq>(B/K)^2>B$ inside one block, which is impossible.
All hypotheses of \cref{supp:lem:simple-incidence} are therefore met.
Modulo $U_x$, a medium column is represented by its restriction to the
remaining rows, so
\begin{align}\label{supp:eq:surplus-K}
 \rank A_x
 &\ge t_x+\frac{(E_x-t_x)_+}{K+1}-1\notag\\
 &\ge \left(1+\frac{H_K-2}{K+1}-o(1)\right)\pi(B).
\end{align}
For the last step, if $E_x\le t_x$ use $t_x\ge E_x$; otherwise use
$t_x\ge(1-o(1))\pi(B)$ and \eqref{supp:eq:medium-mass}.
The parameter can be chosen exactly within this family. Set
$\delta_K=(H_K-2)/(K+1)$. Then
\[
 \delta_{K+1}-\delta_K
   =\frac{3-H_K}{(K+1)(K+2)}.
\]
Since $H_{10}<3<H_{11}$ and $H_K$ is strictly increasing, the unique
maximizer over integers $K\ge2$ is $K=11$. Consequently we use
\[
 H_{11}=\frac{83711}{27720},\qquad
 \delta_*:=\frac{H_{11}-2}{12}
          =\frac{28271}{332640}>\frac1{12}.
\]
This optimizes only the displayed fixed-cutoff family of bounds,
not the rank surplus among all possible methods.

The local product comparisons hold for $B\ge11^3+1=1332$:
$(B/11)^3>B^2+B$, $B^3/121>B^2+B$ and $(B/11)^2>B$.
In the square-divisibility count one may then use $a\le122$ and at most
seven values per $a$, giving the safe bound $854$. These elementary
checks do \emph{not} make $1332$ a threshold for the complete asymptotic
theorem: the prime-divisor and prime number theorem thresholds still
depend on the desired error margin.

\subsection{Post-quadratic starts and summation}
\begin{lemma}[Post-quadratic defect maximum]\label{supp:lem:pointwise-maximum}
There is an absolute constant $\theta_0$ such that, for all sufficiently
large $B$ and every $x>B^2+2$, the number $m_x$ of $B$-defective
vertices in $V_x$ is at most $B-g_L$, where
\[
 g_L=\frac B{\log B}
       (\log\log B-\log\log\log B-\theta_0),
 \qquad g_L/\pi(B)\longrightarrow\infty.
\]
In particular $\PP(J_{x,L}=1)\le2^{-g_L}$.
\end{lemma}
\begin{proof}
We use the Balasubramanian--Shorey exclusion
\cite{BalasubramanianShorey1993} in the form recorded by Shorey
\cite[pp.~55--56, equation (15) with exponent $2$ and the subsequent
square-case statement]{Shorey1994BakerApplications}.
It has an absolute constant $\theta_0$ and, for every fixed $\varepsilon>0$,
a threshold $k_0(\varepsilon)$ such that, for $k\ge k_0(\varepsilon)$,
\[
 \prod_{i=1}^t(z+d_i')=by^2,\quad
 0\le d_1'<\cdots<d_t'<k,\quad b>0,\quad P^+(b)\le k,
 \quad z>e^{1-\theta_0+\varepsilon}\frac{k\log k}{\log\log k}
\]
forces $t<\mu_k(\theta_0)$, where
\[
 \mu_k(\theta)=k\left(1-\frac{\log\log k}{\log k}
       +\frac{\log\log\log k}{\log k}+\frac\theta{\log k}\right).
\]
The smoothness condition on $b$ is part of (15); it is not a size
bound on $b$. All thresholds are uniform in $z,b,y$ and the shifts.
To obtain the form used here, first fix $\varepsilon=1$. Increase the
absolute threshold until
\[
 k^2>e^{2-\theta_0}\frac{k\log k}{\log\log k}.
\]
For $m>k^2$ and $1\le d_1<\cdots<d_t\le k$, set
$z=m+1$ and $d_i'=d_i-1$. These substitutions preserve every factor
$m+d_i=z+d_i'$, put all shifts in $[0,k)$, and satisfy the required
height bound. Hence the product equation forces $t<\mu_k(\theta_0)$.
Write $V_x=\{m+1,\ldots,m+B\}$ with $m=x-2>B^2$.
The product of its $B$-defective vertices is a square times a
$B$-smooth integer. There is no extra upper bound on that smooth
coefficient in the imported statement. Its contrapositive gives
$m_x<\mu_B(\theta_0)=B-g_L$ beyond the fixed absolute threshold.
The affine defect bound is
$\PP(J_{x,L}=1)\le2^{-L}2^{\max(m_x-1,0)}$, which is at most
$2^{-g_L}$ for large $B$.
\end{proof}

Partition the starts $2\le x\le2L^2$ into the forbidden early range,
the three transition starts, $B+2\le x\le B^2+2$, and the remaining
post-quadratic starts. Their total probabilities are respectively
\[
 0,\qquad 3\,2^{-(2-o(1))\pi(B)},\qquad
 O(B^2)2^{-(1+\delta_*-o(1))\pi(B)},\qquad
 O(L^2)2^{-g_L}.
\]
Since $\log B=o(\pi(B))$ and $\pi(B)=\pi(L)+O(1)$, their sum is
at most $2^{-(1+\delta_*-o(1))\pi(L)}$.
Adding the exact border event proves
\[
 q_L=2^{-\pi(L)}\bigl(1+O_{\delta_0}(2^{-\delta_0\pi(L)})\bigr)
 \quad(0<\delta_0<28271/332640),
\]
including the displayed choice $\delta_0=1/12$.
The same interior sum divided by $q_L$ bounds the conditional
probability that the exact border is not the unique microscopic start.

The cutoff-invariance and mesoscopic estimates use one further
observation. At heights above $2L^2$, two $B$-defective vertices at
distance at most $L$ have distinct squarefree $B$-smooth parts; equal
parts would place consecutive squares too far apart. The number of
choices of these parts is $\exp(O(L/\log L))$, and the uniform Pell
bound gives the same type of bound on the number of corresponding windows in the entire admissible population
up to twice the ambient height. Multiplication by $2^{-g_L}$ remains
$o(2^{-\pi(L)})$, without an exterior height-slice factor. Starts with at most one defect
cost only their number times $2^{-L}$. This proves the bounds in the
article with cutoff $K\le2^L$, using $2^L$ as an artificial upper
ambient scale; it does not require a dyadic lower endpoint comparable
to that artificial scale.

\subsection{The prime clock}
Section~\ref{sec:prime-clock} of the article proves the exact prime-clock
law and explains why integer overshoots need not be tight. We record here
the conventions needed for the conditional comparisons. Let
$J_\partial=\min\{j:f(p_j)=-1\}$. On $C_L=1$ and the full-measure
event $J_\partial<\infty$, put $G_L=J_\partial-\pi(L)-1$;
assign $G_L=0$ elsewhere. Then
\[
 \PP(G_L=k\mid C_L=1)=2^{-k-1},\qquad
 \PP(\ell_\partial-L\ge h\mid C_L=1)
             =2^{-[\pi(L+h)-\pi(L)]}.
\]
Since $\{C_L=1\}\subseteq\{B_L^{\rm mic}>0\}$, the conditional
law of $G_L$ given microscopic non-vacancy is a mixture whose
$C_L=1$ component is exactly geometric. Its distance from that law is
at most $\PP(C_L=0\mid B_L^{\rm mic}>0)=O(2^{-\delta_0\pi(L)})$.
The next subsection constructs the finite-cylinder representative used
to carry this mark through the stable product comparison.

\subsection{Relative bulk estimates and affine conditioning}
\label{supp:sec:boundary-conditioning}
Let $\lambda=M2^{-L}\to0$, $L=O(\log M)$ and
$U=[\lceil M^\delta\rceil,M-L+1]\cap\Z$. Fix $\delta$ and an
upper bound for $L/\log M$. With $Y=Y_M^\sharp$, truncate the signed
exact marks at $E$ and put $Q=L+E+1$. The averaged conditional
process error is at most
\[
 C\left[p\mathcal M_B+p|D_Y(Q)|
       +p^2\bigl(MQ+E_Y(Q)+R_{2,\delta}(M,L)\bigr)\right].
\]
The local ranks are those in Appendix C. Summing signs and exact marks
before the separated-pair estimate leaves the relation sum at the base
length $L$; only the bad supports and shared-prime graph use $Q$.
After division by $Mp=\lambda$, the contributions are
\begin{center}\small
\begin{tabularx}{\textwidth}{@{}lX@{}}\toprule
Source & Normalized contribution\\\midrule
One-window defects & $M^{-1/2+o(1)}$\\
Bad maximal supports & $e^{-V_M+o(\nu_M)}$\\
Diagonal and local pairs & $pQ$\\
Shared-prime edges & $\lambda(e^{-V_M+o(\nu_M)}+M^{-1+o(1)})$\\
Base-length relations & $M^{-1/3+\varepsilon}(1+\lambda)$\\
Actual and target mark tails & $O(2^{-E-1})(1+M^{-1/2+o(1)})$\\\bottomrule
\end{tabularx}
\end{center}
For the last row, the exact tail event is a start at length $L+E+1$;
the masked first-moment estimate at that length gives the displayed
bound after averaging the conditional tail probability. Choose
$E=\lceil V_M/\log2\rceil$. Then $Q=O(\log M)$,
$pQ=O(\log M/M)$ for large $M$, and every row tends to zero.
This proves the signed, marked relative error in
\eqref{eq:relative-bulk-rate}, not merely its unmarked projection.

The microscopic vector is measurable at the same prime cutoff because
$2L^2+L<Y$ for large $M$. For a fixed $K$, define $G_{L,K}$ from the
first $K$ prime signs after $L$: count consecutive positive signs, capped
at $K$, and put the value zero off $C_L=1$.
This finite-cylinder variable agrees almost surely with $G_L\wedge K$;
this makes measurability on the finite prime sigma-field explicit,
independently of the convention on the null nonterminating event. Bertrand's postulate bounds the $K$th
prime after $L$ by $2^K L<Y$ eventually, for fixed $K$.
The stable lift retains $(\mathbf B_L,G_{L,K})$ jointly with the full
signed marked bulk field, at cost $\Delta_M=o(\lambda)$.

Write $\alpha=2^{-\pi(L)}$, $b=|U|p$, and let $u_M$ and $v_M$ denote
the interior microscopic first moment and the mesoscopic hit probability.
The source bounds give $u_M+v_M=o(\alpha+b)$.
The one-hit calculation in the proof of \cref{thm:two-clock}
gives sparse-target mass $e^{-b}[\alpha+(1-\alpha)b]$ and a change
of mixture weights at most $b$ relative to \eqref{eq:two-clock-target}.
The conditional bulk mark, sign and position retain their full laws;
only the boundary clock is truncated.
Simultaneous sources and multiple bulk points have probabilities at most
$\alpha b$ and $b^2/2$. Since the sparse-event mass is asymptotic to
$\alpha+b$, normalizing the source subprobability measures gives
\[
 \dTV\bigl(\Law(\Gamma_{M,L}^{[K]}\mid R_M\ge L),Q_{M,L}^{[K]}\bigr)
 \ll\frac{\Delta_M+u_M+v_M+\alpha b+b^2}{\alpha+b}+b.
\]
The additional $b$ is already absorbed by the last two numerator terms.
For the actual first-hit law the discarded boundary tail is at most
$2^{-K-1}$: its unnormalized mass is at most $\alpha2^{-K-1}$ and the
hit probability is at least $\alpha$. The target tail is also at most
$2^{-K-1}$. First let $M\to\infty$, then let $K\to\infty$.
No lower bound on either mixture weight, convergence of their ratio, or
exact finite-size independence is required. The grid-to-Lebesgue passage
is a separate weak limit.

Now let $A=\{GX=b\}$ be a compatible affine cylinder on the exposed
prime signs. Write $r=\rank G$, $I=r\log2$, and
$U_L=\Span\{e_p:p\le L\}$. If the stack with $C_L$ is compatible,
its rank is $r+\pi(L)-\kappa$, where
$\kappa=\dim(\operatorname{row}(G)\cap U_L)$.
Consequently
\[
 \PP(C_L=1\mid A)=2^{-(\pi(L)-\kappa)}
\]
exactly, whenever the exposed cutoff contains the border coordinates.
This finite identity needs no rare-intensity regime. If the stack is
incompatible, that probability is zero.
The parameter $\kappa$ counts consequences entirely supported on the
border coordinates, not input equations that merely mention them.

For the asymptotic conclusions assume $M,L\to\infty$,
$L\le A_0\log M$, $M2^{-L}\to0$, and eventually
$I\le V_M-c\nu_M$ for fixed $c>0$.
The matrices, their dimensions and right-hand sides may vary with $M$.
Multiply the quantitative normalized bulk ledger by $e^I$ before taking
limits. The cutoff term is at most $e^{-c\nu_M+o(\nu_M)}$; all polynomial
terms still vanish because $I=O(V_M)=o(\log M)$.
Thus the amplified bulk error is $o(\lambda)$; amplification of an
unspecified little-oh alone would not justify this conclusion.
Also $I=O(V_M)=o(\pi(L))$ and $\kappa\le r$. The conditioned
microscopic interior mass, relative to $2^{-\pi(L)+\kappa}$, is at
most
\[
 \exp(I)2^{-\delta_0\pi(L)-\kappa}=o(1).
\]
The mesoscopic first term is $\lambda M^{\delta-1}$ and remains
$o(\lambda)$ after multiplication by $e^I$; its other term is dominated
by $2^{-g_L}$ and remains negligible relative to the border mass.
The same one-hit calculation, normalized by $2^{-\pi(L)+\kappa}+b$,
therefore proves the affine-conditioned crossover with $\pi(L)$ replaced
by $\pi(L)-\kappa$. No assumption on future prime coordinates is used
for the rare tail, border concentration or location laws.

Finally let $q_1<\cdots<q_K$ be the first primes after $L$, and put
$Q_{L,K}=\Span\{e_{q_1},\ldots,e_{q_K}\}$ and
$H_L=\operatorname{row}(G)+U_L$. If the extended stack is compatible,
\[
 \PP(G_L\ge K\mid A,C_L=1)
                 =2^{-[K-\dim(H_L\cap Q_{L,K})]};
\]
otherwise this tail probability is zero. Thus the ordinary geometric
mark is preserved when, for every fixed $K$, the condition
$H_L\cap Q_{L,K}=\{0\}$ holds beyond a threshold that may depend on $K$.
This is separate from the parameter $\kappa$ controlling the border mass.
Fix $K$, use the finite-cylinder representative and joint stable lift,
and take the size limit before $K\to\infty$. No common threshold in $K$
and no neutrality for a growing truncation $K(M)$ is assumed.
The exact rank formula remains the alternative when neutrality fails;
no relative estimate for a disappearing mixture component is inferred.

% END ORIGINAL SOURCE: companion/E_boundary.tex

% BEGIN ORIGINAL SOURCE: companion/F_microscopic_pivots.tex
\section{Microscopic comparison by largest-odd-prime pivots}\label{supp:sec:microscopic-pivots}
We prove Theorem~\ref{thm:micro-run-tv}. Throughout this appendix the hypotheses are
\eqref{eq:micro-normalization} and \eqref{eq:micro-budget}, with $\Lambda\to\infty$.
Suppress the index $M$ on $I,V,\nu,Y$, and put $H=\log M$, $u=u(\nu)$.
Write $\PA,\EA$ for conditioning on $\mathcal A_M$.

All arithmetic inputs in this proof are established in the present
article--companion pair and are identified in the local register below.
The final reciprocal bound also follows from the classical
largest-prime-factor asymptotics of \cite{ErdosIvicPomerance1986}.
We retain a direct uniform Rankin proof using the prime number theorem
\cite{Tenenbaum2015}, then spell out the alternative deduction.
Neither the reciprocal bound nor the general Palm--Stein principle
is claimed as a new analytic tool. Palm--Stein comparison is classical
\cite{Barbour1988SteinProcess,ChenXia2004}; the finite total-variation
inequality is included explicitly.

\subsection{The local signed input register}\label{supp:pivot:sec:inputs}
Put $E_*=E_M^*$ from \eqref{eq:micro-cutoff}, $Q=L+E_*+1$, and
$U=\{x\in\Z:\lceil\sqrt M\rceil\le x\le n+1\}$.
A maximal support at $x=j+1$ is $\{j,\ldots,j+Q\}$, with $Q+1$
raw vertices. Its base support has $L+1$ vertices; these two lengths
are not interchanged.
Let $\mathcal D_Y(Q)$ be the indices $j$ with $j+1\in U$ whose maximal
support contains a $Y$-defective integer. Put
\[
 \mathcal M_{L+1}=\sum_{x\in U}(2^{m_{L+1}(x)}-1),\qquad
 R^{\rm val}_{2,U}(M,Q+1)=
 \sum_{\substack{x,y\in U\\|x-y|>Q}}
           (2^{\rho^{\rm val}_{Q+1}(x,y)}-1).
\]
The full-value kernel is the one defined in
\cref{sec:absolute-relations}, with no block-parity restriction.

\begin{lemma}[Signed inputs and an independent shifted scalar tail]
\label{supp:pivot:lem:inputs}
Under \eqref{eq:micro-normalization} and \eqref{eq:micro-budget},
$Q=\log_2M+O(V)$ and $Y>2(Q+1)$ eventually. Uniformly in this band,
\begin{align}
 \mathcal M_{L+1}&\le M^{1/2+o(1)},&
 |\mathcal D_Y(Q)|/M&\le e^{-V+o(\nu)},
                                  \label{supp:pivot:eq:input-deletion}\\
 R^{\rm val}_{2,U}(M,Q+1)&\ll_\varepsilon
   M^\varepsilon\bigl(M^{5/3}+M^{2/3}2^{Q+1}\bigr).
                                  \label{supp:pivot:eq:input-profile}
\end{align}
On every retained support the signed low-type marginal is exactly
$p\gamma_{e,\sigma}$, on every environment in $\mathcal F_Y$.
For two retained sites with $0<|j-k|\le Q$, the conditional joint mass
is at most $4p^2\gamma_{e,\sigma}\gamma_{f,\tau}$.
For separated sites its unconditional counterpart is at most
\begin{equation}\label{supp:pivot:eq:input-pairs}
 p^2\gamma_{e,\sigma}\gamma_{f,\tau}
                2^{\rho^{\rm val}_{Q+1}(j+1,k+1)}.
\end{equation}
Finally, for $L'=L+E_*+1$ and $\lambda'=M2^{-L'}$,
\begin{equation}\label{supp:pivot:eq:scalar-tail}
 \PP\!\left(\exists\,2\le x\le n+1:J_{x,L'}=1\right)
 \ll\lambda'+e^{-c_0\log M/\log\log M},\qquad
 e^I\lambda'\le e^{-V}.
\end{equation}
This scalar tail is established without the microscopic comparison.
\end{lemma}
\begin{proof}
Write $\lambda=Mp$ for the ambient intensity in the cutoff, whereas
$\Lambda=np$ is the exact target intensity in the hypothesis. Since
$L=O(\log M)$, eventually $\Lambda\le\lambda\le2\Lambda$.
For any fixed $0<c_1<c$, the cost $\log2$ is absorbed by
$(c-c_1)\nu$, giving $I+\log\lambda\le V-c_1\nu$ eventually.
Thus no stronger conditioning hypothesis is introduced. The cutoff gives
$2^{E_*}\ge e^{I+V}(2+\lambda)$ and, in particular,
$e^I\lambda2^{-E_*-1}\le e^{-V}/2$.
The budget also gives $I,\log\Lambda,a_M,E_*=O(V)=o(\log M)$.
\Cref{prop:defects,cor:first-moment}, including the macroscopic form
\eqref{eq:macro-defects}, give the weighted mass and, for a deterministic
mask $A\subset U$, the bound
\[
 \sum_{x\in A}\PP(J_{x,L}=1)
 \le p|A|+p\mathcal M_{L+1}.
\]
\Cref{prop:cutoffs,supp:prop:cutoff-calculation} apply to raw heights
$M+O(\log M)$ and give the bad-support count. The factor $Q+1$ and
changes of this size in the ambient height have logarithm $o(\nu)$.
\Cref{prop:absolute-profile} restricted to $U$ gives the displayed
full-value envelope. It is specialized to $M^{5/3+o(1)}$ only after
$Q=\log_2M+O(V)$ has been checked.

For the signed marginals and local pairs, apply
\cref{supp:sec:local-marks}. A prime above $Y>2(Q+1)$ occurring oddly at
one vertex is private throughout a local union. Its raw values are thus
independent fair signs conditionally on the small-prime environment.
For $j<k$, $a=L+e$ and $d=k-j$, the ratios to
$p^2\gamma_{e,\sigma}\gamma_{f,\tau}$ are
\[
 0\ (d<a),\quad4\1_{\{\tau=-\sigma\}}\ (d=a),\quad
 2\1_{\{\tau=\sigma\}}\ (d=a+1),\quad1\ (a+1<d\le Q).
\]
Zero extension to the maximal supports gives
\eqref{supp:pivot:eq:input-pairs}; it is an unconditional, not a
fibrewise, profile bound. For a general $\mathcal A_M$, nonnegative
averaged errors may be multiplied by $e^I$ as in
\cref{lem:stable-lift}. The local bound by four times the rate product
already holds on every small-prime fibre and therefore also after mixing
under $\mathcal A_M$, without that factor. The later common factor
$e^I$ is a valid, coarser way of writing the combined ledger.

\Cref{prop:deep-starts} gives the independent deletion of
$x<\lceil\sqrt M\rceil$, of mass
$O(\Lambda M^{-1/2}+e^{-c_0\log M/\log\log M})$.
For the shifted tail, $L'$ remains in a fixed logarithmic band.
Apply \cref{thm:prefix-poisson} to its count over $2\le x<M$ and then
test non-vacancy. Since $1-e^{-\lambda'}\le\lambda'$, this gives
\eqref{supp:pivot:eq:scalar-tail}; the retained indices are a subset of
that range. Its proof uses the scalar graph estimate, the two-window
profile and \cref{prop:deep-starts}, not the theorem being proved here.
Lastly \eqref{eq:micro-cutoff} gives
$e^IMp2^{-E_*-1}\le e^{-V}$ directly. Summing signs does not enlarge
the tail event.
\end{proof}

\subsection{A finite conditional coupling inequality}
Let $\mathcal J$ be finite, let $W=(W_a)_{a\in\mathcal J}$ be indicators with positive means $\lambda_a$, and for each $a$ couple $W^a$ to $W$ with
$\Law(W^a)=\Law(W\mid W_a=1)$. Denote a coordinate unit vector by $e_a$.
\begin{lemma}[Dimension-free Palm--Stein bound]\label{supp:pivot:lem:palm}
With the supremum convention for total variation,
\begin{equation}\label{supp:pivot:eq:palm}
 \TV\left(\Law(W),\bigotimes_a\Pois(\lambda_a)\right)
 \le\sum_a\lambda_a\EE\|W-(W^a-e_a)\|_1.
\end{equation}
No dependency graph is assumed.
\end{lemma}
\begin{proof}
The immigration--death generator with invariant product-Poisson law $\mathsf Q$ is
\[
 \mathcal A g(w)=\sum_a\lambda_a\Delta_a g(w)
                +\sum_a w_a\{g(w-e_a)-g(w)\}.
\]
For $0\le h\le1$, its semigroup solution
$g_h(w)=-\int_0^\infty(\EE h(Z_t^w)-\mathsf Qh)\,dt$
satisfies $\mathcal A g_h=h-\mathsf Qh$.
The integral is finite by coupling the initial particles to a stationary initial configuration; the expected number of unmatched survivors is at most
$(|w|+\sum_a\lambda_a)e^{-t}$.
Couple two extra initial particles independently, each with death rate one. Their simultaneous survival probability is $e^{-2t}$ and the absolute second difference of an indicator is at most two. Hence
\[
 |\Delta_a\Delta_b g_h(w)|\le\int_0^\infty2e^{-2t}\,dt=1,
 \qquad
 |\Delta_a g_h(w)-\Delta_a g_h(v)|\le\|w-v\|_1.
\]
The first inequality includes $a=b$. For the second, put
$u=w\wedge v$, the coordinatewise minimum. Telescope from $u$ to $w$
and from $u$ to $v$ along coordinate steps. Both paths stay in the
nonnegative lattice and their total length is
$\|w-u\|_1+\|v-u\|_1=\|w-v\|_1$.
Thus comparability of $w$ and $v$ is not required. The conditional-law identity gives
\[
 \EE\mathcal A g_h(W)=\sum_a\lambda_a
       \EE\{\Delta_a g_h(W)-\Delta_a g_h(W^a-e_a)\}.
\]
Here $W^a-e_a\ge0$. Bound the differences and take the supremum over indicator tests. All sums are finite.
\end{proof}

For categorical indicators $W_{j,\alpha}$, suppose forcing $(j,\alpha)$ makes it the unique occupied type at $j$ and leaves every site outside a deterministic set $\mathscr D_j$ unchanged. Put $s_j=\sum_\alpha\lambda_{j,\alpha}$. The same-site part of \eqref{supp:pivot:eq:palm} is $\sum_j s_j^2$. The off-site part is at most
\begin{equation}\label{supp:pivot:eq:categorical}
 \sum_j\sum_{k\in\mathscr D_j\setminus\{j\}}
 \left\{s_js_k+\sum_{\alpha,\alpha'}
           \PP(W_{j,\alpha}=W_{k,\alpha'}=1)\right\}.
\end{equation}
Indeed $|W_{k,\alpha'}-W^{j,\alpha}_{k,\alpha'}|
\le W_{k,\alpha'}+W^{j,\alpha}_{k,\alpha'}$, and
$\lambda_{j,\alpha}\EE W^{j,\alpha}_{k,\alpha'}
=\PP(W_{j,\alpha}=W_{k,\alpha'}=1)$.
Thus a very small type probability cancels, rather than appearing as an inverse minimum probability.
The analytic solution and its Hessian bound are the explicit input used
by the associated Lean proof of \cref{supp:pivot:lem:palm}; the semigroup
construction printed above is not reconstructed by that formal proof.
This distinction concerns formal coverage, not an extra hypothesis in
the written coupling argument.

\subsection{A uniform reciprocal-pivot estimate}
For $m\ge1$ define $\kappa_{\rm odd}(m)$ as in the article and put
$C(X,z)=\#\{1\le m\le X:\kappa_{\rm odd}(m)\le z\}$.
For $z\ge1$, the condition is exactly that every valuation at a prime
above $z$ is even, namely the earlier defective-integer predicate.
The integer zero is never included.
The pivot is not the ordinary largest prime factor: for instance
$\kappa_{\rm odd}(2\cdot13^2)=2$.
Equivalently $\kappa_{\rm odd}(m)=P^+(\operatorname{core}(m))$,
with $P^+(1)=1$; this is $P^*(m)$ in
\cite[eq.~(5)]{deLaBretecheWangXu2026}.
Writing $m=a^2v$ with $v$ squarefree gives, for $s>1/2$,
\begin{equation}\label{supp:pivot:eq:rankin-finite}
 \sum_{\kappa_{\rm odd}(m)\le z}m^{-s}
   =\zeta(2s)\prod_{q\le z}(1+q^{-s}),\qquad
 C(X,z)\le X^s\zeta(2s)\prod_{q\le z}(1+q^{-s}).
\end{equation}
The square part includes primes larger than $z$ with even exponent and cannot be omitted.

\begin{lemma}[Uniform Rankin estimate]\label{supp:pivot:lem:rankin}
For $X=M+O(\log M)$, uniformly for $V-2\le w\le3V+2$,
\begin{equation}\label{supp:pivot:eq:rankin}
 C(X,e^w)\le X\exp\{-D(H/w)+o(\nu)\}.
\end{equation}
The remainder is deterministic and uniform in this band.
\end{lemma}
\begin{proof}
Put $v=u(H/w)$ and $s=1-v/w$. Throughout the band,
$H/w\asymp\nu$, $v=u+O(1)$, and $s\to1$, so eventually $s\ge3/4$.
Therefore
\[
 \log\left\{\zeta(2s)\prod_{q\le e^w}(1+q^{-s})\right\}
    =\sum_{q\le e^w}q^{-s}+O(1).
\]
The sum is $\operatorname{Ei}(v)+o(\nu)$ uniformly. To check the required precision, split at $e^{w/2}$. Chebyshev's bound and partial summation bound the lower part by $O(\log w+e^{v/2})=o(\nu)$.
On the upper part the relative error in $\pi(t)\sim t/\log t$ is uniform for $t\ge e^{(V-2)/2}$ and tends to zero. Partial summation compares the sum to
\[
 \int_{e^{w/2}}^{e^w}\frac{t^{-s}}{\log t}\,dt
       =\operatorname{Ei}(v)-\operatorname{Ei}(v/2).
\]
This integral is $O(\nu)$. The boundary terms
$O(e^v/w+e^{v/2}/w)$, the replacement of a partial-summation factor $s$ by one, and $\operatorname{Ei}(v/2)$ are all $o(\nu)$.
Since $\log X=H+o(1)$, the logarithm of the normalized bound in
\eqref{supp:pivot:eq:rankin-finite} is
\[
 (s-1)\log X+\operatorname{Ei}(v)+o(\nu)
   =-(H/w)v+\operatorname{Ei}(v)+o(\nu)
   =-D(H/w)+o(\nu).
\]
\end{proof}

\begin{lemma}[Convex saddle envelope]\label{supp:pivot:lem:saddle}
In the preceding band,
\begin{equation}\label{supp:pivot:eq:saddle}
 w+D(H/w)\ge2V-V\left(\sqrt{\nu u/V}-1\right)^2
              =2V-O(\nu/u)=2V-o(\nu).
\end{equation}
\end{lemma}
\begin{proof}
A parameter calculation gives the tangent inequality directly.
Put $\nu(t)=e^t/t$ and $D_{\rm par}(t)=e^t-\operatorname{Ei}(t)$
for $t>1$. At the fixed saddle parameter $u$,
\[
 \frac{d}{dt}\{D_{\rm par}(t)-u\nu(t)\}
       =\frac{e^t(t-1)(t-u)}{t^2}.
\]
This expression is nonpositive up to $u$ and nonnegative afterwards,
so the bracket has its minimum at $u$. Since
$D_{\rm par}(u)=V$ and $\nu(u)=\nu$, evaluation at the parameter
corresponding to $H/w$ gives
$D(H/w)\ge V+u(H/w-\nu)$, without differentiating an inverse function.
With $a=\nu u/V$ and $t=w/V$, add $w$ and minimize $t+a/t$:
\[
 w+D(H/w)\ge V(t+1-a+a/t)
                 \ge V(1-a+2\sqrt a)=2V-V(\sqrt a-1)^2.
\]
Now $\nu u-V=\operatorname{Ei}(u)=\nu(1+O(1/u))$ and $V\sim\nu u$, so
\[
 V(\sqrt a-1)^2
 =\frac{\operatorname{Ei}(u)^2}{(\sqrt{\nu u}+\sqrt V)^2}
 =O(\nu/u).
\]
The minimizer need not be exactly $w=V$; the displayed precision, not exact optimality of that choice, preserves the fixed margin $c\nu$.
\end{proof}

\begin{proposition}[Reciprocal-pivot sum]\label{supp:pivot:prop:reciprocal}\label{C-PIVOT-RECIP}
For $X=M+O(\log M)$ and $Y=\lfloor e^V\rfloor$,
\begin{equation}\label{supp:pivot:eq:reciprocal}
 S_M(X):=\sum_{\substack{m\le X\\\kappa_{\rm odd}(m)>Y}}
                 \frac1{\kappa_{\rm odd}(m)}
       \le X\exp\{-2V+o(\nu)\}.
\end{equation}
\end{proposition}
\begin{proof}
Terms with $\kappa_{\rm odd}(m)>e^{3V}$ contribute at most $Xe^{-3V}$.
Divide the remaining logarithmic range into intervals $[b,b+1)$, starting at
$\log Y=V+o(1)$. Each contributes at most
\[
 e^{-b}C(X,e^{b+1})
 \le X\exp\{-b-D(H/(b+1))+o(\nu)\}
 \le X\exp\{-2V+o(\nu)\},
\]
by Lemmas~\ref{supp:pivot:lem:rankin} and \ref{supp:pivot:lem:saddle}.
Indeed $-b-D(H/(b+1))=1-\{b+1+D(H/(b+1))\}$: the next-slice
shift costs the fixed factor $e$ before absorption in $o(\nu)$.
For an integer pivot, $\kappa_{\rm odd}>\lfloor e^V\rfloor$ is
exactly $\kappa_{\rm odd}>e^V$; no threshold mass is lost. There are $O(V)$ intervals, with $\log V=o(\nu)$. Their uniform remainders can be summed. The omitted tail is smaller since $V/\nu\to\infty$.
\end{proof}

\paragraph{Classical antecedent and square-part reduction.}
The bound in \cref{supp:pivot:prop:reciprocal} is not an autonomous
analytic novelty. With $P^+(1)=1$, write
$S_+(x)=\sum_{n\le x}1/P^+(n)$.
Equation~(1.3) of \cite{ErdosIvicPomerance1986}, at $r=1$, gives
\[
 S_+(x)=x\exp\{-G(\log x)+o(\nu_x)\},\qquad
 G(H)=\sqrt{2H(\log H+\log\log H-\log2-2)}.
\]
The expansion of $V_x$ in \cref{prop:cutoffs} gives
$G(\log x)=2V_x+o(\nu_x)$. In the classical formula, the omitted
terms after the first correction have size
$O(\sqrt{H\log H}(\log\log H)^2/(\log H)^2)=o(\nu_x)$.
This identifies the required second-scale precision, not just the
leading square-root exponent.

Here is the deduction for odd-prime pivots. Uniquely write $n=a^2v$
with $v$ squarefree. Positivity gives
\[
 \sum_{n\le x}\frac1{\kappa_{\rm odd}(n)}
 =\sum_{a\le\sqrt x}\ \sum_{\substack{v\le x/a^2\\v\text{ squarefree}}}
                           \frac1{P^+(v)}
 \le\sum_{a\le\sqrt x} S_+(x/a^2).
\]
The terms $a>\exp(3V_x)$ are $O(xe^{-3V_x})$ by $S_+(t)\le t$.
For the other terms, put $H=\log x$ and $t=x/a^2$. Uniformly here,
$\log t=H-O(V_x)$, $\nu_t\sim\nu_x$, and the remainder in the
classical asymptotic is $o(\nu_x)$. The explicit smooth function $G$
satisfies $G'(H)=O(\sqrt{\log H/H})=o(1)$ throughout this range.
The mean-value theorem therefore yields
\[
 S_+(x/a^2)
 \le x\exp\{-2V_x+o(\nu_x)\}\,a^{-2+o(1)}.
\]
Only the explicit function $G$ has been differentiated, not an
unspecified asymptotic remainder. The weights sum to $O(1)$, so
\[
 \sum_{n\le x}\frac1{\kappa_{\rm odd}(n)}
       \le x\exp\{-2V_x+o(\nu_x)\}.
\]
Restricting to $\kappa_{\rm odd}(n)>Y$ only decreases this sum.
At $x=M+O(\log M)$, the hard scales agree at the stated precision,
recovering \cref{supp:pivot:prop:reciprocal}.
This is a deduction from the cited asymptotic, not an assertion that
its statement uses the present pivot notation. The direct Rankin
argument above remains a complete proof of the estimate used below.

\subsection{Directed change sets and exact forcing}\label{supp:pivot:sec:forcing}
Take $E_*=E_M^*$ from \eqref{eq:micro-cutoff}, $Q=L+E_*+1$, and
$\mathcal T_* =\{0,\ldots,E_*\}\times\sgnset$.
Use the original deterministic good set
\begin{equation}\label{supp:pivot:eq:good}
 G_0=\{j:1\le j\le n,\ j+1\ge\lceil\sqrt M\rceil,
                    \ \kappa_{\rm odd}(j+a)>Y\ (0\le a\le Q)\}.
\end{equation}
For $j\in G_0$, put $q_{j,a}=\kappa_{\rm odd}(j+a)$.
These primes are distinct and greater than $Y>2(Q+1)$: a selected prime divides its own vertex to an odd exponent and no other vertex of that maximal support.
For this privacy assertion alone, exceeding $Q$ suffices: a common
prime divisor of two distinct vertices would divide their nonzero
difference of size at most $Q$. Odd valuation, not valuation exactly
one, supplies the parity coordinate. The stronger $Y>2(Q+1)$ is
retained for the separate local-pair argument.

The finite cylinder must include every prime up to an integer
$R\ge\max\{Y,n+Q\}$. This covers all marked vertices and the entire
conditioning event, not only the terminal prefix $M$. Restricting the
infinite prime-sign source to this cylinder preserves the conditional
laws of all the finite events below. Define
\begin{equation}\label{supp:pivot:eq:change-sets}
 \mathscr D_j=\{k\in G_0:q_{j,a}\mid k+b
                   \text{ for some }0\le a,b\le Q\},
 \qquad \mathscr B_M=\sum_{j\in G_0}|\mathscr D_j|.
\end{equation}
These are the possible changes for a specified forcing, not neighbourhoods asserted to be a new dependency graph. Divisibility, including even exponents, is a harmless overestimate of possible change.

\begin{proposition}[The total directed footprint]\label{supp:pivot:prop:footprint}
With $X=n+Q=M+O(\log M)$,
\begin{align}
 |\mathscr D_j|&\le(Q+1)\sum_{a=0}^Q(n/q_{j,a}+1),\label{supp:pivot:eq:footprint-point}\\
 \mathscr B_M&\le n(Q+1)^2(1+S_M(X))
          \ll n^2e^{-2V+o(\nu)}+n(Q+1)^2.\label{supp:pivot:eq:footprint}
\end{align}
\end{proposition}
\begin{proof}
For a fixed prime and target offset $b$, at most $n/q+1$ grid indices satisfy the divisibility; no bound on $b/q$ is required. Sum over offsets and selected primes. When summing reciprocals over $j,a$, each raw integer $m=j+a$ occurs at most $Q+1$ times. Apply Proposition~\ref{supp:pivot:prop:reciprocal}, use $X\asymp n$, and absorb $\log(Q+1)=O(\log H)=o(\nu)$.
\end{proof}

For example, let $Q=3$, $Y=11$, $j=246$ and $k=101$. The source vertices
$246,247,248,249$ have pivots $41,19,31,83$.
The all-large-prime graph connects them to $101,102,103,104$ through $13$, since
$247=13\cdot19$ and $104=8\cdot13$. None of the four source pivots divides a target vertex, so $k\notin\mathscr D_j$. Conversely the pivot $13$ of $104$ divides $247$; the direction matters.

\begin{proposition}[Exact one-word conditional law]\label{supp:pivot:prop:forcing}\label{C-PALM-FORCING}
Let $W=(I_{j,\alpha})_{j\in G_0,\alpha\in\mathcal T_*}$.
For every $a=(j,e,\sigma)$, there is a deterministic modification of prime signs under $\PA$ whose resulting vector $W^a$ has law
\[
 \Law_{\mathcal A}(W^a)=\Law_{\mathcal A}(W\mid I_{j,e,\sigma}=1).
\]
Only $q_{j,0},\ldots,q_{j,L+e+1}$ are modified. The small-prime environment and every coordinate at a site outside $\mathscr D_j$ remain unchanged. If the word already holds, the modification is the identity.
\end{proposition}
\begin{proof}
Write $f(q)=(-1)^{\eta_q}$ and $\sigma=(-1)^\tau$.
The required raw word has bits
$b_0=b_{L+e+1}=1-\tau$ and $b_a=\tau$ for $1\le a\le L+e$.
By privacy, the raw bit at $j+a$ is
$\eta_{q_{j,a}}+\ell_{j,a}(\eta_{\rm nonpivots})$, where $\ell_{j,a}$ uses none of the pivot bits being forced. Keep the other bits fixed and set
\begin{equation}\label{supp:pivot:eq:forcing}
 \eta'_{q_{j,a}}=b_a+\ell_{j,a}(\eta_{\rm nonpivots})\pmod2,
 \qquad 0\le a\le L+e+1.
\end{equation}
Every other bit is left unchanged, including the pivots with
$L+e+2\le a\le Q$ in the maximal support. Conditional on all the
nonpivot bits and on $\mathcal A_M$, the distinct pivot bits are
independent and fair. Thus the event has probability
$2^{-(L+e+2)}=p\gamma_{e,\sigma}$ for every nonpivot configuration. Conditioning on it leaves the complete nonpivot law unchanged and fixes exactly the values in \eqref{supp:pivot:eq:forcing}. This proves the joint conditional law of the entire finite vector.
All forced primes exceed $Y$. Outside $\mathscr D_j$, none divides any of the raw vertices of a low-type event, so that event is unchanged. Categorical exclusivity gives the unique forced atom at the source site.
\end{proof}

This is the step that deals with higher-order dependence: a complete conditional law is constructed, not inferred from pair covariances. The estimates below use pairs only after that law and its possible changes have been established.

\subsection{Finite signed ledger and completion}
At each $j\in G_0$, every low-type conditional mean is $p\gamma_\alpha$ on every small-prime environment and hence under $\PA$.
Apply Lemma~\ref{supp:pivot:lem:palm} and \eqref{supp:pivot:eq:categorical} to the preceding forcing.
The same-site sum is at most $np^2$ and the product-of-means sum is at most $p^2\mathscr B_M$.
For $0<|j-k|\le Q$, the signed local calculation in \cref{supp:pivot:lem:inputs} bounds the joint mass by $4p^2\gamma_\alpha\gamma_{\alpha'}$, already uniformly over environments. There are $O(nQ)$ such pairs.
For separated sites the unconditional bound is
\[
 \PP(I_{j,\alpha}=I_{k,\alpha'}=1)
 \le p^2\gamma_\alpha\gamma_{\alpha'}
       2^{\rho^{\rm val}_{Q+1}(j+1,k+1)}.
\]
Bound its conditional version by $e^I$ times this nonnegative quantity, split $2^\rho=1+(2^\rho-1)$, and enlarge only the excess to all separated pairs. The constant term
stays on the directed change sets; extending it to all pairs would add
an unnecessary quadratic baseline. Since $\sum_{\alpha\in\mathcal T_*}\gamma_\alpha\le1$, this proves
\begin{equation}\label{supp:pivot:eq:finite-ledger}
 \TV\left(\Law_{\mathcal A}(W),
             \bigotimes_{j\in G_0,\alpha\in\mathcal T_*}\Pois(p\gamma_\alpha)\right)
 \ll np^2+e^Ip^2\{nQ+\mathscr B_M+R^{\rm val}_{2,U}(M,Q+1)\}.
\end{equation}
Neither the number of exact types nor their smallest probability appears.

Embed the finite field by zero elsewhere. The total actual and target deletion-and-tail cost is bounded by a fixed multiple of
\begin{equation}\label{supp:pivot:eq:deletion}
 d_{0,M}=e^I\left[
 p\mathcal M_{L+1}+p|\mathcal D_Y(Q)|+\Lambda M^{-1/2}
 +e^{-c_0H/\log H}+Mp\,2^{-E_*-1}\right].
\end{equation}
This follows from the masked first moment, independent deep-start bound and shifted scalar tail in \cref{supp:pivot:lem:inputs}. The actual upper-tail event is a base hit at $L+E_*+1$; signs do not enlarge it. The independent target cost is its omitted intensity. Filling is unnecessary for \eqref{supp:pivot:eq:finite-ledger}, because the Stein bound is dimension free for the reduced deterministic target. Thus
\begin{equation}\label{supp:pivot:eq:complete-ledger}
 \Delta_M^{\rm mic}\ll d_{0,M}+np^2+
        e^Ip^2\{nQ+\mathscr B_M+R^{\rm val}_{2,U}(M,Q+1)\}.
\end{equation}

By \eqref{supp:pivot:eq:footprint}, the footprint term is at most
\[
 e^{I+2\log\Lambda-2V+o(\nu)}+M^{-1+o(1)}
 \le e^{-I-2c\nu+o(\nu)}+M^{-1+o(1)}.
\]
The deletion term is bounded by
\[
 e^{I+\log\Lambda-V+o(\nu)}+M^{-1/2+o(1)}
             +e^{I-c_0H/\log H}+O(e^{-V}).
\]
Here $H/\log H\gg V$ makes the deep-start term negligible. Since
$I,\log\Lambda,E_*=O(V)=o(H)$ and $Q=\log_2 M+O(V)$, the full-value input gives
\[
 e^Ip^2R^{\rm val}_{2,U}(M,Q+1)
       \ll_{\varepsilon_0}M^{-1/3+\varepsilon_0+o(1)}.
\]
The local and same-site terms are $M^{-1+o(1)}$. Choose
$\varepsilon_0<\varepsilon$ and absorb the uniform $o(\nu)$ terms with
$c'<c$. This proves \eqref{eq:micro-tv}.

\paragraph{Dependency boundary.}
This proof uses a single forced word at a time. It does not condition on a Poisson-size plant, estimate an exponentially small planted void, or bound a global absolute cumulant activity. The constructions in Appendix~\ref{supp:sec:palm-complements} are separate consequences and comparisons. The selected-pivot footprint replaces the all-large-prime baseline $e^{I+2\log\Lambda-V+o(\nu)}$; there is no additional inverse-intensity Stein factor.

% END ORIGINAL SOURCE: companion/F_microscopic_pivots.tex

% BEGIN ORIGINAL SOURCE: companion/G_palm_complements.tex
\section{Private kernels, signed Palm voids, and the absolute-cumulant obstruction}\label{supp:sec:palm-complements}
This appendix is a methodological complement for the same binary model.
It retains three distinct conclusions: regular configurations drawn from
the target, finite signed Palm identities, and a counterexample to a
global absolute-cumulant smallness condition. All arguments are given
below. The one-sided Palm deficit follows \emph{from}
\cref{thm:micro-run-tv}, never the reverse. None of these constructions
is an input to \cref{supp:sec:microscopic-pivots}.
Keep the hard scales, cutoff $E_*$, maximal support diameter $Q$, original
good set $G_0$ and conditional law $\PA$ used there, and put $\beta=\log2$.

\subsection{Odd rough kernels and a stronger good set}\label{supp:palm:subsec:regular}
Define
\[
 r_Y(m)=\prod_{\substack{q>Y\\v_q(m)\text{ odd}}}q,
 \quad Z_Y(s)=\zeta(2s)\prod_{q\le Y}(1+q^{-s}),
 \quad B(X,Y;s)=X^{s-1}Z_Y(s),\quad\tfrac12<s<1.
\]
The product $r_Y(m)$ and the single pivot $\kappa_{\rm odd}(m)$ have different roles. Every integer has a unique form $m=a^2vr$, with $v$ squarefree and $Y$-smooth and $r=r_Y(m)$ squarefree with primes above $Y$. If
$D_Y(x)=\#\{m\le x:r_Y(m)=1\}$, Rankin gives $D_Y(x)\le x^sZ_Y(s)$.
The exact convolutions over squarefree rough $r$ imply
\begin{align}
 \frac1X\#\{m\le X:r_Y(m)\le T\}
 &\le B(X,Y;s)\left(1+\frac{T^{1-s}}{1-s}\right),\label{supp:palm:eq:kernel-count}\\
 \frac1X\sum_{\substack{m\le X\\r_Y(m)>1}}
       \frac{z^{\omega(r_Y(m))}}{r_Y(m)}
 &\le B(X,Y;s)\left\{\prod_{q>Y}(1+zq^{-1-s})-1\right\}\notag\\
 &\le B(X,Y;s)\{\exp(z/(sY^s))-1\},\qquad z\ge1.\label{supp:palm:eq:kernel-reciprocal}
\end{align}
For the first inequality sum $D_Y(X/r)$ over rough $r\le T$ and enlarge to all positive integers. For the second sum $z^{\omega(r)}D_Y(X/r)/r$; the empty kernel is removed before forming the Euler product. Finally
$\sum_{q>Y}q^{-1-s}\le\sum_{m>Y}m^{-1-s}\le Y^{-s}/s$.

At $s=1-u/V$ and $X=M+O(Q)$, the calculation of Lemma~\ref{supp:pivot:lem:rankin} at $w=V$ gives
\begin{equation}\label{supp:palm:eq:hard-kernel}
 B(X,Y;s)\le e^{-V+o(\nu)},\qquad
 Y^{-s}=e^{-V+u+o(1)}.
\end{equation}
Fix $0<\vartheta<c$, and put
\begin{equation}\label{supp:palm:eq:strong-good}
 T_M=\exp(\vartheta H/u),\qquad
 G_\vartheta=\{j\in G_0:r_Y(j+a)>T_M\ (0\le a\le Q)\}.
\end{equation}
This is a threshold on a product of primes, not a change of the conditioning cutoff $Y$. Eventually $T_M>Y$ and $T_M=M^{o(1)}$.
\begin{proposition}[Cost of stronger deletion]\label{supp:palm:prop:deletion}
One has
\begin{equation}\label{supp:palm:eq:added-deletion}
 p|G_0\setminus G_\vartheta|
 \le \exp\{\log\Lambda-V+\vartheta\nu+o(\nu)\}=o(1),
\end{equation}
and replacing these extra deleted low-type coordinates by independent target coordinates costs at most $2p|G_0\setminus G_\vartheta|$ under $\PA$. Moreover
\begin{equation}\label{supp:palm:eq:cardinality}
 |G_\vartheta^c|\le O(\sqrt M)+M e^{-V+\vartheta\nu+o(\nu)}
                       =o(M/H).
\end{equation}
Completing the original deletions and mark tails has cost
$O(d_{0,M}+p|G_0\setminus G_\vartheta|)=o(1)$.
\end{proposition}
\begin{proof}
Each integer is used in at most $Q+1$ maximal supports.
Use \eqref{supp:palm:eq:kernel-count}--\eqref{supp:palm:eq:hard-kernel} and
$(1-s)\log T_M=\vartheta\nu$; the logarithms of $Q+1$ and $1/(1-s)$ are $O(\log H)=o(\nu)$. This proves the count and the first bound, whose exponent is at most $-I-(c-\vartheta)\nu+o(\nu)$.
At a site of $G_0$, each low-type mean is exactly $p\gamma_\alpha$ on every environment. Thus the added real and target masses are each at most $p$ per site without an extra factor $e^I$.
The original deletion and high marks are covered by \eqref{supp:pivot:eq:deletion}. The last equality in \eqref{supp:palm:eq:cardinality} follows from $V/\nu\sim u\to\infty$ and $\log H=o(\nu)$.
\end{proof}

Call a collection of maximal supports \emph{private-pivot regular} if each raw vertex occurrence has a prime above $Y$, occurring there to an odd exponent, that divides no other vertex occurrence of the collection. Repeated vertices preclude regularity.
\begin{lemma}[CRT allocation of all odd prime factors]\label{supp:palm:lem:crt}
Assume $n\ge1$, $Q\le n$ and $Y>Q$, as holds eventually in the ambient regime.
Let $J_1,\ldots,J_k$ be independent uniform indices in $\{1,\ldots,n\}$. Conditional on $J_i=j$ and on a source vertex $m=j+a$ with $r=r_Y(m)>1$, the probability that every prime of $r$ is hosted in one of the other $k-1$ supports is at most
\begin{equation}\label{supp:palm:eq:crt}
 \frac{\{3(k-1)(Q+1)\}^{\omega(r)}}r.
\end{equation}
For $k\ge2$ and $z=3k(Q+1)<Y$, the probability that all indices lie in $G_\vartheta$ but their supports are not regular is at most
$k(Q+1)T_M^{-1+\log z/\log Y}$. A single good support is regular.
\end{lemma}
\begin{proof}
Assign each prime of $r$ to a target block and offset where it is hosted. There are at most $((k-1)(Q+1))^{\omega(r)}$ assignments. For a fixed assignment, CRT fixes each target index modulo the product
$r_h$ of its assigned primes. A target block receiving no prime has
probability exactly one. For a nonempty block,
$r_h\le r\le m\le n+Q\le2n$, so that residue has probability at most
$1/r_h+1/n\le3/r_h$. There are at most $\omega(r)$ nonempty blocks;
their nontrivial moduli have product $r$. Independence therefore bounds
the assignment probability by $3^{\omega(r)}/r$, not by a factor
three for every empty block as well. Summing assignments gives
\eqref{supp:palm:eq:crt}.
If $r>T_M$, then $\omega(r)\le\log r/\log Y$ and
$z^{\omega(r)}/r\le T_M^{-1+\log z/\log Y}$.
Union-bound over the $k(Q+1)$ source occurrences. A prime above $Y>Q$ cannot be hosted elsewhere inside its own support.
\end{proof}

Let $\mathscr R_M$ be the target configurations of mass at most $K_M=\lceil2\Lambda\rceil$, with low types $e\le E_*$, sites in $G_\vartheta$, and regular maximal supports. Equivalently, enumerate all its atoms,
with their multiplicities, and impose regularity on every raw vertex
occurrence of that enumeration. This definition is invariant under
permutation, so it passes from the ordered sample to the point
configuration without losing repeated occurrences. The empty
configuration is included.
\begin{theorem}[Private pivots for a Poisson-size target cloud]\label{supp:palm:thm:regular}
For the full same-grid target law $\mathsf Q_M$,
\begin{align}\label{supp:palm:eq:regular-error}
 \mathsf Q_M(\mathscr R_M^c)\le\varepsilon_{\rm reg,M}
 \ll{}&\Lambda M^{-1/2}+e^{\log\Lambda-V+\vartheta\nu+o(\nu)}
           +\Lambda2^{-E_*-1}\notag\\
 &+e^{-(2\log2-1)\Lambda}+e^{-c\vartheta\nu^2/(4u)}=o(1).
\end{align}
For every regular $z=\{(j_i,\alpha_i):1\le i\le k\}$,
\begin{equation}\label{supp:palm:eq:presence}
 \PP(\operatorname{Occ}(z)\mid\mathcal F_Y)
       =\prod_{i=1}^k p\gamma_{\alpha_i}
\end{equation}
on every small-prime environment, and therefore under $\PA$.
\end{theorem}
\begin{proof}
Conditional on target mass $N=k$, sites are independent uniform grid indices and types have law $\gamma$. Exponential Markov bounds give
$\PP(N>K_M)\le e^{-(2\log2-1)\Lambda}$.
The bad-site and high-mark expected counts give the first three terms of
\eqref{supp:palm:eq:regular-error}.
For $k\le K_M$, eventually
\[
 1-\frac{\log(3K_M(Q+1))}{\log Y}\ge\frac{c\nu}{2V}.
\]
Apply Lemma~\ref{supp:palm:lem:crt}, obtaining a remaining bound
$K_M(Q+1)e^{-c\vartheta\nu^2/(2u)}$.
Since $\nu^2/(uV)\to\infty$ and $\log(K_M(Q+1))\le V+O(\log H)$, this is at most the final term of \eqref{supp:palm:eq:regular-error}.
For a regular configuration choose one private odd prime per raw vertex. The selected columns of the valuation matrix form an identity minor over $\mathbb F_2$ even conditional on all unchosen signs. The raw values are therefore independent and fair. Each exact signed word fixes $L+e_i+2$ of them; integrating unused maximal-support vertices proves \eqref{supp:palm:eq:presence}.
\end{proof}
The first assertion is about configurations drawn from the target. The second prescribes presence, not absence of further occurrences. For instance, if $X_{2i-1}=X_{2i}=B_i$ with independent Bernoulli-$p$ variables, then target configurations using at most one site per pair have asymptotic probability one when $np^2\to0$ and their prescribed probabilities equal $p^k$. Yet when $np\to\infty$, the actual field lies in that regular set only when empty, of probability $(1-p)^{n/2}\to0$. Its microscopic distance to the Poisson target tends to one. This surrogate is not the arithmetic model.

\subsection{Exact configuration masses and ordinary Palm deletion}\label{supp:palm:subsec:palm}
Put $G=G_\vartheta$, $g=|G|$, $\mu=gp$, and restrict the full arithmetic and target fields to $G$, without truncating exact lengths or filling. Denote their laws by $\mathsf P_M^G,\mathsf Q_M^G$, and their distance by $\Delta_M^G$.
Let $d_A=\PA(\text{a base start occurs outside }G)$ and $\delta_G=p(n-g)$; both tend to zero by Proposition~\ref{supp:palm:prop:deletion} and the original tail estimates. Projection and the deletion coupling give
\begin{equation}\label{supp:palm:eq:projection}
 0\le\Delta_M^{\rm mic}-\Delta_M^G\le d_A+\delta_G.
\end{equation}
Let $\mathscr R_M^G$ be the same regularity conditions inside $G$, with mass bounded by $K_M$. Its target exceptional probability $\varepsilon_{\rm reg,G}=o(1)$ follows by restriction from Theorem~\ref{supp:palm:thm:regular}.
For regular $z$ put $\lambda_z=\prod_{(j,\alpha)\in z}p\gamma_\alpha$ and
\[
 v(z)=\PA(\mathbf I_M=z\mid\operatorname{Occ}(z)),\qquad
 v_G(z)=\PA(\mathbf I_M|_G=z\mid\operatorname{Occ}(z)).
\]
The regular configurations have distinct sites, so
$\mathsf Q_M^G(\{z\})=e^{-\mu}\lambda_z$ and
$\mathsf P_M^G(\{z\})=\lambda_zv_G(z)$.
The normalized retained void is
\begin{equation}\label{supp:palm:eq:normalized-void-definition}
 R_z(t)=\frac{\EA\!\left[
       \prod_{j\in G\setminus\supp z}(1-tJ_{j+1,L})
                       \mid\operatorname{Occ}(z)\right]}
                    {(1-tp)^{g-|z|}},\qquad0\le t\le1.
\end{equation}
It is nonnegative and finite; $p<1$ and the plant is regular.
\begin{proposition}[Averaged deletion and the normalized void deficit]\label{supp:palm:prop:palm}
One has
\begin{align}\label{supp:palm:eq:averaged-deletion}
 &\EE_{\mathsf Q_M^G}\ind_{\mathscr R_M^G}(z)e^\mu(v_G(z)-v(z))\notag\\
 &\hspace{8mm}=\PA(\mathbf I_M|_G\in\mathscr R_M^G,
                     \text{a start occurs outside }G)\le d_A.
\end{align}
For the normalized void $R_z$ in \eqref{supp:palm:eq:normalized-void-definition},
\begin{equation}\label{supp:palm:eq:normalized-deficit}
 \left|\Delta_M^{\rm mic}
 -\EE_{\mathsf Q_M^G}\ind_{\mathscr R_M^G}(z)(1-R_z(1))_+\right|
 \le d_A+\delta_G+\varepsilon_{\rm reg,G}+O(\Lambda^2/n).
\end{equation}
\end{proposition}
\begin{proof}
Presence factorization implies
$\lambda_z(v_G-v)=\PA(\mathbf I_M|_G=z,\mathbf I_M\ne z)$.
Multiplication by the target mass and summation over disjoint events proves
\eqref{supp:palm:eq:averaged-deletion}. For any countable laws,
$\TV(\mathsf P,\mathsf Q)=\sum_z(\mathsf Q(z)-\mathsf P(z))_+$.
Thus, with
$D_G=\EE_{\mathsf Q_M^G}\ind_{\mathscr R_M^G}(1-e^\mu v_G)_+$,
\[
 0\le\Delta_M^{\rm mic}-D_G\le d_A+\delta_G+\varepsilon_{\rm reg,G}.
\]
Once the exact planted words are fixed, no further retained atom is,
almost surely, the event that every remaining base-start indicator is
zero. Simultaneous finiteness of the runs gives this equivalence, and
both positive-probability conditionings preserve that full-measure event. Hence
$v_G(z)=(1-p)^{g-k}R_z(1)$, where $k=|z|\le K_M$.
Uniformly on regular configurations,
\[
 |\mu+(g-k)\log(1-p)|
     \le kp+\frac{gp^2}{2(1-p)}=O(\Lambda^2/n).
\]
For $b>0,x\ge0$,
$|(1-bx)_+-(1-x)_+|\le1-e^{-|\log b|}\le|\log b|$;
no bound on $x=R_z(1)$ is needed. This proves \eqref{supp:palm:eq:normalized-deficit}.
\end{proof}
The argument pays $d_A$, not $e^\Lambda d_A$. For the full target, the same mass identity gives directly
\[
 0\le\Delta_M^{\rm mic}
   -\EE_{\mathsf Q_M}\ind_{\mathscr R_M}(z)(1-e^\Lambda v(z))_+
   \le\varepsilon_{\rm reg,M}.
\]
Consequently Theorem~\ref{thm:micro-run-tv} makes both one-sided deficits vanish. In the full-target form, setting
$L_z=(-\log v(z)-\Lambda)_+$ on regular configurations when $v(z)>0$,
$L_z=+\infty$ there when $v(z)=0$, and $L_z=0$ otherwise, gives
$(1-e^\Lambda v(z))_+=1-e^{-L_z}$ on the regular set, with
$e^{-\infty}=0$. The extended convention is essential at a zero void. Its expectation vanishes exactly when $L_z\to0$ in target probability. This is not uniform control of every Palm void, and it does not result from dividing an absolute approximation by $e^{-\Lambda}$.

\begin{corollary}[Completion of the one-sided Palm deficit]\label{supp:palm:cor:palm-completion}
Under Theorem~\ref{thm:micro-run-tv},
\begin{equation}\label{supp:palm:eq:palm-completion}
 \EE_{\mathsf Q^G_M}\left[
        \ind_{\mathscr R^G_M}(z)(1-R_z(1))_+\right]\longrightarrow0.
\end{equation}
Equivalently the lower relative-void defect tends to zero in target-configuration probability. This is an averaged, one-sided statement, not a uniform relative estimate over every plant.
\end{corollary}
\begin{proof}
\Cref{supp:palm:prop:palm} identifies the left side with
$\Delta_M^{\rm mic}+o(1)$, using exact configuration masses and ordinary deletion costs. Apply Theorem~\ref{thm:micro-run-tv}, whose proof does not use that identity.
\end{proof}

\subsection{The finite signed polynomial, including the environment}\label{supp:palm:subsec:signed-void}
Fix a regular plant $z$ and write $\mathcal B_z=G\setminus\operatorname{supp}z$, $m=g-|z|$.
Let $\epsilon$ be the bits of primes at most $Y$, and $\xi$ those above $Y$ among the finitely many queried integers. Successive relative signs give matrices
\[
 J_{j+1,L}=\ind_{\{A_j\epsilon+B_j\xi=b\}},\quad
 b=(1,0,\ldots,0)^{\mathsf T}\in\mathbb F_2^L,
 \qquad A_z\epsilon+B_z\xi=b_z
\]
for the base event and the planted raw equations, respectively. Regularity makes $B_z$ full row rank
$q_z=\sum_{(j,e,\sigma)\in z}(L+e+2)$.
Planting has conditional probability $2^{-q_z}$ on every environment, and so leaves the law of $\epsilon\mid\mathcal A_M$ unchanged. Given that environment, $\xi$ is uniform on the displayed affine solution space.
Put
$\widehat\mu_A(s)=\EE[(-1)^{s\cdot\epsilon}\mid\mathcal A_M]$.
For $S\subseteq\mathcal B_z$, $S\ne\varnothing$, define
\begin{equation}\label{supp:palm:eq:signed-coefficient}
 C_z(S)=
 \sum_{\substack{u_j\in\mathbb F_2^L\setminus\{0\},\ j\in S\\
                  v\in\mathbb F_2^{q_z},\ \sum_j u_jB_j=vB_z}}
 (-1)^{\sum_j u_j\cdot b+v\cdot b_z}
 \widehat\mu_A\left(\sum_j u_jA_j+vA_z\right),
 \qquad C_z(\varnothing)=1.
\end{equation}
For a given collection $(u_j)$ there is at most one $v$, by row independence of $B_z$.
\begin{proposition}[Signed avoidance and a target-sign bound]\label{supp:palm:prop:signed-void}
Writing $\EE_z$ for the planted conditional expectation, for $0\le t\le1$,
\begin{align}
 \EE_z\prod_{j\in S}(J_{j+1,L}-p)&=p^{|S|}C_z(S),\label{supp:palm:eq:centered-plant}\\
 R_z(t)&=1+\sum_{\varnothing\ne S\subseteq\mathcal B_z}
           \left(-\frac{tp}{1-tp}\right)^{|S|}C_z(S).\label{supp:palm:eq:signed-polynomial}
\end{align}
These finite identities include singleton corrections and all overlaps. For fixed unsigned regular geometry $\zeta$, expand the sign dependence as
\[
 R_{\zeta,\boldsymbol\sigma}(1)=\sum_{D\subseteq[k]}a_D(\zeta)
                          \prod_{i\in D}\sigma_i,
 \qquad\mathcal V(\zeta)=\sum_{D\ne\varnothing}a_D(\zeta)^2.
\]
Under the independent fair target signs,
\begin{equation}\label{supp:palm:eq:walsh}
 \EE_{\boldsymbol\sigma}(1-R_{\zeta,\boldsymbol\sigma}(1))_+
 \le\min\left\{1,(1-a_\varnothing(\zeta))_+
                             +\tfrac12\sqrt{\mathcal V(\zeta)}\right\}.
\end{equation}
\end{proposition}
\begin{proof}
The character expansion is
$J_{j+1,L}-p=p\sum_{u_j\ne0}(-1)^{u_j\cdot b}
 (-1)^{(u_jA_j)\cdot\epsilon+(u_jB_j)\cdot\xi}$.
On the planted affine space a character $w\cdot\xi$ has zero mean unless
$w=vB_z$, in which case its mean is
$(-1)^{v\cdot b_z+(vA_z)\cdot\epsilon}$.
Multiplication and environment averaging prove \eqref{supp:palm:eq:centered-plant}.
Expand
$\prod_j(1-tJ_{j+1,L})/(1-tp)^m
 =\prod_j\{1-t(J_{j+1,L}-p)/(1-tp)\}$ to obtain
\eqref{supp:palm:eq:signed-polynomial}.

In planted block $i$, $b_{z,i}=c_i+\tau_i\mathbf1$ with
$\sigma_i=(-1)^{\tau_i}$, where $c_i$ is one at the two borders and zero inside. Thus grouping terms by the block parities of $v$ gives the Walsh expansion. Orthogonality gives mean $a_\varnothing$ and variance $\mathcal V$.
For any integrable $R$ of mean $a$,
\[
 \EE(1-R)_+\le(1-a)_++\EE(a-R)_+
       =(1-a)_++\tfrac12\EE|R-a|
       \le(1-a)_++\tfrac12\sqrt{\Var R}.
\]
Here $R\ge0$, so the left side is also at most one. This proves \eqref{supp:palm:eq:walsh}.
\end{proof}
For the unconditioned model $\widehat\mu_A(s)=\ind_{\{s=0\}}$; for a general admissible event it is not that indicator. Nor are the planted marginal means assumed to equal $p$. The coefficients and their signs must be retained before any norm is taken. Theorem~\ref{thm:micro-run-tv} controls the deficit in \eqref{supp:palm:eq:normalized-deficit}, but does not establish that
$(1-a_\varnothing)_+$ or $\mathcal V$ vanish separately. No such stronger assertion is needed.

Two other exact formulations are useful for separating sufficient conditions from the concluded deficit.
Cumulants below are defined by finite moment--partition identities.
Deterministic recentering leaves orders at least two unchanged, but
$\operatorname{cum}_z(J_{j+1,L}-p)=\EE_zJ_{j+1,L}-p$ at order one.
Since $p$ is a reference rate, not necessarily the planted mean, these
singleton terms must be kept. If $W_j=J_{j+1,L}-p$ and
\[
 A_z(t)=\sum_{\varnothing\ne S\subseteq\mathcal B_z}
       \left(\frac{t}{1-tp}\right)^{|S|}
         |\operatorname{cum}_z(W_j:j\in S)|,
\]
then the finite moment-partition expansion gives
$|R_z(t)-1|\le e^{A_z(t)}-1$, including singleton Palm corrections.
Its per-site weight at $t=1$ is $(1-p)^{-1}$, not the global weight $2$ discussed below.
Also, with $N_z=\sum_{j\in\mathcal B_z}J_{j+1,L}$ and the planted law tilted by
$(1-t)^{N_z}/\EE_z(1-t)^{N_z}$ for $t<1$,
\[
 -\frac{d}{dt}\log R_z(t)
     =\frac{\EE_{z,t}N_z}{1-t}-\frac{mp}{1-tp}.
\]
Integration to one is an improper identity, with value $+\infty$ when the void is zero. We assert no smallness estimate for either right-hand side.

\subsection{The absolute conditional pair layer}\label{supp:palm:subsec:pairs}
There is a useful pair estimate independent of the microscopic proof. For two good disjoint prescribed raw words of total length $q_1+q_2=q$, write
$A\epsilon+B\xi=b$, $r_0=2^{-q}$,
$\rho_>=q-\operatorname{rank}B$ and
$\rho_{\rm all}=q-\operatorname{rank}[A\ B]$.
If the full system is compatible, a fraction
$a=2^{\rho_{\rm all}-\rho_>}$ of environments is admissible, and on those environments its conditional probability is $r_02^{\rho_>}$, otherwise zero. Thus
\begin{equation}\label{supp:palm:eq:two-ranks}
 \EE_\epsilon|\PP(B\xi=b-A\epsilon\mid\epsilon)-r_0|
 =r_0\{2^{\rho_{\rm all}}-1+2(1-2^{\rho_{\rm all}-\rho_>})\}.
\end{equation}
If the full system is incompatible the left side is $r_0$, and in both cases it is at most
$r_0\{(2^{\rho_{\rm all}}-1)+2\ind_{\{\rho_> >\rho_{\rm all}\}}\}$.
In the incompatible case $\rho_{\rm all}\ge1$.

For a concrete distinction, take $L=3$, $e=f=0$, both signs positive, $Y=11$, and starts $x=92,y=114$. The ten raw vertices are $91,\ldots,95$ and $113,\ldots,117$. Each individual rough row rank is five, their combined rough rank is seven, and their full rank is ten. Thus the unconditioned joint mass is $2^{-10}$, but its conditional value is $2^{-7}$ on one eighth of environments and zero elsewhere. The mean absolute conditional covariance is $7/4096$.

Let $H_2^>(M,Q;Y)$ count ordered separated good maximal-support pairs with nonzero rough nullity. Such a relation uses a vertex $m$ in one support, and every prime of $r_Y(m)$ must be hosted by the other support. Assign those primes to target offsets and use CRT. With $X=M+Q+1$ this gives
\[
 H_2^>(M,Q;Y)\ll(Q+1)X
  \sum_{\substack{m\le X\\r_Y(m)>1}}
              \frac{(Q+1)^{\omega(r_Y(m))}}{r_Y(m)}.
\]
There is only one target block, whose residue count is $O(X/r_Y(m))$ since $r_Y(m)\le X$.
Equations \eqref{supp:palm:eq:kernel-reciprocal}--\eqref{supp:palm:eq:hard-kernel} at $z=Q+1$ imply
\begin{equation}\label{supp:palm:eq:rough-host}
 H_2^>(M,Q;Y)\le M^2e^{-2V+o(\nu)}.
\end{equation}
Indeed $(Q+1)Y^{-s}\to0$ and $u+\log Q=o(\nu)$.
\begin{proposition}[The full signed pair activity]\label{supp:palm:prop:pair-activity}
Uniformly for deterministic $G\subseteq G_0$, the low-type pair activity obeys
\begin{align}\label{supp:palm:eq:pair-activity}
 \mathcal K_{2,M}
 &:=2\sum_{\substack{j<k\\j,k\in G}}\sum_{\alpha,\alpha'\in\mathcal T_*}
          |\Cov_{\mathcal A}(I_{j,\alpha},I_{k,\alpha'})|\notag\\
 &\ll e^Ip^2\{nQ+R^{\rm val}_{2,U}(M,Q+1)+H_2^>(M,Q;Y)\}\notag\\
 &\ll_\varepsilon e^{I+2\log\Lambda-2V+o(\nu)}+M^{-1/3+\varepsilon}=o(1).
\end{align}
\end{proposition}
\begin{proof}
Good-word marginal means are constant on environments. Hence absolute covariance under $\mathcal A_M$ is at most $e^I$ times mean absolute covariance over the unconditioned small-prime environments. Apply \eqref{supp:palm:eq:two-ranks}. Zero extension bounds the full-kernel excess by the maximal full-value profile; the second term requires a rough relation host. Local pairs cost $O(e^Ip^2nQ)$ with signed factor at most four. Sum $\gamma_\alpha\gamma_{\alpha'}$ before applying \eqref{supp:palm:eq:rough-host} and the profile. The budget bounds the first exponential by $e^{-I-2c\nu+o(\nu)}$.
\end{proof}
This is an absolute conditional covariance result, not a graph theorem: zero pair covariance does not imply independence from families of other sites. In particular it cannot replace the exact one-word coupling in Appendix~\ref{supp:sec:microscopic-pivots}.

\subsection{The finite cumulant bound and its arithmetic obstruction}\label{supp:palm:subsec:cumulants}
Let $G\subseteq G_0$ be a deterministic retained set and let $\mathcal T_*=\{0,\ldots,E_M^*\}\times\sgnset$. Define the global distinct-site activity
\begin{equation}\label{supp:palm:eq:absolute-activity}
 \mathcal K_{\rm abs,M}
 =\sum_{\substack{S\subseteq G\\|S|\ge2}}2^{|S|-1}
   \sum_{\boldsymbol\alpha\in\mathcal T_*^S}
       \left|\operatorname{cum}(I_{j,\alpha_j}:j\in S)\right|.
\end{equation}
Here the cumulants are under the stated arithmetic law, not under a product target.


For a finite family of categories $Y_j\in\{\partial\}\cup\mathcal T_j$ with marginals $\nu_j$, let $I_{j,a}=\ind_{\{Y_j=a\}}$ and let $\mathcal K$ be the activity \eqref{supp:palm:eq:absolute-activity} with these types and their actual law.
\begin{lemma}[A finite categorical cumulant bound]\label{supp:palm:lem:cumulant}
One has
\begin{equation}\label{supp:palm:eq:cumulant-tv}
 \TV\left(\Law((Y_j)_j),\bigotimes_j\nu_j\right)
             \le\min\{1,(e^{2\mathcal K}-1)/2\}.
\end{equation}
For $t\ge0$, if $B_j$ are Bernoulli variables of means $r_j$, and
$F_t=\EE\prod_j\{1+t(B_j-r_j)\}$, then
\begin{equation}\label{supp:palm:eq:polynomial-envelope}
 |F_t|\le\exp\!\left\{
   \sum_{\substack{S\\|S|\ge2}}t^{|S|}
                |\operatorname{cum}(B_j:j\in S)|\right\}.
\end{equation}
For categorical occupancies $B_j=\sum_a I_{j,a}$ with $2\max_jr_j<1$,
$\mathcal K\ge\tfrac12\log F_2$.
\end{lemma}
\begin{proof}
The exact identity
$\delta_{Y_j}=\nu_j+\sum_a(I_{j,a}-\EE I_{j,a})(\delta_a-\delta_\partial)$
gives, after tensor expansion, centered moment coefficients on subsets $S$.
The signed tensor has zero-mass supremum norm $2^{|S|-1}$. Singletons vanish.
Expand each centered moment into cumulant partitions, with blocks of size at least two. If
$b_B=2^{|B|-1}\sum_{\boldsymbol a}|\operatorname{cum}(I_{j,a_j}:j\in B)|$,
a partition into $m$ blocks contributes at most
$2^{m-1}\prod_B b_B$. Sum disjoint block collections, then drop disjointness to obtain
$\frac12\sum_{m\ge1}(2\sum_B b_B)^m/m!=(e^{2\mathcal K}-1)/2$.
The identical finite partition expansion with weights $t^{|B|}$ proves
\eqref{supp:palm:eq:polynomial-envelope}. Multilinearity gives
$2\mathcal K\ge\sum_{|S|\ge2}2^{|S|}|\operatorname{cum}(B_j:j\in S)|$.
When $2\max r_j<1$, all factors in $F_2$ are positive, so taking logarithms proves the last assertion. No analytic infinite cumulant series is assumed.
\end{proof}
The law on the right of \eqref{supp:palm:eq:cumulant-tv} is the product of categorical marginals, not yet independent Poisson coordinates. Poissonizing occupied mass $s_j\le p$ costs at most $\sum_js_j^2\le np^2$ by coupling the total count and using the same type kernel. The finite inequality remains true, but its global-smallness hypothesis fails as follows.

\begin{theorem}[Small microscopic distance with divergent absolute activity]\label{supp:palm:thm:cumulant-obstruction}
Let $q$ tend to infinity through odd primes and set
\[
 a_q=\lfloor\log_2q\rfloor,\qquad M_q=2^{q-1+a_q},\qquad L_q=q-1.
\]
Take $\mathcal A_{M_q}$ to be the whole space. If the retained sets have the exact low-type marginals and omit $o(M_q/\log M_q)$ sites, then
\begin{equation}\label{supp:palm:eq:cumulant-obstruction}
 \liminf_{q\to\infty}\frac{\log M_q}{M_q}\mathcal K_{\rm abs,M_q}
 \ge\frac{\log2}{2}(\log3-1)>0.
\end{equation}
This applies to $G_0$ of Appendix~\ref{supp:sec:microscopic-pivots} and to the stronger retained sets in Appendix~\ref{supp:sec:palm-complements}. The two-site contribution tends to zero, so the activity of orders at least three diverges. Nevertheless $\Delta_{M_q}^{\rm mic}\to0$, already within the earlier sufficient labelled range.
\end{theorem}
\begin{proof}[Proof of Theorem~\ref{supp:palm:thm:cumulant-obstruction}]
For $M=M_q$, $L=q-1$, $a_q=\lfloor\log_2q\rfloor$,
\[
 H\sim\beta q,\qquad
 \Lambda=2^{a_q}-(q-1)2^{-(q-1)}\asymp q.
\]
Thus $a_q=o(\log M)$ and both $\log\Lambda$ and
$\log\Lambda+\log(1+\Lambda)$ are $O(\log H)=o(V-c\nu)$ for every fixed $c$.
Let $g=|G|$, $S_*=1-2^{-E_*-1}$ and
$B_j=\sum_{\alpha\in\mathcal T_*}I_{j,\alpha}$.
Their common exact mean is $r=pS_*\le p$, and $gr\le\Lambda$.

Consider the event
\[
 \mathcal E_q=\{f(q)=-1,\ f(\ell)=1
              \text{ for every prime }\ell\le M+Q,\ \ell\ne q\}.
\]
Its probability is $2^{-\pi(M+Q)}$ in the original unconditioned model;
it is not a new admissible conditioning event. The cutoff $M+Q$ covers
the queried vertices and has $Q=o(M)$. A cylinder enlarged to $2M$
would change the witness cost and cannot be substituted here. On this event every queried value is $(-1)^{v_q(m)}$.
For $(t+1)q\le M$ and $q\nmid t(t+1)$, the endpoints $tq,(t+1)q$ are negative and the $q-1$ interior values are positive. Therefore the site $j=tq$ carries the low type $(0,+1)$.
With $T=\lfloor M/q\rfloor-1$, the number of certified indices is exactly
\[
 T-\lfloor T/q\rfloor-\lfloor(T+1)/q\rfloor
       =(1-2/q)M/q+O(1).
\]
The omissions are counted in $[1,M]$, including the $L$ terminal
indices outside $G\subseteq\{1,\ldots,M-L\}$. Their additional cost
$L=O(\log M)$ is negligible. Omitting $o(M/H)=o(M/q)$ sites leaves at least
$N_0=(1-o(1))M/q$ retained low-type atoms.
For $N=\sum_{j\in G}B_j$ and sufficiently large $q$,
\[
 F_2=(1-2r)^g\EE\left(\frac{3-2r}{1-2r}\right)^N
 \ge 2^{-\pi(M+Q)}(1-2r)^g
                   \left(\frac{3-2r}{1-2r}\right)^{N_0}.
\]
The prime number theorem gives $\pi(M+Q)=(1+o(1))M/H$;
$g\log(1-2r)\ge-O(\Lambda)$; and
$\log((3-2r)/(1-2r))=\log3+O(r)$.
Since $\Lambda\asymp H=o(M/H)$, it follows that
\[
 \log F_2\ge\{\beta(\log3-1)+o(1)\}M/H.
\]
Lemma~\ref{supp:palm:lem:cumulant} proves \eqref{supp:palm:eq:cumulant-obstruction}.
The omission condition holds for $G_\vartheta$ by \eqref{supp:palm:eq:cardinality} and for $G_0$ by its smaller deletion. Proposition~\ref{supp:palm:prop:pair-activity} gives the vanishing pair layer, and hence divergence of the remaining orders. These parameters lie even in the graph-based labelled budget. In particular \cref{thm:micro-run-tv} gives the asserted vanishing microscopic distance, independently of the activity calculation.
\end{proof}
For a fixed $t>0$, the same witness has leading lower exponent
$\beta\{\log(1+t)-1\}M/H$. It diverges when $t>e-1$; this is a threshold for the witness, not an optimal boundary of the model. Restricting the occupancy to $N\le b\Lambda$, $b>1$, can have vanishing ordinary total-variation cost, but it changes its means, cumulants, and conditional forcing identities. No such nonlocal truncation is used to retain an invalid absolute-smallness claim.

% END ORIGINAL SOURCE: companion/G_palm_complements.tex

\section*{Availability and declaration}
The scope, explicit literature inputs and limitations of the Lean development,
together with its seven Palomar registrations, are described in the article's
section \hyperref[sec:formalization]{``Formalization and Palomar registrations''}.
The article also records the \hyperref[sec:ai-assistance]{use of AI assistance and
the author's responsibility}. This companion and the article form one proof
package, with their sources and bibliography available together.
\printbibliography[heading=bibintoc,title={References}]

\end{refsection}
\end{document}