% Options for packages loaded elsewhere \PassOptionsToPackage{unicode}{hyperref} \PassOptionsToPackage{hyphens}{url} \PassOptionsToPackage{dvipsnames,svgnames,x11names}{xcolor} \documentclass[ 10pt, a4paper, ]{article} \usepackage{xcolor} \usepackage[margin=22mm]{geometry} \usepackage{amsmath,amssymb} \usepackage{mathrsfs} \usepackage{graphicx} \setcounter{secnumdepth}{-\maxdimen} % remove section numbering \usepackage{iftex} \ifPDFTeX \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{textcomp} % provide euro and other symbols \else % if luatex or xetex \usepackage{unicode-math} % this also loads fontspec \defaultfontfeatures{Scale=MatchLowercase} \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1} \fi \usepackage{lmodern} \ifPDFTeX\else % xetex/luatex font selection \setmonofont[]{lmmono10-regular.otf} \fi % Use upquote if available, for straight quotes in verbatim environments \IfFileExists{upquote.sty}{\usepackage{upquote}}{} \IfFileExists{microtype.sty}{% use microtype if available \usepackage[]{microtype} \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts }{} \makeatletter \@ifundefined{KOMAClassName}{% if non-KOMA class \IfFileExists{parskip.sty}{% \usepackage{parskip} }{% else \setlength{\parindent}{0pt} \setlength{\parskip}{6pt plus 2pt minus 1pt}} }{% if KOMA class \KOMAoptions{parskip=half}} \makeatother \usepackage{longtable,booktabs,array} \newcounter{none} % for unnumbered tables \usepackage{calc} % for calculating minipage widths % Correct order of tables after \paragraph or \subparagraph \usepackage{etoolbox} \makeatletter \patchcmd\longtable{\par}{\if@noskipsec\mbox{}\fi\par}{}{} \makeatother % Allow footnotes in longtable head/foot \IfFileExists{footnotehyper.sty}{\usepackage{footnotehyper}}{\usepackage{footnote}} \makesavenoteenv{longtable} \setlength{\emergencystretch}{3em} % prevent overfull lines \providecommand{\tightlist}{% \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} % Reading-draft typography only; no mathematical definitions or substitutions. \usepackage{microtype} \usepackage{fvextra} \usepackage{seqsplit} \DefineVerbatimEnvironment{Highlighting}{Verbatim}{breaklines,breakanywhere,commandchars=\\\{\}} \DefineVerbatimEnvironment{verbatim}{Verbatim}{breaklines,breakanywhere} \setlength{\emergencystretch}{3em} \allowdisplaybreaks[2] \usepackage{bookmark} \IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available \urlstyle{same} % fallback for those not using the hyperref driver hyperxmp: \makeatletter \@ifundefined{xmpquote}{\newcommand{\xmpquote}[1]{#1}}{} \makeatother \hypersetup{ pdfauthor={Idris Ali Shaik}, colorlinks=true, linkcolor={Maroon}, filecolor={Maroon}, citecolor={Blue}, urlcolor={Blue}, pdfcreator={LaTeX via pandoc}} \author{Idris Ali Shaik} \date{} \hypersetup{pdftitle={Positive density of Collatz convergence at every rate above 3/log(4/3)},pdfsubject={Revised source draft, 1 October 2026}} \begin{document} \section{\texorpdfstring{Positive density of Collatz convergence at every rate above \(3/\log(4/3)\)}{Positive density of Collatz convergence at every rate above 3/\textbackslash log(4/3)}}\label{positive-density-of-collatz-convergence-at-every-rate-above-3log43} Idris Ali Shaik\\ Independent researcher (unaffiliated)\\ \href{mailto:shaikidris@apache.org}{\nolinkurl{shaikidris@apache.org}}\\ \href{https://orcid.org/0009-0009-9699-9712}{ORCID: 0009-0009-9699-9712}\\ Revised source draft, 1 October 2026\\ Based on \href{https://doi.org/10.5281/zenodo.22871622}{published version 1.0.0, 21 September 2026}\\ © 2026 Idris Ali Shaik. Licensed under CC BY 4.0. \subsection{Abstract}\label{abstract} We prove that a positive proportion of positive integers reach \(1\) within \(c\log x\) ordinary Collatz steps for every fixed \(c>3/\log(4/3)\), with one positive lower natural-density bound independent of \(c\). For each fixed positive target not divisible by three, the same range of times holds with a target-dependent bound, irrespective of its subsequent orbit. Targets divisible by three have predecessor basins of natural density zero. Diagonalization gives one positive-density set with a vanishing excess above the limiting time coefficient. The proof extends weighted inverse trajectories from a fixed finite family of roots and preserves positive weighted mass at every sufficiently large scale. We derive an explicit Fourier-decay estimate, at frequencies not divisible by three, for an auxiliary distribution on residue classes. Its proof uses paired geometric valuations and their exact binomial first-passage law. At order \(6409\), the coefficient \(C\) satisfies \(\log_2\log_2\log_2 C<176\), compared with \(>2^{172}\) for the explicit coefficient in Mazur's work. Fractional-moment certificates, residue-capacity bounds, and an enlarged family of roots give explicit density estimates after harmonic and dyadic counting. Further refinements locate surviving mass within the root family, retain conditional residue information, and recover additional distinct sources with a bounded extra step cost. The primitive-decay theorem and the main qualitative density conclusions are formalized in Lean 4. The coefficient-size comparison and the complete optimized density bounds remain paper proofs. \textbf{Keywords:} Collatz problem; hitting times; lower natural density; inverse trajectories; Fourier decay; formal verification. \textbf{2020 Mathematics Subject Classification:} 11B83, 37P99. \subsection{Introduction and main results}\label{introduction-and-main-results} \subsubsection{Ordinary hitting times and lower density}\label{ordinary-hitting-times-and-lower-density} Write \(\mathbb N=\{1,2,3,\ldots\}\). The ordinary Collatz map is \[ \operatorname{Col}(x)= \begin{cases} x/2,&x\text{ even},\\ 3x+1,&x\text{ odd}. \end{cases} \] The Collatz conjecture asserts that every positive integer eventually reaches \(1\). Throughout this paper, convergence means reaching \(1\), and each application of \(\operatorname{Col}\) counts as one step. For a positive target \(y\), let \(\tau_y(x)\) be the least nonnegative integer \(j\) such that \(\operatorname{Col}^j(x)=y\), with \(\tau_y(x)=\infty\) when there is no such \(j\). All unmarked logarithms are natural, and \(\log_2\) denotes base two. Put \[ c_0=\frac3{\log(4/3)},\qquad \mathcal G_c(y)=\{x\ge1:\tau_y(x)\le c\log x\},\qquad \mathcal G_c=\mathcal G_c(1). \] The notation \[ \mathcal G=\{x\ge1:\tau_1(x)\le(10431/1000)\log x\} \] is used for the fixed numerical specialization. For \(S\subseteq\mathbb N\), its lower natural density is \[ \underline d(S)=\liminf_{X\to\infty}\frac{\#(S\cap[1,X])}{X}. \] Our results count both even and odd integers and give lower bounds at every sufficiently large cutoff. A positive lower natural density is weaker than density one and does not assert that the natural density exists. For a starting integer greater than one, reaching one also witnesses descent; our conclusions control the time to reach one, rather than only a drop below the starting value. \subsubsection{Main results}\label{main-results} \textbf{Theorem I.1 (positive density at every rate above \(c_0\)).} There is an explicitly computable constant \(c_*>0\) such that, for every real \(c>c_0\), \[ \liminf_{Y\to\infty}\frac1Y \#\{x\in\mathbb N:x\le Y,\ \tau_1(x)\le c\log x\} \ge c_*>0. \tag{I.main} \] The constant \(c_*\) is independent of \(c\). For each fixed \(c>c_0\) and \(0c_0\). \textbf{Theorem I.2 (every fixed target coprime to three).} For every positive integer \(y\) with \(3\nmid y\), there is an effectively defined \(d_y>0\) such that \[ \forall c>c_0:\qquad \underline d(\mathcal G_c(y))\ge d_y. \tag{I.targets} \] The estimate applies irrespective of the orbit after the target is reached. Its constant is fixed before \(c\). The \hyperref[fixed_targets]{fixed-target construction} gives the root-score recipe and its one-period refinement (D.oneperiodscore). The restriction on \(y\) is exact: when \(3\mid y\), its predecessor basin is \(\{2^j y:j\ge0\}\), which has natural density zero. Disjoint predecessor basins also preclude a positive constant uniform over all admissible targets. An elementary maximum estimate gives the common-clock result without the large finite moment tables: \[ \forall c>c_0:\qquad \underline d(\mathcal G_c)\ge d_{\rm qual}>0. \tag{I.qualitative} \] Both the root score and the coarse residue level in (D.elementarydensity) are chosen before \(c\). The qualitative argument can also use Tao's reference-law mixing estimates {[}1{]}. The common density has the diagonal consequence (D.vanishingclock): there is one positive-density set on which \(\tau_1(x)\le(c_0+\eta(x))\log x\) for a nonincreasing \(\eta(x)\to0\). Section 12 strengthens the quantitative construction while leaving the clock threshold and qualitative scope unchanged. It replaces a worst-case allocation of surviving mass by a bound on its location in the root list, improves the conditional moment input, and counts additional distinct inverse sources. For every eligible seed stage \(n\), it supplies the explicit bound \(\mathcal C_{\rm rev}(W_{\rm pre}(n))\) in (P.density). The final fresh-source step alone improves its preceding complete formula by at least \(4727/4704\); that factor is not the total improvement. Both an exact product-profile alternative and the conditional-moment formula are retained. These additions are paper proofs, outside the eighteen selected statements in the current Lean package. \subsubsection{Root scores and explicit density bounds}\label{root-scores-and-explicit-density-bounds} The construction begins with a finite family of odd \emph{roots} with specified paths to the target. A reference distribution modulo powers of three assigns a marker to each inverse trajectory. A \emph{root score} \(W>0\) is a guaranteed sum of persistent marked weights, each divided by its root. Sections 7--8 construct the baseline rational score \(W_*\in(0,2^{-4096}]\). There are four density estimates for a given score. The formula \(c_{\rm MV}(W)\) uses maximum and centered second-moment bounds; \(c_{\rm FM}(W)\) adds a direct \(3/2\)-moment with twelve-step propagation; \(c_{{\rm FM},16}(W)\) also uses sixteen-step propagation; and \(c_{\rm comp}(W)\) uses complementary residue capacity. Section 10 defines these functions by optimization over the coarse residue level, with a common mixing envelope and counting conversion. For every \(0 2^{\,2^{(a-9)/95}/1000-2a-31}>2. \tag{I.gain} \] At the same score, (Q.sixteengain) gives \(c_{\rm comp}(W)>c_{{\rm FM},16}(W)>(1369/625)c_{\rm FM}(W)\). These comparisons concern the complete optimized lower-bound formulas. Their absolute size is estimated in Section 11. \textbf{Quantitative refinement of the root pool.} The residue-completion construction in \hyperref[appendix_roots]{Appendix C} supplies a score \(W_{\mathcal F}^{\rm joint}=W_{\mathcal F}^{\rm joint}(q_N,P)\), where \(q_N\) and \(P\) are the startup level and residual mass from (R.residual)--(R.score). It satisfies \[ W_{\mathcal F}^{\rm joint}>\mathfrak A_bW_*, \qquad \mathfrak A_b=2.9273409933\ldots\times10^{275}, \qquad b=2^{25}. \tag{I.rootgain} \] For comparisons within the twelve-step formula, define \[ \begin{aligned} c_*&:=c_{\rm FM}(W_*),\\ c_{\rm amp}&:=c_{\rm FM}(\mathfrak A_bW_*),\\ c_{\rm joint}&:=c_{\rm FM}(W_{\mathcal F}^{\rm joint}). \end{aligned} \tag{I.constants} \] The combined density bound is \[ \begin{aligned} \underline d(\mathcal G_c)&\ge c_{\rm comp}(W_{\mathcal F}^{\rm joint})\\ &>c_{\rm joint}>c_{\rm amp}>c_*>0 \qquad(c>c_0). \end{aligned} \tag{I.rootdensity} \] The first comparison follows from (Q.sixteengain), and the remaining ones from (Q.cubicscore) and (C.domainguard). All constants are independent of \(c\). The root construction redistributes valuations along fixed trajectories and subtracts their sum from the incoming inverse step. This preserves the total ordinary clock for each seed and index. Backward residue completion and prefix-height estimates produce residue-complete nonrecurrent root blocks. \subsubsection{The counting mechanism}\label{the-counting-mechanism} Exact affine formulas identify the valuation words that give positive integer trajectories. After normalization by the roots and identification of repeated representations, the aggregate weight of a source \(x\) is at most \(1/x\). Thus source weight \(u\) in \([X,\Lambda X)\) already requires at least \(uX\) distinct integers. Harmonic allocation and counting their disjoint dyadic towers give the sharper conversion used in Section 10. The analytic step bounds primitive Fourier coefficients of the reference law. Appendix E derives an explicit coefficient using the original pair process, its binomial first-passage law, and separated phase triangles. Section 6 converts this estimate into all-level mixing with exponent \(2314/25\) and coefficient \(C_6=2C_{\rm primitive}20^{6409}+2\). This controls the total variation of the root marks and the error in passing to a fixed coarse level. Splitting a reference word gives a mixture of affine permutations. Jensen's inequality therefore bounds a moment at a combined level by the product of the moments at the two component levels, for every real order \(p\ge1\). The finite direct \(3/2\)-moment certificates extend to all levels through this product inequality. Hölder's inequality, applied to both occupied and complementary residue capacity, converts marked mass into source mass. For example, the second-moment bound \(M_2(12)\le6.803\) and mean one give \(M_{3/2}(12)\le\sqrt{6.803}\approx2.608256\), whereas direct calculation gives \(M_{3/2}(12)\le2.113605\). The counting denominators from the second-moment block at sixteen and fractional block at twelve have rates \[ 8.685^{1/16}\approx1.14465106,\qquad 2.113605^{1/6}\approx1.13284538. \] The slower growth of the latter compensates for its extra power of the residual marked mass, as (I.gain) shows. The direct depth-sixteen fractional certificate further reduces its denominator rate to \(1.10650942918\ldots\). The seed words have exact depth \(b_j\) and total valuation in \([2b_j-\lceil b_j^{3/5}\rceil,2b_j+\lceil b_j^{3/5}\rceil]\). Their reference survival is at least \(24/25\). A summable variation bound then supplies a fixed positive root score. Varying the continuation and terminal parameters gives every fixed \(c>c_0\) with this same score; \(10431/1000=10.431\) is one numerical specialization. \subsubsection{Relation to previous work}\label{relation-to-previous-work} Classical stopping-time and density results include the work of Terras {[}9{]}, Everett {[}10{]}, and Korec {[}11{]}. These concern descent along an orbit, whereas reaching a prescribed fixed target is the stronger event considered here. Lagarias {[}12{]} surveys the problem and its generalizations; Inselmann {[}13{]} studies approximation of the map and average total stopping time. These provide context for the distinction between a descent witness, a hitting-time bound, and a counting theorem for convergence. Krasikov and Lagarias {[}6{]} use difference inequalities and a computer-assisted argument to obtain at least \(X^{0.84}\) predecessors below \(X\) for each fixed positive target coprime to three, once \(X\) is sufficiently large. Tao {[}1{]} proves that the minimum of a Collatz orbit is below any prescribed function tending to infinity for almost all starting integers in logarithmic density. His proof develops the reference-law and Fourier methods used here. Mazur {[}3{]} obtains convergence to \(1\) within \((523/50)\log x\) ordinary steps on a set of positive lower natural density, with an explicit finite cutoff. We build on his weighted inverse-trajectory and terminal-counting framework. The analytic and trajectory estimates used in our density proof are derived in this paper, including the primitive Fourier bound in Appendix E. The pairing and separated-triangle method follows Tao {[}1, Section 7{]} and its explicit development in {[}3{]}; our first-passage calculation works directly with the original pair process and supplies the coefficient used throughout. The main clock result gives one density constant for every fixed \(c>3/\log(4/3)\). The moment estimates, complementary capacity, and root completion provide its quantitative refinements. Section 11 compares their formulas with the bound in {[}3{]}, including its distinct finite-cutoff recipe. Appendix A compares the two seed constructions at a common analytic coefficient. Mazur {[}4{]} also proves positive lower natural density of predecessors for each fixed positive target coprime to three. Theorem I.2 supplies ordinary hitting times throughout \(c>c_0\), with one effective target-dependent constant. Its root construction applies independently of the orbit after the target is reached. The author's earlier preprints treat two related descent questions. The polylogarithmic-descent paper {[}7{]} gives natural-density-one conclusions with a common landing for the shortcut and ordinary orbits, logarithmic passage times, and control of the preceding trajectories. The endpoint-transport paper {[}8{]} proves an exceptional-set pullback estimate and applies it to stretched-logarithmic descent. Their target scales still grow with the starting value, so those results do not by themselves imply convergence to one. Here the target is fixed and the conclusion is positive lower natural density, rather than density one. The earlier papers are comparisons, not inputs to the proof below. \subsubsection{Terminology and recurring notation}\label{notation_overview} A \emph{source} is a starting integer being counted, a \emph{target} is the integer to be reached, and a \emph{root} is a selected odd intermediate integer with a specified finite path to that target. A \emph{valuation word} records the positive numbers of halvings between successive odd states. The finite initial word families form the \emph{seed}. A \emph{graft} is the finite transition to the stopped blocks used to continue the inverse trajectories. A \emph{persistent mark} is a lower bound valid at every later stage. Its weighted sum, rather than each individual mark, is positive. A \emph{residue-complete block} has one root in every unit residue class at the chosen modulus; \emph{unit} means coprime to three. A root family is \emph{nonrecurrent} when no positive accelerated iterate of a member belongs to the family, including the same member. {\def\LTcaptype{none} % do not increment counter \begin{longtable}[]{@{} >{\raggedright\arraybackslash}p{(\linewidth - 2\tabcolsep) * \real{0.5000}} >{\raggedright\arraybackslash}p{(\linewidth - 2\tabcolsep) * \real{0.5000}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedright Notation \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright Meaning \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot \(\operatorname{Col},\tau_y(x)\) & Ordinary Collatz map and first hitting time; an odd step and every halving are counted separately. \\ \(c,c_0\) & Ordinary-step coefficient and the ratio \(3/\log(4/3)\). \\ \(c_*,c_{\rm amp},c_{\rm joint}\) & Density constants in (I.constants). \\ \(G_q,\mathcal U_q\) & All residues and unit residues modulo \(3^q\). \\ \(f_q,\rho_q,H_q\) & Mean-one reference density, its rescaling \((2/3)f_q\), and the reference fan. \\ \(W,W_*\) & A root score and the baseline score. \\ \(W_{\rm pre}(n),\mathcal C_{\rm rev}\) & Prefix-based root score and retained quantitative alternatives in Section 12. \\ \(c_{\rm MV},c_{\rm FM},c_{{\rm FM},16},c_{\rm comp}\) & Maximum/variance, twelve-step fractional, sixteen-step fractional, and complementary-capacity density formulas. \\ \(m,k(X)\) & Fixed coarse residue level and finer level growing with the cutoff. \\ \end{longtable} } Symbols confined to a construction have local meanings. The schedule coefficient \(L\) in Sections 3--5 is distinct from \(L=2\log2\) in counting; the seed sizes \(b_j\) are distinct from the growing terminal size \(b\); and the accelerated map \(U\) in Section 1 is distinct from the source mass \(U\) in Section 10. Local notation tables accompany the terminal and counting arguments. \subsubsection{Proof architecture and order of choices}\label{proof_architecture} The argument passes from analytic mixing to persistent root scores, then to integer source families and density: \[ \begin{array}{c} \text{Primitive Fourier decay (Appendix E)}\\ \downarrow\\ \text{Reference mixing and a fixed persistent root score}\\ \downarrow\\ \text{Source families at every large scale, for each }c>c_0\\ \swarrow\qquad\searrow\\ \text{Elementary maximum bound}\qquad\text{Direct moment bounds}\\ \searrow\qquad\swarrow\\ \text{Harmonic and dyadic counting}\\ \downarrow\\ \text{Positive lower natural density}. \end{array} \] The \hyperref[quantitative_reduction]{quantitative reduction} states the source-family theorem and counting estimates first. Sections 1--2 establish the exact word identities and transfer bounds. Sections 3--5 construct all-scale sources from mixing and persistent-root estimates, proved in Sections 6--8. Sections 9--10 give the moment and counting arguments, and Section 11 compares their explicit bounds. Section 12 then revisits the seed, root allocation, terminal shell and counting stages in that order, keeping all costs needed by the final density consumer. The forward references from the construction are thus discharged before its final application. The principal dependencies are summarized below. Equations retain their mnemonic labels; theorem numbers follow the section numbers, with I and R for the introduction and the preliminary reduction. {\def\LTcaptype{none} % do not increment counter \begin{longtable}[]{@{} >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}} >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}} >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedright Conclusion \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright Inputs used \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright Proof location \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot Primitive decay & Exact pair law, first passage, phase geometry & Appendix E \\ Reference mixing & Primitive decay and finite Fourier decomposition & Section 6 \\ Persistent positive root score & Mixing, seed survival, finite root period & Sections 7--8 \\ Sources at every large scale & Exact word laws, persistent score, finite graft & Sections 1--5 \\ Common-clock convergence and fixed-target hitting & Source families, capacity bounds, harmonic and dyadic counting & Section 10 \\ Quantitative improvements & Direct moment certificates, coefficient comparison, larger root scores & Sections 9--11; Appendices B--C \\ Profile and entropy consequences & Affine mixtures, projectivity, summable mixing & Appendix D \\ Revised quantitative family & Coupled seed and root prefixes; radius-four sources; conditional moments, joint allocation and fresh inverse sources & Section 12 \\ \end{longtable} } Appendix A compares fixed-depth and first-crossing seeds. Appendix B proves monotonicity in the analytic coefficient. \hyperref[appendix_roots]{Appendix C} establishes the root refinement; \hyperref[appendix_reference_consequences]{Appendix D} develops concentration-profile and entropy consequences. \hyperref[appendix_primitive_local]{Appendix E} contains the primitive-decay proof required for the explicit theorem. The analytic coefficient, seed, finite root pool, and moment certificates are fixed first. The root score and mixing envelope determine the coarse counting level, independently of the clock. Next choose \(c>c_0\), its positive terminal parameter, an error tolerance, and a finite graft. The counting cutoff then tends to infinity. The root pool and graft remain fixed while the terminal trajectories and their precision grow. The final counting argument uses finitely many disjoint intervals before taking its remaining limits. \subsection{From source families to density}\label{quantitative_reduction} This section isolates the estimates used in counting. The intervening construction will supply the source families in the theorem below; its proof can be read after the quantitative purpose of those families is clear. \subsubsection{The reference distribution and source-family theorem}\label{the-reference-distribution-and-source-family-theorem} For an integer \(m\ge0\), let \(G_m=\mathbb Z/3^m\mathbb Z\), and write \(\langle h\rangle_m=3^{-m}\sum_{r\in G_m}h(r)\). Take independent positive integers \(K_i\) with \(\Pr(K_i=a)=2^{-a}\), and put \(S_i=K_1+\cdots+K_i\). The reference law \(\mu_m\) is the distribution of \(\sum_{i=1}^m3^{i-1}2^{-S_i}\) in \(G_m\), where powers of two are inverted modulo \(3^m\). Its density relative to the uniform measure is \(f_m(r)=3^m\mu_m(r)\), and \(\rho_m=(2/3)f_m\). The empty sum gives \(f_0=1\). Section 2 relates this reference law to weighted integer trajectories by exact affine transfer identities. The weighted sum needed after reducing terminal trajectories is \[ F_j(x)=4^jx+(4^j-1)/3,\qquad H_m(x)=\sum_{j\ge0}4^{-j}\rho_m(F_j(x)). \] We call this sum the reference fan. Each \(F_j\) permutes \(G_m\), so \(\langle H_m\rangle_m=8/9=:\mu\). The \hyperref[chapter09]{reference-moment section} proves its exact relation to \(f_{m+1}\) and its moment estimates. \textbf{Theorem R.1 (all-scale source families).} The root construction supplies a fixed rational score \(W_*>0\). For every fixed \(c>c_0\), every \(\eta>0\) and every fixed \(\Lambda>16\), a single finite extension of the construction works at all sufficiently large real \(X\). It gives a finite odd-source set \(S_X\subseteq\mathcal G_c\cap[X,R_XX)\), weights \(0\le a_x\le1/x\), and an integer \(k(X)\), such that \[ W_*-\eta\le\sum_{x\in S_X}a_xH_{k(X)}(x),\qquad R_X<\Lambda,\quad R_X\longrightarrow16,\quad k(X)\longrightarrow\infty,\quad 3^{k(X)}/X\longrightarrow0. \tag{I.sources} \] The roots and score are independent of \(c,\eta,\Lambda,X\). This is (N.everyclock), with the actual radius supplied by (N.selector) and (N.scales); (R.graftinput) proves its root and mixing premises. The score is defined by the finite recipe (R.score). \subsubsection{General moments give lower bounds for physical source mass}\label{general-moments-give-lower-bounds-for-physical-source-mass} Fix a coarse level \(m\ge2\). Let \(\epsilon_m\) be the mixing-error envelope (M.envelope), with its explicitly specified positive integer coefficient \(C_6\). In particular, for every \(n\ge m\), \[ \langle|f_n-f_m\circ\pi_{n,m}|\rangle_n\le\epsilon_m \le C_6m^{-2314/25}, \] where \(\pi_{n,m}\) is reduction from \(G_n\) to \(G_m\). Put \(L=2\log2\), \(\kappa=(8/9)L\), and \(p_m=(W-\kappa\epsilon_m)_+\), where \(t_+=\max\{t,0\}\) and \(W\) is a score for which the source-family conclusion holds. For \(s>1\), suppose \(B_{s,m}\ge\langle H_m^s\rangle_m\). The total source mass \(U=\sum_xa_x\) then has the limiting lower bound \[ v_{s,m}(W)= \frac{p_m^{s/(s-1)}}{(LB_{s,m})^{1/(s-1)}}. \tag{I.generalp} \] More precisely, every strict smaller nonnegative mass is available at every sufficiently large scale, after fixing the error allowance. Here is the key inequality. With \(Q=3^m\), the actual residue masses \(w_r=\sum_{x\equiv r\ (Q)}a_x\) obey \[ w_r\le\frac{T_X}{Q},\qquad T_X=\frac QX+\frac12\log R_X\longrightarrow L. \] Indeed, odd representatives in a residue class have spacing \(2Q\). Weighted Hölder therefore gives \[ \sum_rw_rH_m(r) \le U^{1-1/s}\left(\sum_rw_rH_m(r)^s\right)^{1/s} \le U^{1-1/s}(T_XB_{s,m})^{1/s}. \] The fine-to-coarse error tends to at most \(\kappa\epsilon_m\). Taking limits with \(m,s\) fixed yields (I.generalp). The proof of (D.generalp) below supplies the precise tolerance and all-scale quantifiers. The finite-cutoff estimates use \(T_X\); their limiting coefficient is \(L\). Distinct odd sources have disjoint sets of multiples by powers of two, and these multiples remain in \(\mathcal G_c\), since \(c\log2>1\). Keeping their weights while counting gives the conversion \[ \mathcal D(u)=\frac{32}{225}(e^{2u}-1),\qquad 0\le u\le\tfrac12\log(3/2),\qquad \underline d(\mathcal G_c)\ge\mathcal D(u). \tag{I.conversion} \] Here \(u\) must be available in every sufficiently late interval of every fixed ratio greater than sixteen. The proof is (D.radial). For the constructed range \(0 \frac4{1125}W^3\left(\frac{57}{50}\right)^{-m_\circ}. \tag{I.rational} \] The choice gives \(p_{m_\circ}>W/2\); (D.explicit) proves the remaining constants and the integer-power recipe for \(m_\circ\). This convenient rational bound is weaker than the optimized formula. In particular it applies to \(W=W_*\), giving an explicit lower bound without first carrying out the full optimization. \subsection{1. Exact valuation words and physical trajectories}\label{chapter01} We first relate a finite list of valuations to a congruence class of trajectories and an exact affine formula. Reversing that formula constructs predecessors of a known convergent integer. The resulting weight bound, after histories with the same starting integer are combined, will be the input to the final count. For a prime \(p\) and a positive integer \(n\), write \(\nu_p(n)\) for the largest integer \(a\ge0\) such that \(p^a\) divides \(n\). Let \(U(x)=(3x+1)/2^{\nu_2(3x+1)}\) on positive odd integers. An ordinary Collatz step sends odd \(x\) to \(3x+1\), and even \(x\) to \(x/2\). One \(U\)-step of valuation \(k\) comprises \(k+1\) ordinary steps. \subsubsection{Exact local certificates}\label{exact-local-certificates} For a word \(w=(k_1,\ldots,k_d)\) of positive integers, put \[ S_j=\sum_{i\le j}k_i,\quad S_0=0,\qquad B_0=0,\quad B_j=3B_{j-1}+2^{S_{j-1}}. \tag{W.data} \] A certificate \(\operatorname{WordCert}(r,E,w)\) consists of a positive odd representative \(r\), an integer \(E\ge S_d+1\), and the finite checks \(\nu_2(3U^{j-1}(r)+1)=k_j\) for \(1\le j\le d\). An empty word is allowed, with \(E\ge1\). Here a \emph{source} is the starting integer of the forward trajectory, and its \emph{endpoint} is the value reached after the word. The congruence condition \(x\equiv r\pmod{2^E}\) is called a \emph{cylinder}. Specifying a value to \(e\) bits of precision means fixing its residue modulo \(2^e\). \textbf{Theorem 1.1 (word invariance, affine endpoint, and precision).} Every positive \(x\equiv r\pmod{2^E}\) has exactly the certified word and satisfies \[ U^j(x)=\frac{3^jx+B_j}{2^{S_j}}\quad(0\le j\le d). \tag{W.affine} \] For \(x=r+2^Eh>0\), \[ U^d(x)=U^d(r)+3^d2^{E-S_d}h. \tag{W.output} \] The guaranteed output precision is \(E-S_d\) bits, and cannot uniformly be increased by one bit on this entire cylinder. If \(2^{S_d}>3^d\), descent holds for every such source with \(x>B_d/(2^{S_d}-3^d)\). The word takes exactly \(S_d+d\) ordinary steps. \emph{Proof.} Suppose the two trajectories agree in valuations through step \(j-1\). Their next pre-division numerators differ by \(3^j(x-r)/2^{S_{j-1}}\), divisible by \(2^{E-S_{j-1}}\). Since \(E-S_{j-1}\ge k_j+1\), this does not change the known valuation \(k_j\). Division gives the next affine formula and difference, completing induction. The coefficient of \(h\) in (W.output) has exact valuation \(E-S_d\), since \(3^d\) is odd. Two consecutive sufficiently large nonnegative \(h\)'s give positive sources and endpoints differing by that valuation, so no stronger uniform precision is possible. Descent is exactly \((2^{S_d}-3^d)x>B_d\). Summing \(k_j+1\) gives the clock. \(\square\) \textbf{Lemma 1.2 (one source cylinder per word).} Every positive valuation word is realized by exactly one odd residue class modulo \(2^{S_d+1}\). That cylinder has Haar probability \(2^{-S_d}\) among odd 2-adic integers. In finite terms, exactly one of the \(2^{S_d}\) odd residue classes modulo \(2^{S_d+1}\) realizes the word. Haar probability assigns equal mass to these classes. \emph{Proof.} The empty word is the odd class modulo two. Suppose a prefix of total valuation \(S\) has its unique class modulo \(2^{S+1}\). By (W.output), refining it modulo \(2^{S+k+1}\) makes its current endpoint run bijectively over odd residues modulo \(2^{k+1}\), because the variable coefficient is twice a unit. Exactly one residue obeys \(3y+1\equiv2^k\pmod{2^{k+1}}\), the condition for valuation \(k\). That residue is odd, and the selected source class has positive representatives. Induction proves existence and uniqueness. The product of the successive Haar fractions \(2^{-k}\) is \(2^{-S_d}\). The later counting argument uses the actual weights of integer sources. \(\square\) \subsubsection{Orientation and affine composition}\label{orientation-and-affine-composition} An inverse word \((k_1,\ldots,k_d)\) attached to an odd root \(M\) means \[ x_0=M,\qquad x_i=(2^{k_i}x_{i-1}-1)/3. \tag{W.inverse} \] It is physically realized when every \(x_i\) is positive, odd, and integral. We use \emph{physical} only to distinguish these actual integer trajectories from the auxiliary residue-class calculations introduced later. Then \(U(x_i)=x_{i-1}\), and the chronological forward word from \(x_d\) is \((k_d,\ldots,k_1)\), not the displayed inverse order. Direct substitution gives \[ M=3^d2^{-S_d}x_d+C_w,\qquad C_w=\sum_{i=1}^d3^{i-1}2^{-S_i}. \tag{W.offset} \] The totals \(S_d,d\) and the clock \(S_d+d\) are reversal invariant; prefix tests and additive offset formulas are not. \emph{Proof.} Solve the next inverse step as \(x_{i-1}=3\,2^{-k_i}x_i+2^{-k_i}\). Substitution into \(M=3^{i-1}2^{-S_{i-1}}x_{i-1}+C_{i-1}\) adds \(3^{i-1}2^{-S_i}\) to the offset and gives the next slope. Induction proves (W.offset). Oddness of \(x_{i-1}\) in \(3x_i+1=2^{k_i}x_{i-1}\) proves the valuation is exactly \(k_i\). \(\square\) If \(M=\omega_1y+C_1\) and \(y=\omega_2x+C_2\), then \[ M=\omega_1\omega_2x+C_1+\omega_1C_2. \tag{W.compose} \] This composition formula will be used when new inverse blocks are attached to a finite history. It retains the order of the blocks and their offsets. \subsubsection{Convergent roots and uniqueness of source weights}\label{convergent-roots-and-uniqueness-of-source-weights} The \emph{pre-1 odd basin} of a root consists of the odd integers whose forward trajectory visits that root before reaching one. A family of words is \emph{prefix-free} if no selected word is a proper initial segment of another. \emph{Parsing} a concatenation means identifying the cuts between its component words. These two properties will prevent a trajectory from being counted more than once. For \(s\ge1\), let \(R_s=(4^s-1)/3\). It is positive and odd, \(3R_s+1=2^{2s}\), and \(U(R_s)=1\). For \(s>1\), its first ordinary hitting time of one is \(2s+1\); \(R_1=1\) has hitting time zero. Distinct \(R_s>1\) have disjoint pre-1 odd basins: after visiting one of them a trajectory has next odd value one, where it remains. It cannot visit another root, or visit the same root twice before reaching one. A physically realized complete history ending at \(R_s>1\), with source \(x\) and physical weight \(\omega=3^d2^{-S_d}\), satisfies \[ x\omega\le R_s \tag{W.charge} \] by the nonnegative offset in (W.offset). Under a deterministic prefix-free stopping/parsing rule, a source and root admit at most one complete retained history: the orbit and valuations are unique, the root is visited at most once, and parsing has no choice. Thus the same weight bound applies after grouping by actual source, provided compression preimages have first been identified rather than counted as additional reduced histories. Normalize the weight at root \(R_s\) by dividing by \(R_s\), and pool distinct such roots. Basin disjointness gives the aggregate weight bound \(a_x\le1/x\), not one copy of that bound per root. The later terminal-compression proof must establish unique reduced-history parsing before invoking this conclusion. \textbf{Lemma 1.3 (source weights for a nonrecurrent root family).} Let \(\mathcal P\) be a finite set of distinct positive odd roots such that \[ U^j(M)\ne L \quad(M,L\in\mathcal P,\ j\ge1), \tag{W.antichain} \] including \(M=L\). Then a source visits the pool at most once. Under the same deterministic parsing and identification of compression preimages, \[ a_x=\sum_{M\in\mathcal P}\frac1M \sum_{\substack{z\ {\rm reduced\ from}\ M\\x_z=x}}\omega_z \le\frac1x. \tag{W.poolcharge} \] Indeed, two visits in chronological order would violate (W.antichain). The single visit fixes its valuation word and its parsed history, and \(x\omega_z\le M\) follows from (W.offset). Distinct convergent roots form such a pool whenever none is a forward iterate of another; a root greater than one that reaches one cannot recur. The nonrecurrence condition is imposed on the actual orbit. \(\square\) \textbf{Lemma 1.4 (one-step roots of an arbitrary odd unit).} For \(y>0\) odd, \(3\nmid y\), define \[ e(y)=\begin{cases}2,&y\equiv1\pmod3,\\1,&y\equiv2\pmod3,\end{cases} \qquad X_t(y)=\frac{2^{2t+e(y)}y-1}{3}\quad(t\ge0). \tag{W.constructor} \] These are precisely the positive odd predecessors of \(y\), without repetition, with exact valuation \(2t+e(y)\). Integrality determines the parity of that valuation, and \(3X_t(y)+1=2^{2t+e(y)}y\) proves the claim. There is no odd predecessor when \(3\mid y\). With \(R_0=0\), one also has \(X_t(y)=2^{e(y)}yR_t+(2^{e(y)}y-1)/3\). For \(q\ge1\), the multiplier is invertible modulo \(3^q\), so the root-period proof in Section 2, which also applies to nonnegative indices, shows that every \(3^q\) consecutive \(t\)'s give every residue once. Filtering out nonunits retains exactly \(2\cdot3^{q-1}\) roots. \(\square\) \textbf{Lemma 1.5 (removing a possible cycle predecessor).} For any function \(F\) and fixed \(y\), at most one \(z\) with \(F(z)=y\) is periodic. The nonperiodic points in that fiber satisfy the analogue of (W.antichain). A periodic predecessor puts \(y\) on its cycle, where there is exactly one cyclic predecessor of \(y\). If \(F^j(z)=w\), \(j>0\), for two predecessors, then \(F^j(y)=y\) and \(w=F^{j-1}(y)\); thus \(w\) is periodic. This also rules out recurrence of one nonperiodic predecessor. \(\square\) \textbf{Corollary 1.6 (the complete finite-depth hierarchy).} Define \[ \mathcal C_0=\{1\},\qquad \mathcal C_{d+1} =\{X_t(y):y\in\mathcal C_d,\ 3\nmid y,\ t\ge0,\ X_t(y)>1\}. \tag{W.depthclasses} \] Then \(\mathcal C_d\) is exactly the set of positive odd integers with first accelerated hitting time \(d\) to one. Induct on \(d\), using (W.constructor) and uniqueness of the next odd value. Excluding one prevents extending its cycle, and \(3\mid y\) supplies no predecessors. For \(d\ge1\), different members of \(\mathcal C_d\) cannot lie on one another's forward orbit: such a visit would strictly decrease the remaining first-hitting depth. Thus every finite subset is an antichain. The hierarchy exhausts finite convergent odd trajectories, without asserting that every odd integer belongs to a finite level. \subsection{2. Reference transfer and capacity of physical histories}\label{chapter02} The word formulas now connect actual trajectories to the auxiliary residue distribution of {[}1, 2{]}. The affine transfer follows {[}3, §2{]}, and the selected-family comparison follows {[}3, §3{]}. We give their proofs with the word orientation and normalization used here, then bound the total trajectory weight in each residue class. These capacity bounds will compare weighted sums with the reference distribution. The quantitative mixing estimate is stated here and proved in the later mixing section. \subsubsection{The reference law and its normalization}\label{the-reference-law-and-its-normalization} Write \(G_q=\mathbb Z/3^q\mathbb Z\), including the one-point group \(G_0\), and \(\langle f\rangle_q=3^{-q}\sum_{y\in G_q}f(y)\). A dyadic rational has an unambiguous image in every \(G_q\): a fraction with denominator a power of two is evaluated using the inverse of that denominator modulo \(3^q\). At positive levels, the units are the residues not divisible by three. We call \(q\) the \emph{ternary residue level} of the modulus \(3^q\). For \(n\ge m\), let \(\pi_{n,m}:G_n\to G_m\) be reduction modulo \(3^m\). On an auxiliary probability space let \(K_i\) be independent with \(\Pr(K_i=a)=2^{-a}\), \(a\ge1\), and put \(S_i=K_1+\cdots+K_i\). Define \[ C_n=\sum_{i=1}^n3^{i-1}2^{-S_i}\pmod{3^n},\quad \mu_n(y)=\Pr(C_n=y),\quad f_n=3^n\mu_n,\quad \rho_n=\frac23 f_n. \tag{T.law} \] Thus \(f_0=1\) and \(\rho_0=2/3\). This normalization is applied once. The function \(f_n\) is the probability density relative to the uniform measure on \(G_n\), and \(\rho_n\) is its scaled version. A \emph{marker} means the numerical factor \(\rho_n(x\bmod3^n)\). To mark a trajectory weight, multiply it by this factor at the current end of the inverse construction, which is the source of the corresponding forward trajectory. Its \emph{marked mass} is the sum of those products, whereas \emph{unmarked mass} is the sum of the weights themselves. Markers are not indicator functions. In particular \(\langle f_n\rangle_n=1\) and \(\langle\rho_n\rangle_n=2/3\). For \(n\ge1\) their support consists of units. Reversing the iid word shows that \(C_n\) has the same law as \(n\) iterations of \(y\mapsto(3y+1)/2^K\) starting at zero. We use the inverse- prefix order because its affine offset is exactly the one in (W.offset). For \(n\ge m\), the first \(m\) terms of \(C_n\) are \(C_m\) and every later term is zero modulo \(3^m\). Consequently \(\mu_n\) projects to \(\mu_m\). Equivalently, the average of \(f_n\) over each fiber of \(G_n\to G_m\) is \(f_m\). This is the projectivity identity for the reference law. The only analytic premise in this section is a fixed nonnegative envelope \(\epsilon_m\) satisfying, for every \(n\ge m\ge1\), \[ \langle|f_n-f_m\circ\pi_{n,m}|\rangle_n\le\epsilon_m. \tag{T.mix} \] Section 6 specifies the explicit coefficient and mixing envelope used in the density formula. The left side is the unhalved L1 distance after uniform lifting, averaged over the full residue group. For \(\rho\), the distance is multiplied by \(2/3\). \subsubsection{One word as an affine injection}\label{one-word-as-an-affine-injection} Fix an inverse word \(w\) of length \(d\) and total valuation \(A\). Let \(\omega_w=3^d2^{-A}\) and \(C_w\) be its offset from (W.offset). If \(q\ge d\), the map \[ J_{w,q}:G_{q-d}\longrightarrow G_q,\qquad J_{w,q}(z)=C_w+3^d2^{-A}z \tag{T.injection} \] is well defined and injective. Its image is the single fiber \(y=C_w\bmod3^d\). Indeed, changing \(z\) by \(3^{q-d}\) changes the expression by a multiple of \(3^q\), and \(2^{-A}\) is a unit. This also proves bijectivity onto that fiber. For a real function \(g\) on \(G_{q-d}\), define \[ (\mathcal T_{w,q}g)(y)= \begin{cases} \omega_w g(J_{w,q}^{-1}(y)),&y\in\operatorname{im}J_{w,q},\\ 0,&\text{otherwise}. \end{cases} \tag{T.operator} \] \textbf{Lemma 2.1 (exact transfer norms).} This operator is linear and positive, and \[ \langle\mathcal T_{w,q}g\rangle_q=2^{-A}\langle g\rangle_{q-d}, \qquad \langle|\mathcal T_{w,q}g|\rangle_q =2^{-A}\langle|g|\rangle_{q-d}. \tag{T.norm} \] \emph{Proof.} Sum over the affine image using its unique preimage. The scaling is \(3^{-q}\omega_w3^{q-d}=2^{-A}\). Positivity makes the same calculation valid for absolute values. The empty word gives the identity, including \(q=0\). \(\square\) At an odd integer root \(M\), membership in this affine image is equivalent to integrality of the last inverse endpoint. It also implies integrality of every earlier endpoint. More explicitly, the inverse recurrence has denominator \(3^i\) at step \(i\), and the full inverse numerator reduced modulo \(3^i\) is its step-\(i\) numerator multiplied by the unit \(2^{A-S_i}\). Each integral endpoint is odd by induction, since the numerator \(2^{k_i}x_{i-1}-1\) is odd. Positivity is a separate condition. If \(M>C_w\), then \(M>C_v\) for each prefix \(v\), because the offset partial sums increase. The identities \(x_i=(M-C_{w|i})/\omega_{w|i}\) prove every endpoint positive. Under this condition the value of (T.operator) at \(M\) is exactly the physical inverse word weight times its endpoint marker. The condition \(M>C_w\) therefore identifies the residue transfer with positive integer trajectories. \subsubsection{Prefix-free stopping and its exact deficit}\label{prefix-free-stopping-and-its-exact-deficit} Let \(V\) be a finite prefix-free set of inverse words, each of length at most \(h\). Set \(p(V)=\sum_{w\in V}2^{-A(w)}\) and \(q=h+k\), where \(k\ge1\). A length-\(q\) iid word can have at most one prefix in \(V\); therefore \(p(V)\le1\). Put \[ A_{V,q}=\sum_{w\in V}\mathcal T_{w,q}\rho_{q-|w|},\qquad L_{V,q,k}=\sum_{w\in V}\mathcal T_{w,q} (\rho_k\circ\pi_{q-|w|,k}). \tag{T.selected} \] \textbf{Lemma 2.2 (selected reference density).} Pointwise \(0\le A_{V,q}\le\rho_q\), and \[ \langle A_{V,q}\rangle_q= \langle L_{V,q,k}\rangle_q=\frac23p(V),\qquad \langle\rho_q-A_{V,q}\rangle_q=\frac23(1-p(V)). \tag{T.deficit} \] \emph{Proof.} Condition the length-\(q\) iid word on having prefix \(w\). Its remaining \(q-|w|\) letters are independent geometric variables, and its offset is \(C_w+\omega_w C_{\rm tail}\). Its contribution to the full density is therefore \(\mathcal T_{w,q}\rho_{q-|w|}\). The prefix events are disjoint. Their union gives the selected nonnegative subdensity, proving the pointwise inequality. Its mean and that of \(L\) follow from (T.norm) and \(\langle\rho_r\rangle=2/3\). Subtract the means for the last identity. This also covers \(V\) empty or containing the empty word. \(\square\) \textbf{Theorem 2.3 (finite stopping transfer).} For \(1\le\ell\le q\), \[ \left\langle\left|L_{V,q,k}-\rho_\ell\circ\pi_{q,\ell}\right| \right\rangle_q \le\frac23\{p(V)\epsilon_k+\epsilon_\ell+1-p(V)\} \le\frac23\{\epsilon_k+\epsilon_\ell+1-p(V)\}. \tag{T.stopped} \] \emph{Proof.} Replace each coarse \(\rho_k\) by \(\rho_{q-|w|}\). The triangle inequality, (T.norm), and (T.mix) bound the replacement cost by \((2/3)\epsilon_k\sum_w2^{-A(w)}\). The selected density differs from \(\rho_q\) by the exact nonnegative deficit in (T.deficit). Finally replace \(\rho_q\) by the lifted \(\rho_\ell\) at cost \((2/3)\epsilon_\ell\). The family retains its original reference weights. \(\square\) The same statements apply to any fixed subfamily of \(V\). If the family is chosen differently for different physical parents, this theorem alone does not control that choice: partition the parents by their chosen family and pay the comparison for each nonempty group, or prove a different coupling. \subsubsection{Capacity of actual integer histories}\label{capacity-of-actual-integer-histories} Let \(\mathcal H\) be a finite family of physically realized inverse histories from a fixed positive odd root \(M\). We call a pair consisting of the word length and total valuation a \emph{tag}. At most \(T\ge0\) distinct total tags \((d,A)\) occur. Assume that each history has weight \(\omega=3^d2^{-A}\le\sigma\), where \(\sigma\ge0\), offset in \([c_-,c_+]\), where \(c_-\le c_+\), and that within each tag an integer source supports at most one history. Write \(C=c_+-c_-\) and \[ H_q(a)=\sum_{h\in\mathcal H:\,x_h\equiv a\ (3^q)}\omega_h. \] \textbf{Theorem 2.4 (offset-window capacity).} For every \(q\ge0\) and every residue \(a\), \[ 3^qH_q(a)\le T(C+3^q\sigma),\qquad \left|\sum_aH_q(a)\psi(a)\right| \le T(C+3^q\sigma)\langle|\psi|\rangle_q. \tag{T.capacity} \] \emph{Proof.} Fix a tag. Its slope \(\omega\) is constant, and \(M=\omega x_h+C_h\) confines all sources to an interval of length \(C/\omega\). Sources in a residue class modulo \(3^q\) have spacing at least \(3^q\). With no duplicate source within the tag, there are at most \(1+C/(\omega3^q)\) of them. Multiplication by \(\omega\) bounds their mass by \(\sigma+C/3^q\). Sum at most \(T\) tags. Pairing the pointwise bound with \(|\psi|\) proves the second inequality. \(\square\) Any restriction of \(\mathcal H\) obeys the same bound, without renormalization. If total depth and valuation are both at most \(S\), one may take \(T=(S+1)^2\). For \(B\) concatenated blocks of slopes at most \(\beta\), one may take \(\sigma=\beta^B\). The boundary term \(3^q\sigma\) is retained at every level. Pairing (T.stopped) with such a parent histogram yields a physical marked variation bound of \[ \frac23T(C+3^q\sigma) (\epsilon_k+\epsilon_\ell+1-p(V)). \tag{T.pairing} \] To identify this pairing with an orbit extension, check both integrality and positivity as above for every selected child. All comparisons remain exact when the formal residue class contains just one occupied physical source. \subsubsection{A finite root period}\label{a-finite-root-period} \textbf{Lemma 2.5 (root residues).} For \(q\ge0\), every interval of \(3^q\) consecutive positive indices \(s\) contains exactly one \(R_s=(4^s-1)/3\) in each residue modulo \(3^q\). For \(q\ge1\), \(R_s\) is a unit exactly when \(3\) does not divide \(s\). \emph{Proof.} For \(t\ge1\), \(\nu_3(4^t-1)=1+\nu_3(t)\). If \(3\nmid t\), then \((4^t-1)/3=1+4+\cdots+4^{t-1}\) is \(t\) modulo three. Next, if \(a\equiv1\pmod3\), then \(a^2+a+1\) has valuation one: writing \(a=1+3z\) gives \(3(1+3z+3z^2)\). Factoring \(a^3-1\) shows that multiplying \(t\) by three raises the valuation by one. This proves the formula for all \(t\). Now \(R_{s+t}-R_s=4^s(4^t-1)/3\) has valuation \(\nu_3(t)\). Two indices have the same root residue exactly when they agree modulo \(3^q\). Finally \(R_s\equiv s\pmod3\). \(\square\) A finite root family can therefore realize a full ternary average exactly. To use this average uniformly at later stages, we must also control the total subsequent change of the marks at each fixed root. \subsection{3. Stopped reference blocks and a global continuation schedule}\label{chapter03} The preceding identities apply to any suitable finite word family. We now construct families by stopping when the affine slope first becomes small enough. Grouping several stopped words controls both size growth and the ordinary step count. Crucially, these families depend on the prescribed block index, not on the integer to which they are attached. Their probability space is the reference law of (T.law). \subsubsection{A constant-slope stopping rule}\label{a-constant-slope-stopping-rule} Let \(\lambda=\log_2 3\) and \(\beta=1/8\). For an inverse word \(w\) define its depth \(d\), total valuation \(A\), displacement \(Y=A-\lambda d\), and ordinary step count \(R=A+d\). Its slope is \(2^{-Y}\), so the displacement is the negative base-two logarithm of that slope. Let \(\tau\) be the first positive depth with \(\sum_{i\le\tau}(K_i-\lambda)\ge3\). Write \(Y,R\) also for the stopped totals. For integer \(E\ge1\), let \(V(E)\) consist of first-crossing words with \(A0\). The vector \((Y,R)\) has an exponential moment in a neighborhood of zero. \emph{Proof.} Put \(X=K-\lambda\). Its mean is \(2-\lambda>0\). Its MGF is finite around zero, so for some \(t>0\), \(q=\mathbb E e^{-tX}<1\). If \(\tau>n\), the displacement at depth \(n\) is below three. Markov's inequality gives \[ \Pr(\tau>n)\le e^{3t}q^n. \tag{S.tautail} \] For small \(s>0\), let \(g(s)=\mathbb E e^{2s(K+1)}\); choose \(s\) with \(g(s)q<1\). On \(\tau=n\), \(R=\sum_{i\le n}(K_i+1)\). By Cauchy--Schwarz and \(\Pr(\tau=n)\le\Pr(\tau>n-1)\), \[ \mathbb E(e^{sR};\tau=n) \le g(s)^{n/2}e^{3t/2}q^{(n-1)/2}. \] Summing the geometric majorant proves the claim. Since \(|Y|\le(1+\lambda)R\), a sufficiently small joint exponential moment follows. \(\square\) \textbf{Lemma 3.2 (exact stopped means and concentration).} Set \(\mu_Y=\mathbb EY\) and \(\mu_R=\mathbb ER\). Then \[ \mu_Y=(2-\lambda)\mathbb E\tau\ge3,\quad \mu_R=3\mathbb E\tau,\quad \frac{\mu_R}{(\log2)\mu_Y}=\frac3{\log(4/3)}. \tag{S.means} \] For each fixed \(\delta>0\) there are finite positive \(C_\delta,c_\delta\) such that independent stopped copies obey \[ \Pr\left(\left|\sum_{i=1}^lY_i-l\mu_Y\right|>\delta l \ \text{or}\ \left|\sum_{i=1}^lR_i-l\mu_R\right|>\delta l\right) \le C_\delta e^{-c_\delta l}. \tag{S.corridor} \] \emph{Proof.} The event \(\tau\ge i\) depends only on \(K_1,\ldots,K_{i-1}\). Independence and absolute integrability allow summing \(\mathbb E[K_i1_{\tau\ge i}]=2\Pr(\tau\ge i)\). Thus \(\mathbb E\sum_{i\le\tau}K_i=2\mathbb E\tau\), which gives the two means without an unbounded optional-stopping assertion. The stopped displacement is at least three pointwise. The ratio follows from \((2-\lambda)\log2=\log(4/3)\). For a centered coordinate \(Z=Y-\mu_Y\) or \(R-\mu_R\), its log-MGF is differentiable at zero with derivative zero. Choose \(t>0\) small enough that \(\log\mathbb Ee^{tZ}\tfrac15N-\tfrac32 \ge\tfrac2{25}E-\tfrac{17}{10}\ge\tfrac1{15}E. \] The last inequalities hold for \(E\ge256\). The first probability in (S.binomial) is therefore at most \(e^{-E/75}\). In the second event the downward distance is \((E-1)/2-(N-1)\ge E/10+1/2\), so its probability is at most \(e^{-E/50}\). Their sum is at most \(2e^{-E/75}\). Finally \(\log2<7/10\): the first four terms of \(e^{7/10}\) sum to more than two. Thus \(E(1/75-(\log2)/100)\ge256(19/3000)>\log2\), proving the second bound. \(\square\) This conservative exponent fixes an effective continuation schedule. It does not replace the sharper reference survival certificate used for the initial word family. In the next section we show that changing to this continuation after a sufficiently long initial history loses arbitrarily little marked mass, with that initial choice fixed before the cutoff limit. \subsubsection{Global indices, finite parsing, and bounded offsets}\label{global-indices-finite-parsing-and-bounded-offsets} Fix once and for all \[ E_i=\lceil1000\log_2(i+2)\rceil\quad(i\ge1). \tag{S.precision} \] Then \(E_i\ge256\) and (S.timeout) gives \(\varepsilon(E_i)\le(i+2)^{-10}\). For any word of depth \(d\), (W.offset) and \(K_i\ge1\) give \[ 0\le C_w\le\sum_{i=1}^d3^{i-1}2^{-i} =(3/2)^d-1. \tag{S.offset} \] For concatenated words \(w_i\in V(E_i)\), successive slopes are at most \(\beta\). The composition identity (W.compose) bounds the complete offset by the fixed finite number \[ C_\infty=\sum_{i\ge1}\beta^{i-1} ((3/2)^{E_i-1}-1)<\infty. \tag{S.offsetsum} \] The bracket grows at most polynomially in \(i\), while \(\beta^{i-1}\) is geometric. A ratio bound on a polynomial-geometric tail gives effective truncation. The factors \(\beta^{i-1}\) provide the accumulated contraction in this sum. Concatenations of a prescribed finite number of prefix-free blocks have a unique parsing. Parse the first word at its first crossing, remove it, and repeat with the next prescribed precision. If one concatenation properly extended another with the same number of component blocks, this parsing would force every component to agree and leave no extension. Thus the concatenation family is prefix-free, as is every restriction of it. \subsubsection{Rational corridors and finite selection}\label{rational-corridors-and-finite-selection} Fix a rational outer tolerance \(\delta>0\). Effective approximation of (S.means) supplies rational centers satisfying \[ |\widetilde\mu_Y-\mu_Y|<\delta/4,\qquad |\widetilde\mu_R-\mu_R|<\delta/4. \tag{S.centers} \] For a concatenation of \(l\) stopped blocks retain the rational inner bounds \[ |Y-\widetilde\mu_Yl|\le(\delta/2)l,\qquad |R-\widetilde\mu_Rl|\le(\delta/2)l. \tag{S.innercorridor} \] The true \(\delta/4\)-corridor is contained in this inner corridor, which is contained in the true \(\delta\)-corridor. Both inclusions follow from the triangle inequality. Thus selection preserves the outer geometric bounds while its rejected mass is controlled by (S.corridor) at tolerance \(\delta/4\). This is a decidable finite selection. The \(R\)-comparison is rational. For a rational endpoint \(a/b\), \(b>0\), comparison of \(Y=A-d\log_2 3\) with \(a/b\) reduces, when \(bA-a\ge0\), to comparing the integers \(2^{bA-a}\) and \(3^{bd}\). If \(bA-a<0\), its sign decides the comparison; depth zero is rational. Both inequalities defining the absolute-value bound are handled this way. For the continuation construction, use the effective constants \(C_{\delta/4},c_{\delta/4}\) from (S.corridor). Every transition and macroblock below uses these concentration constants, while its geometric conclusions use the outer tolerance \(\delta\). \subsubsection{Parent-independent macroblocks at the actual current count}\label{parent-independent-macroblocks-at-the-actual-current-count} Fix \(\delta=1/20000\) for the later clock, choose the centers above, and take a fixed \(L>0\) with \(c_{\delta/4} L>10\). For cumulative original-block count \(B\ge0\), define \[ \ell(B)=\lceil L\log(B+2)\rceil,\qquad B_{j+1}=B_j+\ell(B_j). \tag{S.schedule} \] A \emph{macroblock} groups \(\ell(B)\) successive stopped blocks. The next family \(\mathcal C_B\) consists of concatenations \(w_i\in V(E_i)\), \(B10\), the final polynomial bound has constants independent of the starting count \(B_0\). \(\square\) In particular \(p_B>0\) eventually. At a fixed starting \(B_0\), every retained continuation satisfies the summed corridors, with only its fixed incoming prefix contributing additive constants. The map from displacement and step count to valuation and depth is invertible: \[ d=(R-Y)/(1+\lambda),\qquad A=(\lambda R+Y)/(1+\lambda). \tag{S.tags} \] Windows of width \(O(B)\) in both \(Y,R\) therefore permit \(O((B+1)^2)\) total integer tags, with constants depending only on the fixed corridors. Applying (T.capacity) retains this tag cost, the offset envelope, and the boundary term \(3^q\beta^B\). At \(k=\lceil B/1000\rceil\) and \(q=k_{\rm next}+h_B\), \(q/B\) tends to \(1/1000<\log_3 8\), so that boundary term decays exponentially. For the full history from block one, if \(\epsilon_k\le C_6/k^6\), (T.pairing) has summable increments: the unrestricted \(O(B^2\log^2 B)\) tag bound times \(O(B^{-6}+B^{-9})\) is summable even over every integer \(B\). The tighter corridor tag count removes the harmless \(\log^2 B\) factor when its literal hypotheses are available. For the literal application, set \(k_{\rm old}=\lceil B/1000\rceil\), \(k_{\rm new}=\lceil(B+\ell(B))/1000\rceil\), and \(q=k_{\rm new}+h_B\). In (T.stopped), use \(k=k_{\rm new}\) and its \(\ell=k_{\rm old}\). The hypotheses \(1\le k_{\rm old}\le q\) hold for \(B\ge1\). Any initial \(B=0\) stage is separate finite bookkeeping; positive-level mixing is used only for the eventual tail above. When the continuation starts after a large count \(B_0\), the schedules (S.precision) and (S.schedule) use that actual count. Restarting a fixed small macro index would leave an initial rejection probability without a bound. The next section also retains the number of incoming tags in the error bound and proves that its product with the continuation tail tends to zero as the initial history is extended. \subsection{4. A finite graft with uniformly vanishing weighted loss}\label{chapter04} We need to extend a family with positive marked mass while retaining control of the trajectories' sizes and step counts. An initial family of words is called a \emph{seed family}. A \emph{graft} keeps a finite initial segment of seed blocks, appends a finite transition, and then continues with the stopped macroblocks of the preceding section. Its \emph{splice index} is the number of seed blocks retained. The estimate below bounds the marked mass lost in this change of continuation, uniformly in the root. Quantitative reference mixing and lower bounds on the seed marks are explicit premises here. They are established in the mixing, seed-mark and persistent-root sections. All estimates below retain the actual inverse-history weights \(\omega=3^d2^{-A}\). \subsubsection{The seed words and their geometric interface}\label{the-seed-words-and-their-geometric-interface} Fix an integer \(b_0\ge32^5\), and define \[ b_{t+1}=b_t+\lfloor b_t/100\rfloor,\quad m_b=\lceil b^{3/5}\rceil,\quad \gamma=\frac{201}{200},\quad g=\frac{2013}{2000}. \tag{G.seed} \] Here \(t\) counts seed blocks. At stage \(t\), take exactly the words \[ V_t=\{w:|w|=b_t,\quad |A(w)-2b_t|\le m_{b_t}\}. \tag{G.fixedseed} \] The parameter \(b_t\) also specifies the parent-size threshold \(16^{b_t}\). Every family is finite, since its positive letters have bounded total valuation, and prefix-free, since all its words have the same length. The seed fixes the depth and restricts the total valuation. The later stopped blocks retain their own first-crossing rule from \hyperref[chapter03]{Section 3}; their survival and graft estimates are established separately below. For a positive odd root \(M\ge16^{b_0}\), let \(\mathcal P_t(M)\) contain the physically realized concatenations of the first \(t\) such words. The empty concatenation at \(t=0\) is included. Histories retain their complete words and cuts; identical copies are not counted as different histories. Write \(x_h,\omega_h,C_h\) for source, slope and offset. All integral positive odd sources are admitted, including nonunits; markers at positive level vanish on the latter. \textbf{Lemma 4.1 (seed geometry and actual capacity).} Set \[ C_{\rm tag}=20(4b_0+1),\quad C_{\rm in}=C_{\rm tag}(2^{b_0+1}+16^{b_0}),\quad P_t=C_{\rm in}(t+1)^3g^t . \tag{G.capacityconstant} \] Every history in \(\mathcal P_t(M)\) satisfies \[ x_h\ge16^{b_t},\quad \omega_h\le16^{b_0-b_t},\quad 0\le C_h\le2^{b_0+1},\quad b_0\gamma^t\le b_t\le b_0(101/100)^t. \tag{G.prefix} \] There are at most \(C_{\rm tag}(t+1)^3g^t\le P_t\) total depth/valuation tags. For \(H^{\rm in}_{t,q}(a)=\sum_{h:x_h\equiv a\ (3^q)}\omega_h\), \[ 3^q H^{\rm in}_{t,q}(a)\le P_t\quad(0\le q\le2b_t). \tag{G.incapacity} \] These bounds are independent of the root and persist under restriction of the history family without renormalization. \emph{Proof.} Write \(\lambda=\log_2 3<8/5\), using \(3^5<2^8\). For \(b\ge32^5\), \(m_b/b\le2b^{-2/5}\le1/512\), so every selected word satisfies \[ Y=A-\lambda b\ge(2-\lambda)b-m_b>b/3,\qquad C_w\le(3/2)^b-1<2^b. \tag{G.seedgeometry} \] The offset estimate is (S.offset). At a parent \(R\ge16^b\), it gives \(R-C_w>R/2\), and hence \[ x=\frac{R-C_w}{\omega_w}>2^{b/3-1}R,\qquad \omega_w<2^{-b/3}. \] Since \(b/3-1\ge4\lfloor b/100\rfloor\) on the stated range, \(x\ge16^{\lfloor b/100\rfloor}R\) and \(\omega_w\le16^{-\lfloor b/100\rfloor}\). This also gives positivity for every compatible word and its prefixes. Multiplying proves the endpoint and slope bounds in (G.prefix). The composition identity (W.compose) bounds the total offset by \[ \sum_{j0\) at tolerance \(\delta/4\). Choose one integer \(L\ge1\) with \(c_{\delta/4} L>10\). The global precisions are \(E_i=\lceil1000\log_2(i+2)\rceil\), and macro lengths are \(\ell(B)=\lceil L\log(B+2)\rceil\). All these choices precede the splice index \(t\), and \(t\) will precede any physical counting cutoff. \subsubsection{The transition and its error bound}\label{the-transition-and-its-error-bound} At a proposed splice \(t\), write \[ E=\lfloor\sqrt{b_t}\rfloor,\quad B_0=\lfloor E/4\rfloor,\quad k_0=\lceil B_0/1000\rceil,\quad Q=2b_t. \tag{G.transition} \] Take \(B_0\) words from \(V(E)\), and retain the concatenation only if its total displacement and step count satisfy \[ |Y-\widetilde\mu_YB_0|\le(\delta/2)B_0,\qquad |R-\widetilde\mu_RB_0|\le(\delta/2)B_0. \] These decidable inner bounds imply the outer \(\delta\)-corridors about the true means, by (S.centers)--(S.innercorridor). This fixed transition family is finite, prefix-free and independent of the physical parent. It has maximum depth at most \(B_0(E-1)\le b_t/4\). For \(E\ge256\), its reference mass \(p_t\) obeys \[ 1-p_t\le B_0\,2^{-E/100} +C_{\delta/4} e^{-c_{\delta/4} B_0}. \tag{G.transitionfailure} \] Indeed sample \(B_0\) complete stopped blocks first, of total reference mass one. The union of their precision failures has probability at most \(B_0\varepsilon(E)\); (S.timeout) and the quarter-tolerance concentration bound give the two terms. Retained words keep their original reference weights. Apply this same family to each incoming endpoint, keeping its original weight. Mark the resulting physical endpoint at level \(k_0\), and call the total \(Z^{\rm hyb}_{t,0}(M)\). Physical realization for every compatible branch will be established below before using the comparison. For sufficiently large \(t\), \(k_0\ge1\), \(a_t\le Q\), and the maximum transition depth plus \(k_0\) is at most \(Q\). Use (T.stopped) with the possibly enlarged depth bound \(h=Q-k_0\), rather than requiring \(Q\) to be the minimal possible level. Then (G.incapacity) gives \[ |Z^{\rm hyb}_{t,0}(M)-Z_t(M)| \le D_t^{\rm tr}:= \frac23P_t\left\{\frac{C_6}{k_0^6}+\frac{C_6}{a_t^6} +B_0\,2^{-E/100} +C_{\delta/4} e^{-c_{\delta/4} B_0}\right\}. \tag{G.transitiondebit} \] The bound includes both mixing terms and the rejected reference mass. In particular, the incoming estimate is used only at its allowed level \(Q=2b_t\). This error bound tends to zero uniformly in the root. To see the dependence explicitly, for \(E\ge256\) one has \[ E\ge\sqrt{b_t}/2,\quad \sqrt{b_t}/16\le B_0\le\sqrt{b_t}/4,\quad k_0\ge\sqrt{b_t}/16000,\quad a_t\ge b_t/8. \] For fixed \(c>0\), \(e^{-cu}\le r!c^{-r}u^{-r}\), by the \(r\)-th term of the exponential series. Use \(r=7\) on the precision term, including its factor \(B_0\), and \(r=6\) on the corridor term. It follows that \(D_t^{\rm tr}\le C_{\rm tr}P_t b_t^{-3}\) for a fixed effective \(C_{\rm tr}\). Since \(g<\gamma^3\), \((t+1)^3(g/\gamma^3)^t\to0\), as required. \subsubsection{Composed offsets and realization of every retained branch}\label{composed-offsets-and-realization-of-every-retained-branch} It is useful to replace the stopped section's infinite offset envelope by a finite rational upper bound. Let \[ \overline C_\infty= \sum_{i=1}^{1997}8^{1-i}(i+2)^{1000} +\frac43\,8^{-1997}2000^{1000}. \tag{G.offsetconstant} \] Then \(C_\infty\le\overline C_\infty\). In fact \((3/2)^{E_i-1}-1<(i+2)^{1000}\). For \(D_i=8^{1-i}(i+2)^{1000}\) and \(i\ge1998\), the binomial theorem gives \[ \frac{D_{i+1}}{D_i} =\frac18\left(1+\frac1{i+2}\right)^{1000} \le\frac1{8(1-1000/(i+2))}\le\frac14. \] Summing this geometric tail proves the assertion. The transition counts as the first \(B_0\) appended stopped blocks. Its exceptional common precision \(E\) is not substituted into the later logarithmic schedule. At every later macro start use its actual cumulative appended count: \[ B_{j+1}=B_j+\ell(B_j),\qquad \mathcal C_{B_j}\text{ uses the indices }B_j1\) and \(2\delta<1\), there are at most \((B+1)^2\) appended integer tags. Together with the incoming tag count this permits at most \(P_t(B+1)^2\) combined total tags. Within a combined total tag, a source determines its complete forward word. Its reversal determines the first \(t\) seed cuts, then the \(B_0\) transition first crossings and all later prescribed macro cuts. Hence there is at most one history per source in a tag, even if multiple pairs of incoming/appended tags have the same sum. Applying (T.capacity) anew, with the composed offset width and \(\omega_{\rm pre}\le1\), gives \[ 3^qH^{\rm hyb}_{t,B,q}(a) \le P_t(B+1)^2(C_*+3^q\beta^B) \qquad(q\ge0). \tag{G.hybridcapacity} \] Unlike (G.incapacity), this fresh spacing estimate is valid at every level. All its constants were fixed before \(t\). For definiteness, the following explicit threshold suffices to control the boundary term: \[ B_{\rm cap}=[400000(L+1)]^2,\quad D_{\rm mac}=L+1+C_{\delta/4},\quad K_{\rm var}=\frac83(C_*+1)(2C_6\,1000^6+D_{\rm mac}). \tag{G.tailconstants} \] At a macro start \(B\), put \(B'=B+\ell(B)\), \(k=\lceil B/1000\rceil\), \(k'=\lceil B'/1000\rceil\), and \[ h_B=\sum_{i=B+1}^{B'}(E_i-1),\qquad q(B)=k'+h_B. \] For every \(B\ge B_{\rm cap}\), \(q(B)\le B/2\), and therefore \(3^{q(B)}\beta^B\le1\). Here is a direct bound with no hidden dependence on the splice. Write \(a=L+1\). For \(B\ge2\), \(\ell(B)\le a\log(B+2)\). The inequality \(\log x\le4x^{1/4}\) for \(x\ge1\) shows \(\ell(B)\le5a B^{1/4}\le B\) at the stated threshold. For the corresponding indices, \(i+2\le2(B+2)\). Since \(\log2>1/2\), one obtains \[ h_B\le4000a\log^2(B+2),\qquad q(B)\le B/1000+4002a\log^2(B+2). \] Also \(\log^2(B+2)\le16\sqrt{B+2}\le24\sqrt B\). Using \(\sqrt B\ge400000a\), the last bound is at most \[ \left(\frac1{1000}+\frac{96048}{400000}\right)B =\frac{96448}{400000}B1/2\); no numerical fit is used. The macro failure estimate (S.macrodeficit) satisfies the explicit bound \[ 1-p_B\le \frac{D_{\rm mac}}{(B+2)^9}. \] Indeed \(\ell(B)\le(L+1)(B+2)\), and \(e^{-c_{\delta/4}\ell(B)}\le(B+2)^{-c_{\delta/4} L}\le(B+2)^{-10}\). Both marker errors are at most \(C_6(1000/B)^6\). Pair (T.stopped) at \(k',k,q(B)\) with (G.hybridcapacity); all these are positive marker levels once \(B\ge B_{\rm cap}\). The marked increments satisfy \[ |Z^{\rm hyb}_{t,j+1}(M)-Z^{\rm hyb}_{t,j}(M)| \le K_{\rm var}P_t B_j^{-4}. \tag{G.increment} \] This uses \((B+1)^2\le4B^2\), the boundary bound, and the two distinct mixing costs. The current marker in \(Z^{\rm hyb}_{t,j}\) is always \(\rho_{\lceil B_j/1000\rceil}\). The visited \(B_j\)'s are increasing integers. For \(B_0\ge2\), \[ \sum_{n\ge B_0}n^{-4} \le\int_{B_0-1}^\infty x^{-4}\,dx =\frac1{3(B_0-1)^3}\le3B_0^{-3}. \] Consequently the total continuation error bound, including its first macro, is at most \(3K_{\rm var}P_tB_0^{-3}\). The bound is uniform in the splice. \subsubsection{The graft theorem and the fixed-root quantifiers}\label{the-graft-theorem-and-the-fixed-root-quantifiers} \textbf{Theorem 4.2 (finite graft).} At all sufficiently late splices \(t\), the construction above is physically realized and, for every \(j\ge0\) and every positive odd root \(M\ge16^{b_0}\), \[ |Z^{\rm hyb}_{t,j}(M)-Z_t(M)|\le\Delta_t,\qquad \Delta_t=D_t^{\rm tr}+3K_{\rm var}P_tB_0^{-3} \longrightarrow0. \tag{G.graft} \] Each hybrid marked sequence has a nonnegative limit, with the same bound. If the fixed pool satisfies (G.marks), then for every \(\eta>0\) there is one finite common splice \(t\ge N\) such that \[ \sum_{M\in\mathcal R}\frac{Z^{\rm hyb}_{t,j}(M)}M \ge W-\eta\qquad(j\ge0). \tag{G.pool} \] The same holds for its limit. Neither the root pool nor \(W\) depends on \(\eta\), the stage or any subsequent counting cutoff. \emph{Proof.} Choose \(t\) beyond the offset threshold, with \(E\ge256\) and \(B_0\ge B_{\rm cap}\). The realization argument validates the finite transition comparison and every continuation comparison. Sum (G.increment) and (G.transitiondebit) to obtain (G.graft). Absolute summability gives the limit and preserves nonnegativity. The transition error bound has already been shown to tend to zero. For the other term, (G.prefix) and \(B_0\ge\sqrt{b_t}/16\) imply \[ P_tB_0^{-3} \le4096C_{\rm in}b_0^{-3/2}(t+1)^3 (g/\gamma^{3/2})^t\longrightarrow0, \] because \[ \gamma^3-g^2=\frac{16263}{8000000}>0 . \] All thresholds and constants are independent of the root and of the chosen later splice; they are effective from the displayed rules and the effective stopped-law constants. Finally fix the pool and let \(S=\sum_{M\in\mathcal R}1/M>0\). Choose a single \(t\ge N\) with \(\Delta_tS<\eta\). Use \(Z_t(M)\ge z_M\) and sum (G.graft) with the positive coefficients \(1/M\). This proves (G.pool) for the pooled marks. If \(\eta{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}} >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}} >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedright Quantities \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright Role and order of choice \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright Definition \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot \(\mathcal R,W\) & Fixed roots and persistent score, chosen before the clock; counting also uses nonrecurrence & (G.marks), (W.antichain) \\ \(y_\star,T_{\mathcal R}\) & Fixed target and finite maximum root-to-target path length; \(y_\star=1\) in the convergence construction & (G.terminalstate), (N.targetfamily) \\ \(\theta,\delta=\theta/20\) & Terminal ratio and concentration tolerance; positive and fixed before the splice & (N.everyclock), (N.parammargin) \\ \(L,\ell(B),E_i\) & Macro-block length \(\ell(B)=\lceil L\log(B+2)\rceil\) and global word precisions; \(L\) depends on \(\delta\) & (S.schedule), (S.precision) \\ \(t,B_0,B\) & Seed splice index, initial appended block count, and current appended block count & (G.transition), (G.terminalstate) \\ \(\mu_Y,\mu_R\) & Mean binary displacement and ordinary step cost per stopped block & (S.means) \\ \(\widetilde\mu_Y,\widetilde\mu_R\) & Rational centers for the inner corridor; concentration is used at tolerance \(\delta/4\) & (S.centers), (S.innercorridor) \\ \(b=\lfloor\theta B\rfloor,\ k=\lceil\theta B\rceil\) & Terminal size and reference level, growing with the stage & (N.stage), (N.everyclock) \\ \(w_b,l_b,h_b,e_b,r_b\) & Crossing window, overshoot scale and range of terminal shifts & (N.parameters) \\ \(u,K=e_b\) & Parent-dependent barrier shift and retained overshoot cap & (N.words), (N.stage) \\ \(I_B,R_X\) & Common interval of counting cutoffs and actual source-shell radius & (N.common), (N.everyclock) \\ \end{longtable} } Here \(B\) counts stopped reference blocks; it is not the number of valuation letters. The seed sizes \(b_j\) and the terminal size \(b\) are distinct. After the clock and tolerance are fixed, the splice and its schedules stay fixed while \(B,b,k\) grow with the counting cutoff. \subsubsection{A finite terminal family and its exact reference deficit}\label{a-finite-terminal-family-and-its-exact-reference-deficit} Let \(\lambda=\log_2 3\) and \(J(v)=\lceil\lambda v\rceil\) for nonnegative integers \(v\), with \(J(0)=0\). For an integer \(b\ge200\), put \[ w_b=\lfloor3b/5\rfloor,\quad l_b=b-w_b,\quad h_b=b+w_b, \quad e_b=\lfloor b/100\rfloor,\quad r_b=2w_b-J(w_b)-2e_b. \tag{N.parameters} \] The integer \(r_b\) is nonnegative. More precisely, \[ \frac{229}{1000}b-2\le r_b \le\frac35(2-\lambda)b,\qquad e_b\ge b/200. \tag{N.radius} \] Indeed \(\lambda<317/200\), as \(3^{200}<2^{317}\). Using \(w_b\ge3b/5-1\), \(J(w_b)\le\lambda w_b+1\), and \(e_b\le b/100\) gives the lower bound with additive loss less than two. The upper bound follows from \(J(w_b)\ge\lambda w_b\) and \(e_b\ge0\). The last floor inequality holds for every \(b\ge200\). For an integer \(0\le u\le2r_b\), define the nondecreasing integer barrier \[ H_{b,u}(s)=2b+J((s-b)_+)-J((b-s)_+)+u-r_b. \tag{N.barrier} \] For an inverse word \(v=(k_1,\ldots,k_d)\), write \(A_s=\sum_{i\le s}k_i\). For an integer cap \(K\ge0\), let \(\mathcal V(b,u,K)\) consist of words satisfying \[ l_bL_b^\sharp(y) \ge\frac{g_b16^b}{4\,3^b}, \] because \(r_b\le2b\). Since \(k\le b+1\), \[ \frac{3^k}{X}\le\frac{12}{g_b}(9/16)^b \longrightarrow0. \tag{N.precision} \] This includes the genuine ceiling case \(k=b+1\). The limit is uniform over all cutoffs in the common interval. Also \(k\to\infty\) with the selected stage. For every terminal word of cap \(K=e_b\), monotonicity of \(J\), \(u-r_b\le r_b\), and the exact radius identity imply \[ A_d\le2b+J(w_b)+r_b+e_b=2b+2w_b-e_b, \quad A_d+d\le3b+3w_b-e_b\le\frac{24}{5}b. \] The family with overshoot at most one has the same last bound, since \(e_b\ge2\). Thus the looser retained coefficient \(481/100\) also bounds its terminal step count. The reduced source itself is eventually at least its current parent: from (N.shell), \(u\ge0\), and (N.radius), \[ \log_2\frac{x}{y} \ge\log_2g_b+(2-\lambda)b-r_b-2 \ge\log_2g_b+\frac25(2-\lambda)b-2>0. \] Combining this size bound with (N.central) gives, uniformly over reduced sources and cutoffs, \[ \limsup \frac{\tau_1(x)}{\log x} \le\frac{\mu_R+\delta+481/100000} {(\log2)(\mu_Y-\delta)}. \tag{N.clock} \] Here the limit runs through the central stages, and fixed prefix/root step counts are already included in \(c_\tau\). The denominator is a lower bound on the logarithm of the reduced source, not that of its larger original. By (S.means), the right side is decreasing in \(\mu_Y\ge3\). At three it is \[ \frac9{(3-\delta)\log(4/3)} +\frac{\delta+481/100000}{(3-\delta)\log2} <\frac{1747479290400}{167532267749} <\frac{10431}{1000}. \tag{N.clockmargin} \] For a short exact verification, the positive atanh series \(\log((1+z)/(1-z))=2\sum_{j\ge0}z^{2j+1}/(2j+1)\) gives \(\log(4/3)>2(1/7+1/(3\cdot7^3)+1/(5\cdot7^5))\) and \(\log2>2(1/3+1/(3\cdot3^3)+1/(5\cdot3^5))>693/1000\). Substituting these lower bounds gives the displayed rational upper bound; the final cross-multiplied gap is \[ 10431\cdot167532267749 -1000\cdot1747479290400=49794489819>0. \] The series follows by integrating the geometric series on \([0,z]\). Because (N.clock) is uniform, increasing the stage threshold for each fixed finite extension ensures that every later reduced source satisfies the literal ordinary step bound \(\tau_1(x)\le(10431/1000)\log x\). \subsubsection{The all-scale physical source-family theorem}\label{the-all-scale-physical-source-family-theorem} \textbf{Theorem 5.5.} Suppose the mixing and finite-pool premises of (G.marks) hold with certified score \(W>0\), and the pool is a convergent antichain as specified at (G.terminalstate). For every \(\varepsilon>0\) and every fixed \(\Lambda>16\), one can choose one finite common graft and a threshold \(X_0\) such that, for every real \(X\ge X_0\), there are a finite set of odd sources \(S_X\), nonnegative weights \(a_x\), a level \(k(X)\), and a normalized original terminal mark \(\mathscr A_X\) satisfying \[ \begin{split} S_X&\subseteq[X,\Lambda X)\cap \{x:\tau_1(x)\le(10431/1000)\log x\},\qquad a_x\le1/x,\\ W-\varepsilon&\le\mathscr A_X\le\sum_{x\in S_X}a_xH_{k(X)}(x),\\ k(X)&\longrightarrow\infty,\qquad 3^{k(X)}/X\longrightarrow0. \end{split} \tag{N.sourcefamily} \] The fixed pool and \(W\) do not depend on \(\varepsilon,\Lambda,X\). The graft and its schedules are fixed before \(X\); its selected terminal stage and shift families may depend on \(X\). \emph{Proof.} Put \(a=\min\{\varepsilon/2,W/4\}>0\). Use (G.pool) to fix one common graft with central normalized marks at least \(W-a\) at every later stage. The reciprocal-root sum \(S=\sum_{M\in\mathcal R}1/M\) is fixed and finite. By (N.payment), \(Sd_Bc_0\), every \(\varepsilon>0\), and every fixed \(\Lambda>16\), the conclusion of (N.sourcefamily) holds with \(c\log x\) in place of \((10431/1000)\log x\). More precisely, the same fixed score \(W\) and root pool admit sources satisfying \[ \begin{split} S_X&\subseteq[X,R_XX)\cap\{x:\tau_1(x)\le c\log x\}, \qquad 0\le a_x\le1/x,\\ W-\varepsilon&\le\mathscr A_X\le \sum_{x\in S_X}a_xH_{k(X)}(x),\\ R_X&<\Lambda,\quad R_X\longrightarrow16,\quad k(X)\longrightarrow\infty,\quad3^{k(X)}/X\longrightarrow0. \end{split} \tag{N.everyclock} \] One finite graft and its schedules are fixed before all sufficiently large real \(X\). They may depend on \(c,\varepsilon,\Lambda\); the root pool and \(W\) do not. \emph{Proof.} We check each estimate affected by the new parameter. Fix a rational \(0<\theta\le1/1000\), put \(\delta=\theta/20\), choose rational centers (S.centers), and choose concentration constants \(C_{\delta/4},c_{\delta/4}\) at tolerance \(\delta/4\), with an integer \(L\ge1\) satisfying \(c_{\delta/4} L>10\). Use the decidable inner selectors (S.innercorridor) in the transition and macroblocks. Retain the stopped law, \(\beta=1/8\), global precisions (S.precision), and seed families. Use the same macro schedule with this \(L,\delta\), and replace every appended marker level by \(\lceil\theta B\rceil\). All these choices precede the splice. At a splice \(t\), retain \(E=\lfloor\sqrt{b_t}\rfloor\), \(B_0=\lfloor E/4\rfloor\), and \(Q=2b_t\), and take \(k_0=\lceil\theta B_0\rceil\). The transition rejection bound (G.transitionfailure) holds with the new concentration constants. Eventually \[ B_0\ge\sqrt{b_t}/16,\quad k_0\ge\theta\sqrt{b_t}/16\ge1,\quad B_0(E-1)+k_0\le Q. \] The unchanged incoming capacity is therefore available at \(Q\). The two substitutions and rejected mass in (G.transitiondebit) give \(D_t^{\rm tr}\le C_{\rm tr}(\theta)P_tb_t^{-3}\to0\), using the same exponential-series bounds as before. The offsets and slopes have not changed, so (G.offsetabsorption) still proves physical realization of every compatible branch at a sufficiently late splice. The new transition and macro corridors still sum to (G.totalcorridor). Since \(2\delta<1\), the appended tag count is at most \((B+1)^2\), and the full capacity remains (G.hybridcapacity), including its incoming multiplier \(P_t\). At a macro start put \(B'=B+\ell(B)\), \(k=\lceil\theta B\rceil\), \(k'=\lceil\theta B'\rceil\), and \(q(B)=k'+h_B\). The same depth bound gives \[ q(B)\le\theta B+4002(L+1)\log^2(B+2)\le B/2 \quad\bigl(B\ge[400000(L+1)]^2\bigr). \] Indeed \(\theta\le1/1000\), so the explicit boundary calculation following (G.tailconstants) applies. With \[ K_{\rm var}(\theta)=\frac83(C_*+1) \bigl(2C_6\theta^{-6}+L+1+C_{\delta/4}\bigr), \] the increment is at most \(K_{\rm var}(\theta)P_tB^{-4}\). Consequently the combined transition and continuation error is at most \[ \Delta_t(\theta)=D_t^{\rm tr} +3K_{\rm var}(\theta)P_tB_0^{-3}\longrightarrow0. \tag{N.paramgraft} \] The crucial incoming-factor estimate is unchanged: \[ P_tB_0^{-3}\le4096C_{\rm in}b_0^{-3/2}(t+1)^3 (g/\gamma^{3/2})^t\longrightarrow0, \qquad \gamma^3-g^2=16263/8000000>0. \] Thus one sufficiently late common splice meets any prescribed positive pool tolerance. The constants may grow as \(\theta\downarrow0\); only a fixed positive \(\theta\) is being used in this limit. Now take \(b=\lfloor\theta B\rfloor\), \(k=\lceil\theta B\rceil\), and \(K=e_b\) at the terminal stage. The state (G.terminalstate) holds with the new \(\delta\) and fixed prefix constants. Since \(\mu_Y-\delta>4\theta\), every central endpoint satisfies \(y\ge16^b\) eventually. Use the same terminal barriers, radii and common intervals (N.common). Their width is \[ D(B)-A(B)\ge \bigl((458/1000)\theta-2\delta\bigr)B-O(1) =\frac{179}{500}\theta B-O(1). \tag{N.paramroom} \] Floors and ceilings now give \(b'-b\le\theta\ell(B)+1\); the changes in both interval endpoints are still \(O_\theta(\log B)\). The positive linear width implies consecutive overlap, and both endpoints tend to infinity by the same bound on \(a(B)\). These intervals therefore cover a half-line at this fixed \(\theta\). The selected-shift partition still has at most \(2b+1\le B\) groups eventually. Since \(k\le b+1\), its level is \(q=h_b+k\le(13/5)b+1\le B/2\) eventually. Formula (N.payment) therefore holds literally with the new \(b,k\). Its right side is \(O_\theta(P_tB^{-3})\) plus polynomial factors times exponentials decaying in \(\theta B\): use \(k\ge\theta B\) and \(b\ge\theta B/2\) eventually. It tends to zero uniformly over the fixed pool and every cutoff in the selected interval. This retains both marker costs and the cost of the parent-dependent choice. Compression, its injection with a retained fan index, and (N.charge) are unchanged. The actual radius is still \(16/(1-8^{-b})\to16\). The ceiling bound \(k\le b+1\) gives (N.precision), so \(3^k/X\to0\) and \(k\to\infty\). The reduced source is still at least its parent eventually. Its terminal ordinary step cost is at most \((481/100)b\le(481/100)\theta B\), hence uniformly \[ \limsup\frac{\tau_1(x)}{\log x}\le c_\theta:={ \mu_R+\theta/20+(481/100)\theta\over (\log2)(\mu_Y-\theta/20)}. \tag{N.paramclock} \] Using (S.means) and \(\mu_Y\ge3\) yields \[ 02/7>3/11\) implies \(c_0<11\), and \(2/3<\log2<7/10\). The numerator of the last fraction is less than \(1049/200\), whereas three times its denominator exceeds \(6-1/10000>1049/200\). Choose the fixed rational \(\theta\) also to satisfy \(3\theta0\) be a fixed integer. Suppose a nonempty finite odd pool satisfies (G.marks) and (W.antichain), and each root has a specified finite ordinary path to \(y_\star\). Retain the same mixing hypothesis as in (N.everyclock). For every fixed \(c>c_0\), every \(\varepsilon>0\), and every \(\Lambda>16\), every conclusion of that theorem holds with \(\mathcal G_c(y_\star)\) in place of \(\mathcal G_c\): \[ \begin{gathered} S_X\subseteq[X,R_XX)\cap\mathcal G_c(y_\star),\qquad 0\le a_x\le1/x,\\ W-\varepsilon\le\mathscr A_X\le\sum_{x\in S_X}a_xH_{k(X)}(x),\\ R_X\to16,\qquad k(X)\to\infty,\qquad3^{k(X)}/X\to0. \end{gathered} \tag{N.targetfamily} \] The pool, target and score precede \(c\); the graft and cutoff may depend on \(c,\varepsilon,\Lambda\). No convergence of \(y_\star\) is assumed. \emph{Proof.} The seed realization, variation and graft proofs are uniform for all odd roots above \(16^{b_0}\). Let \(T_{\mathcal R}\) now be the maximum of the specified root-to-target path lengths. Concatenation gives \(\tau_{y_\star}(x)\le A+d+T_{\mathcal R}\) for a word of total valuation \(A\) and length \(d\). Thus the second line of (G.terminalstate) and (N.central) holds with \(\tau_{y_\star}\) and this finite additive constant; all source-size bounds are unchanged. The terminal selector, shift partition and two marker error bounds depend on those size bounds and incoming capacity. Their proofs are unchanged. Nonrecurrence gives a unique root visit and hence the same complete valuation word and deterministic cuts. After the same compression identification, (W.poolcharge) supplies (N.charge); the finite-image injection still supplies (N.fancharge). The reduced source is at least its parent eventually, and its terminal ordinary cost is still bounded by \((481/100)\theta B\). Consequently (N.paramclock) holds for \(\tau_{y_\star}\), with precisely the same \(c_\theta\). Choose \(3\theta0\) gives \(c_\theta-c_0>0\) in (N.parammargin). Taking \(\theta\) smaller controls this upper-bound gap but does not remove it. Setting \(\theta=0\) would instead give \(b=k=0\), outside the terminal construction, and \(\delta=0\), outside its positive-tolerance concentration estimates. It also removes the positive linear width in (N.paramroom), the parameterized form of (N.room), which makes consecutive cutoff intervals overlap. The graft constants contain \(\theta^{-6}\), and the limiting arguments fix \(\theta\) before the splice and cutoff. A shrinking-parameter construction would require additional uniform control of these errors, coverage and clock slack. The common lower-density bound for \(c>c_0\) does not supply that control or pass automatically to \(c=c_0\). The nonquantitative diagonal consequence (D.vanishingclock) does give an unspecified vanishing clock loss; it requires no uniform shrinking-parameter estimates. \subsubsection{From source families to density}\label{from-source-families-to-density} The result above gives the precise interval coverage, aggregate source weights and ordinary clock needed for counting. Its mixing and persistent-mark premises are inherited through (G.marks). We now establish those premises, then return to this source family to derive the explicit density bound. \subsection{6. Explicit reference mixing from primitive Fourier decay}\label{chapter06} The construction compares reference densities at different ternary levels. We now bound the error in replacing a finer distribution by the uniform lift of a coarser one. The explicit Fourier-decay theorem proved in Appendix E supplies the input. The Fourier decomposition below gives a bound valid at every finer level; we then optimize its auxiliary parameter. This error bound controls both the continuing seed construction and the final conversion from marked mass to source weight. \subsubsection{The primitive theorem and the quantitative target}\label{the-primitive-theorem-and-the-quantitative-target} Use the reference law \(\mu_n\) of (T.law). For a function \(p:G_n\to\mathbb C\), set \[ \widehat p(\xi)=\sum_{x\in G_n}p(x)e^{2\pi i\xi x/3^n}. \] The Fourier transform is unnormalized. A frequency \(\xi\in G_n\) is primitive when \(3\nmid\xi\). We first work with any fixed real \(C\ge1\) satisfying the following input: \[ |\widehat{\mu_n}(\xi)|\le C n^{-6409} \quad(n\ge1,\ \xi\in G_n,\ 3\nmid\xi). \tag{M.primitive} \] The canonical numerical choice of \(C\) and the local theorem asserting this input are given in the final subsection. The proofs up to that instantiation are uniform in every such \(C\). Write \(h=\log2\), \(\gamma=\log(4/3)\), \(\lambda=\log3/\log2\), and fix \[ D=C20^{6409},\quad C_6=2D+2,\quad m_0=2^{80},\quad A=2314/25. \tag{M.constants} \] For real \(80\le v\le679/5\), put \[ a(v)=6407-\tfrac12hv^2,\qquad q_v=hv, \] \[ E_v(x)=2D x^{-a(v)} +\frac{32}{q_v^2\log x}x^{1-q_v} +2x^{-q_v}+2x^{-q_v-1}\qquad(x\ge m_0). \tag{M.widthbound} \] The target is, for every integer \(1\le m\le n\), \[ \operatorname{Osc}(m,n):= \sum_{x\in G_n}|\mu_n(x)-3^{m-n}\mu_m(x\bmod3^m)| \le\epsilon_m, \tag{M.target} \] \[ \epsilon_m= \begin{cases} \min\{2,C_6m^{-A}\},&m4\) and \[ ((3/2)v^2+v)L\le(3/200)n,\qquad ((5/2)v^2+v)L\le(3/10)n. \tag{M.logbudgets} \] Also \(1<\lambda<8/5\), \(693/10002/7\); the upper bound follows from \(\log(1+1/3)<1/3\). Finally \(3>2\) and \(3^5<2^8\) give the bounds on \(\lambda\). For \(x>0\), \(\log x\le\sqrt x\). Indeed, with \(z=\sqrt x/2\), \(\log z\le z-1\) and \(\log2\le1\) imply this inequality. For \(n\ge2^{80}\), therefore \(L\le\sqrt n\) and \(\sqrt n\ge2^{40}\). Both coefficients in (M.logbudgets) increase with \(v\). The exact comparisons \[ (3/2)136^2+136<(3/200)2^{40},\qquad (5/2)136^2+136<(3/10)2^{40} \] prove the result. Also \(L\ge80h>4\). \(\square\) \textbf{Lemma 6.2 (head location).} Every globally typical word has a unique head in the families above. Every nonempty head family satisfies \[ 20k\le17n,\qquad Q0\). The inequality \[ v\sqrt{sL}\le s/10+(5/2)v^2L \tag{M.youngindex} \] follows by completing a square. Applied with \(s=n\), typicality and (M.logbudgets) give \(S_n\ge(8/5)n>Q\). Thus there is a unique first crossing at a depth \(k+1\le n\), and its prefix is typical. For a nonempty head, typicality of its prefix of length \(k\) and \(S_k\le Q\) give, when \(k>0\), \[ (19/10)k\le\lambda n+((3/2)v^2+v)L <(323/200)n. \] Hence \(20k<17n\). The case \(k=0\) also satisfies the asserted bound. The last valuation is at most \(2+v(\sqrt L+L)\le2vL\): use \(\sqrt L\le L/2\), \(L\ge4\), and \(v\ge17\). Consequently \(l\le Q+2vL=\lambda n-v(v-2)L<2n\). This proves both coverage and the stated finite index ranges. \(\square\) \textbf{Lemma 6.3 (small cleared offset).} Every head satisfies \[ 01/14\) and \(e^{1/14}\ge1+1/14>16/15\) suffice. With \(l\le\lambda n-v^2L+2vL\), it follows that \[ B(w)<3^n\exp\{Q_*(v)L+K_*\}, \] \[ Q_*(v)=v^2\left(-h+\frac{h^2}{3\gamma}\right)+3hv, \qquad K_*=-2h+3\gamma/4+16. \tag{M.headmargin} \] The bounds in (M.logbounds) give \[ -h+\frac{h^2}{3\gamma}<-\frac{196637}{1500000}, \qquad Q_*(v)<-\frac{196637}{1500000}v^2+\frac{1041}{500}v. \] The displayed quadratic is decreasing for \(v\ge80\) and is less than \(-672\) at 80. Also \(K_*<17\). Since \(L>4\), the exponent is negative. Positivity of every summand proves the lower bound. \(\square\) \textbf{Lemma 6.4 (head injection and its second moment).} At fixed \(d,l\), the map from a head to \(C(w)\bmod3^n\) is injective. Let \(p_{k,l}\) be the nonnegative sub-probability mass function on \(G_n\) obtained by mapping the \((k,l)\) head family with its original iid weights. Then \[ \sum_{y\in G_n}p_{k,l}(y)^2\le2^{-l}. \tag{M.headL2} \] \emph{Proof.} We first show that \(B(w)\), together with \(d,l\), determines the word. For \(d=1\), its only valuation is \(l\). For \(d\ge2\), \[ B(w)-3^{d-1}=2^{k_d}B(k_1,\ldots,k_{d-1}). \] The shorter cleared offset is odd: its last summand is odd and every earlier summand is even. Thus \(k_d\) is exactly the 2-adic valuation of the positive left side. Divide by \(2^{k_d}\), reduce the total from \(l\) to \(l-k_d\), and induct. Equality of the source residues, multiplied by the common unit \(2^l\), gives equality of the cleared offsets modulo \(3^n\). By (M.headsmall) they are equal as integers, and the preceding decoding applies. Each accepted head has probability exactly \(2^{-l}\). Injection makes every occupied atom of \(p_{k,l}\) equal to \(2^{-l}\); hence its squared sum is \(2^{-l}\sum_y p_{k,l}(y)\le2^{-l}\). Empty families give zero. \(\square\) \subsubsection{Tail levels and the squared Fourier estimate}\label{tail-levels-and-the-squared-fourier-estimate} For any real vector \(p\) on \(G_n\), define its uniform fiber average and its oscillation by \[ (\mathsf A_{m,n}p)(x)=3^{m-n} \sum_{y\equiv x\ (3^m)}p(y),\qquad \mathsf O_{m,n}(p)=\|p-\mathsf A_{m,n}p\|_1. \tag{M.average} \] Projectivity makes \(\mathsf O_{m,n}(\mu_n)=\operatorname{Osc}(m,n)\). The averaging map is positive, mass-preserving and L1-contracting. For real vectors \(p,t\) and convolution \((p*t)(x)=\sum_y p(y)t(x-y)\), the following finite identity is the precise use of Fourier cancellation. \textbf{Lemma 6.5 (finite collision bound).} If \(1\le m\le n\) and \(|\widehat t(\xi)|\le\delta\) at every frequency not divisible by \(3^{n-m}\), with \(\delta\ge0\), then \[ \mathsf O_{m,n}(p*t)^2\le \delta^2 3^n\sum_y|p(y)|^2. \tag{M.collision} \] \emph{Proof.} The geometric character sum on a fiber shows that the Fourier transform of \(\mathsf A_{m,n}\) keeps exactly the frequencies divisible by \(3^{n-m}\), and sets all others to zero. Orthogonality of the same characters gives Parseval in the present normalization: \(\sum_\xi|\widehat p(\xi)|^2=3^n\sum_y|p(y)|^2\). Also \(\widehat{p*t}=\widehat p\,\widehat t\), by changing variables in the finite double sum. If \(z=p*t-\mathsf A_{m,n}(p*t)\), these identities and Cauchy--Schwarz give \[ \|z\|_1^2\le3^n\sum_y|z(y)|^2 =\sum_{3^{n-m}\nmid\xi}|\widehat p(\xi)\widehat t(\xi)|^2 \le\delta^2\sum_\xi|\widehat p(\xi)|^2. \] This proves the claim, including the empty discarded-frequency set when \(m=n\). \(\square\) Now assume \(9n\le10m\le10n\) and consider a nonempty \((k,l)\) head family. Its length \(d=k+1\) satisfies \(d\le m\): by (M.headlocation), \(m-k\ge n/20\ge1\). Set \(T=n-d\). Let \(t_{k,l}\) be the law on \(G_n\) of \(3^d2^{-l}C_T\), using the remaining independent \(T\) valuations. The selected full-word submass is exactly \[ c_{k,l}=p_{k,l}*t_{k,l}. \tag{M.convolution} \] Indeed, (M.split) has this form, and the tail is unrestricted and independent of the head; all heads in this slice have the same \(d,l\). For a discarded frequency, choose its nonzero representative and put \(t=\nu_3(\xi)Q=\lambda n-v^2L\) gives \(3^n2^{-l}0\), \(F(h)=0\), and \(F''(\theta)=-e^\theta(c+1-c\theta)<0\). Concavity proves \(F\ge0\), hence \(g''\le H''\). Integrate twice from zero. For the other sign, \(g''(-\theta)\le2\) for \(\theta\ge0\); the same integration gives \(g(-\theta)\le\theta^2\). \(\square\) \textbf{Lemma 6.8 (two-sided sum bound).} For \(r\ge1,z>0\) and independent copies \(X_1,\ldots,X_r\), \[ \Pr\{|\textstyle\sum_{i=1}^rX_i| \ge2\sqrt{rz}+cz\}\le2e^{-z}. \tag{M.chernoff} \] \emph{Proof.} For either sign apply the exponential Markov inequality at \(\theta=s/(1+cs)L\), use \(z=(q+2)L\). Its threshold is at most \(v(\sqrt{rL}+L)\), because \[ v\ge2\sqrt{q+2}+2c. \] To check this uniformly, use \(h<7/10\), \(c<3/2\); \((v-3)^2>4((7/10)v+2)\) at \(v=80\), with positive derivative thereafter. Each long interval therefore fails with probability at most \(2n^{-q-2}\). Bounding their number by the total \(n(n+1)/2\) gives the remaining two terms. The strict failure events are contained in the non-strict tail events just estimated. \(\square\) Combining (M.high) and (M.failure) gives \[ \operatorname{Osc}(m,n)\le E_v(n) \quad(n\ge m_0,\ 9n\le10m\le10n). \tag{M.localmixing} \] \subsubsection{From nearby levels to the full compact envelope}\label{from-nearby-levels-to-the-full-compact-envelope} For \(m\le r\le n\), projectivity and uniform lifting give \[ \operatorname{Osc}(m,n)\le \operatorname{Osc}(m,r)+\operatorname{Osc}(r,n). \tag{M.triangle} \] To see the normalization, insert \(3^{r-n}\mu_r\) between the two vectors defining \(\operatorname{Osc}(m,n)\). The second difference is the uniform lift of \(\mu_r-3^{m-r}\mu_m\), whose L1 norm is unchanged. Every power appearing in (M.widthbound) exceeds 8 throughout the width range. Indeed \(a(v)\ge a(679/5)>8\) by (M.logbounds), and \(q_v-1\ge80H_--1>8\). The positive terms are decreasing in \(x\). Since \((11/10)^8>2\), \[ 2E_v((11/10)x)90\), \(r\ge(11/10)m>m\). If \(n\le r\), (M.localmixing) and monotonicity give \(E_v(n)\le E_v(m)\). Otherwise \(m2314/25,\qquad v_*H_--1>2314/25. \tag{M.exponentguards} \] For \(m\ge m_0\), (M.allhigh) therefore implies \[ \operatorname{Osc}(m,n) \le m^{-A}\left(2D+\frac{32}{q_{v_*}^2\log m} +\frac2m+\frac2{m^2}\right) <(2D+2)m^{-A}=C_6m^{-A}. \] The bracket after \(2D\) is less than two already with \(q_{v_*}>80(693/1000)>55\), \(\log m>4\), and \(m\ge2^{80}\). For \(m2\,20^{6409}>2^{25637}>2^{8001}> 2(2^{80})^A. \] Thus \(2C_6. \tag{M.sourceadapter} \] The strict comparison follows from (E.coefficientcomparison). Section 11 compares against this fixed coefficient. Appendix A compares fixed-depth and first-crossing seeds at a common primitive coefficient. Appendix E uses the pairing and separated-triangle method of {[}1, Section 7{]} and {[}3{]}. The preceding argument converts its coefficient into the envelope (M.envelope), used by the seed and counting constructions. \subsection{7. Seed marks and variation bounds uniform over roots}\label{chapter07} We return to the seed families defined in the continuation construction. Their reference average can be evaluated exactly, but an average at one stage does not guarantee that any fixed root keeps a positive mark later. Using the preceding mixing estimate, we bound the total change at each root over all subsequent stages. The next section combines this variation bound with positive reference survival to choose the required fixed roots. \subsubsection{The finite mark and its exact normalization}\label{the-finite-mark-and-its-exact-normalization} Fix an integer \(b=b_0\ge32^5\). Use the fixed-depth seed families (G.fixedseed), with \[ b_{j+1}=b_j+\lfloor b_j/100\rfloor,\quad m_j=\lceil b_j^{3/5}\rceil,\quad a_j=\lfloor b_j/4\rfloor. \tag{B.parameters} \] Thus \(V_j=\{w:|w|=b_j,\ |A(w)-2b_j|\le m_j\}\). Set \[ p_j=\sum_{w\in V_j}2^{-A(w)},\qquad D_N=\sum_{jm\) is contained in the union of these two events. In particular \[ 1-p_j\le2\exp\!\left(-\frac{m_j^2}{4b_j+3m_j}\right) \le2\exp\!\left(-\frac{b_j^{1/5}}{4+3/512}\right). \tag{B.failure} \] The positive-tail denominator includes the geometric MGF's \(3m\) correction. No stopping-window or overshoot error term is needed for this seed. \subsubsection{The increment before simplifying its constants}\label{the-increment-before-simplifying-its-constants} All reference mixing below uses the fixed coefficient \(C_6\) from (M.constants), with (M.powers) at order 92. A generic order-six coefficient is not sufficient. At stage \(j\), put \(B=b_j\) and \[ \ell=a_j=\lfloor B/4\rfloor,\quad k=a_{j+1}=\lfloor(B+\lfloor B/100\rfloor)/4\rfloor, \quad q=B+k. \tag{B.incrementlevels} \] Both marker levels are at least one and \(k\ge\ell\ge B/8\). For the last inequality, \(\lfloor B/4\rfloor\ge B/4-1\ge B/8\) when \(B\ge8\). Moreover \[ \ell\le q,\qquad q\le\left(1+\frac{101}{400}\right)B =\frac{501}{400}B<2B. \tag{B.paidconductor} \] Thus the comparison level stays within (G.incapacity)'s allowance. Define \(L=L_{V_j,q,k}\) by (T.selected), and let \(H^{\rm in}_{j,q}\) be the actual incoming histogram of (G.incapacity). The same root-independent family \(V_j\) is applied to each parent. Its affine compatibility is exactly physical realization: (G.prefix) puts that parent at least at \(16^B\), so every compatible extension is positive. Uniqueness of the prescribed cut and multiplication of the slope weights then give the exact signed identity \[ Z_{j+1}(M)-Z_j(M)= \sum_{y\in G_q}H^{\rm in}_{j,q}(y) [L(y)-\rho_\ell(\pi_{q,\ell}y)]. \tag{B.incrementidentity} \] There is no conditioning or division by retained mass in this formula. Combining (T.stopped), (G.incapacity) and (M.powers) proves, simultaneously for every positive odd \(M\ge16^b\) and every \(j\ge0\), \[ |Z_{j+1}(M)-Z_j(M)|\le\frac23P_j \left(\frac{C_6}{k^{92}}+\frac{C_6}{\ell^{92}}+1-p_j\right). \tag{B.rawincrement} \] In particular this retains the actual incoming tag factor \(20(4b+1)(j+1)^3(2013/2000)^j\). Its capacity coefficient is \[ P_j=20(4b+1)(2^{b+1}+16^b)(j+1)^3(2013/2000)^j. \] The \(16^b\) term bounds the finite-modulus boundary error \(3^q\omega\); it has not been discarded as a negligible asymptotic term. The two mixing costs and the valuation-window failure are three separate costs in (B.rawincrement). \subsubsection{An every-generation majorant with the retained coefficient}\label{an-every-generation-majorant-with-the-retained-coefficient} Put \[ \alpha=100999/100000,\quad g=2013/2000,\quad \lambda_{\rm var}=g\alpha^{-92},\quad S_{92}=8\cdot465!\cdot576^{465}. \tag{B.rates} \] For every integer \(B\ge100000\), \[ B+\lfloor B/100\rfloor\ge(101/100)B-1 \ge\alpha B. \] All floors in (B.parameters) meet this condition, so \(b_j\ge b\alpha^j\). There is no omitted early-stage exception. Since \(4+3/512<2333/100\), (B.failure) is at most \(2e^{-b_j^{1/5}/(2333/100)}\). For \(B\ge1\), use \(2333/100<576\), followed by the degree-465 exponential-series term, to get \[ 2e^{-B^{1/5}/(2333/100)} \le8B e^{-B^{1/5}/576} \le8\cdot465!\,576^{465}B^{1-465/5} =S_{92}B^{-92}. \tag{B.polynomialfailure} \] Also \(465!<512^{465}\) and \(576<1024\), whence \(S_{92}<2^{8838}\). The actual coefficient in \hyperref[chapter06]{the reference-mixing section} satisfies \(C_6>2^{25637}\), so \(S_{92}2^{8838}\) for which (T.mix) holds with \[ \widehat\epsilon_m=\min\{2,\widehat C m^{-2314/25}\}. \tag{B.coefficientscope} \] Indeed this supplies both the order-92 and order-six estimates, and \(S_{92}<2^{8838}<\widehat C\) supplies the coefficient in (B.onerate). The remaining bounds are the same scalar inequalities. With \(\widehat F_b=2^{467}b16^b(\widehat C+1)\), the survival theorem in the next section is unchanged, its variation tail still tends to zero, and its finite startup and root-pool recipes give a positive score. The startup inequalities use \(t=\lceil\log_2\widehat F_b\rceil \ge4b\), which is retained. The sufficient condition \(\widehat C>2^{8838}\) is used through \(S_{92}<\widehat C\) in (B.onerate). The startup bound \(t\ge4b=2^{27}>25637\) holds for every \(\widehat C\ge1\). Thus the finite startup argument applies throughout this coefficient range; the explicit construction takes \(\widehat C=C_6\). \subsection{8. Infinite survival and a fixed persistent root score}\label{chapter08} The preceding section bounded the total future variation of the seed marks. We now show that their reference average stays above that loss after a fixed initial stage. A complete finite period of known convergent roots realizes the reference average. Allocating the remaining mass among these roots gives a positive lower bound for their reciprocal-root weighted sum, called the \emph{root score}. This establishes (G.marks), the premise needed by the finite continuation and terminal-source constructions. The survival estimate below uses a summable fixed-time tail and small rational inequalities. \subsubsection{A short proof of reference survival}\label{a-short-proof-of-reference-survival} Fix \(b=32^5=2^{25}\), retaining the fixed-depth stage families, success masses \(p_j\), floors \(b_j\) and marks of Section 7. Put \[ \alpha=\frac{100999}{100000},\qquad r=\frac{504}{503},\qquad u=\frac{511}{64},\qquad d_j=2e^{-ur^j}. \tag{R.deficits} \] The recurrence gives \(b_j\ge b\alpha^j\), and the exact comparisons \[ 100999\,503^5-100000\,504^5=32325056940257>0,\qquad \frac{32}{4+3/512}=\frac{16384}{2051}>\frac{511}{64} \tag{R.survivalguards} \] show that (B.failure) implies \(0\le1-p_j\le d_j\). The function \(f(t)=e^{-ur^t}\) is decreasing on \([0,\infty)\). Thus \(\sum_{j\ge0}f(j)\le f(0)+\int_0^\infty f(t)\,dt\). Substitution \(x=ur^t\), followed by \(\log r>1/504\), gives \[ \sum_{j\ge0}d_j <2e^{-u}\left[ 1+504\left(\frac1u-\frac1{u^2}+\frac2{u^3}\right)\right]. \tag{R.survivalsum} \] Here \(\log(1+x)>x/(1+x)\) for \(x>0\) follows by integration of \(1/(1+t)\). For the integral estimate used in (R.survivalsum), write \[ \int_u^\infty\frac{e^{-x}}x\,dx =e^{-u}\int_0^\infty\frac{e^{-t}}{u+t}\,dt \le e^{-u}\left(\frac1u-\frac1{u^2}+\frac2{u^3}\right). \] The pointwise inequality \(1/(u+t)\le1/u-t/u^2+t^2/u^3\) holds for \(t\ge0\); multiplication by \(u+t\) leaves the nonnegative remainder \(t^3/u^3\). Integrating \(1,t,t^2\) against \(e^{-t}\) gives \(1,1,2\). The required numerical comparisons are: \[ \frac1u-\frac1{u^2}+\frac2{u^3} =\frac{15142976}{133432831}<\frac{227}{2000}, \qquad e^u>2920. \tag{R.exponentialguard} \] For the second use \(u=8-1/64\), \[ e^8>\sum_{k=0}^{16}\frac{8^k}{k!} =\frac{70233453593}{23648625},\qquad e^{-1/64}>63/64, \] and \[ 63\cdot70233453593 -2920\cdot64\cdot23648625=5252536359>0. \] Consequently \[ \sum_{j\ge0}(1-p_j) <\frac2{2920}\left(1+504\frac{227}{2000}\right) =\frac{14551}{365000}<\frac1{25}. \tag{R.failurebudget} \] For a finite product, \(\prod_{j\Pi\ (N\ge0),\qquad \prod_{j\ge0}p_j\ge\Pi.} \tag{R.certificate} \] This proves survival from the displayed rational inequalities. The finite reference-moment tables are separate inputs to the quantitative density formula. \subsubsection{Complete variation tail and earliest positive startup}\label{complete-variation-tail-and-earliest-positive-startup} For the canonical score use \(C_6\) from \hyperref[chapter06]{the reference-mixing section}. The qualitative alternative (B.coefficientscope) may replace it throughout this section by \(\widehat C\), retaining both the order-92 estimate and its scalar size condition. A coefficient supplying only order six does not suffice. Define \[ F_b=2^{467}b16^b(C_6+1),\quad t=\lceil\log_2F_b\rceil,\quad L=\lceil\log_2(t+1)\rceil, \] \[ a=2013\,100000^{92},\quad d=2000\,100999^{92},\quad \lambda=a/d,\quad V(n)=2^tT_3(n,\lambda),\quad T_3(n,z)=\sum_{j\ge n}(j+1)^3z^j. \tag{R.variationdata} \] Thus \(t\) is an integer, \(F_b\le2^t\), \(t\ge4b=2^{27}\), and \(t<2^L\). The canonical arithmetic specification of \(C_6\) makes \(t\) an effective integer; it is not necessary to print its expansion. By (B.tail), for every admissible root \(M\) and every \(j\ge n\), \[ |Z_j(M)-Z_n(M)|\le V(n). \tag{R.latermarks} \] For \(00\). The rational inequalities (B.ratebounds) give \(2/5<\lambda<41/100\). The additional finite integer comparison is \[ 2^{131}2013^{100}100000^{9200} <2000^{100}100999^{9200},\qquad \lambda^{100}<2^{-131}. \tag{R.hundredrate} \] If \(n\ge16\) and \(00 \tag{R.residual} \] make sense, with \(16\le N\le N_{\rm fixed}-1\). Strict monotonicity of \(V\) permits rational binary search on this finite interval. All values below 16 fail the positivity predicate: the tail then contains \(17^3\lambda^{16}>17^3 2^{-32}\), and multiplication by \(2^t\) already exceeds one. This defines the startup by a finite integer recipe. The exact value \(\Pi=24/25\), and then the whole residual \(P\), are retained. For later quantitative comparisons one needs more than \(P>0\). Let \(e=d-a>0\), and for any integer \(n\ge0\) define \[ U_n=d\big[(n+1)^3e^3+3(n+1)^2ae^2+ 3(n+1)a(d+a)e+a(d^2+4ad+a^2)\big]. \] Substitution into (R.exacttail) gives \(T_3(n,\lambda)=a^nU_n/[d^n(d-a)^4]\). If \(P(n)=\Pi-V(n)>0\), its numerator over the common denominator \(25d^n(d-a)^4\) is a positive integer. Because \(d<2^{11+17\cdot92}=2^{1575}\), \[ P(n)\ge[25d^n(d-a)^4]^{-1} >2^{-1575n-6305}>2^{-1575n-7324}. \tag{R.denominator} \] Here \(25<2^5\); the final weaker bound is retained for the startup comparison. No reduced-denominator calculation or lower bound on the numerical distance to a threshold is assumed. The integer \(2^t\) adds no denominator. \subsubsection{The fixed finite convergent root pool}\label{the-fixed-finite-convergent-root-pool} Write \(q=q_N\) as in (B.conductor), and let \[ H=2b+1,\quad \mathcal S_q=\{s\in\mathbb Z:H\le s16^b>1\). Moreover \(3R_s+1=4^s\), so its next accelerated odd iterate is one, and its ordinary stopping time is \(1+2s\). The root pool and its maximum clock are finite and fixed before any physical counting cutoff. Set \(g_s=g_N(R_s\bmod3^q)\). By (B.mean), (B.cap), (B.physical), and the reference survival bound, \[ 0\le g_s\le B_q,\quad Z_N(R_s)=g_s,\quad \sum_{s\in\mathcal S_q}g_s =K_q\prod_{j0\), \(W(q,p)\ge K_qp/\max_s R_s>0\). For the following evaluation only, relabel the roots in decreasing order as \(R_1>\cdots>R_{K_q}\). Write \(h=\lfloor K_qp/B_q\rfloor\), \(r_0=K_qp-hB_q\). Since \(p\le14^\Delta R_s\). Compare this feasible upper bound with \(W(q,p)\ge K_qp/\max_{\mathcal S_q}R_s\). Iterating gives, for \(Q>q\ge2\), \[ W(q,p)> (4/3)^{Q-q}2^{\,3^Q-3^q}W(Q,p). \tag{R.conductorgain} \] This comparison uses the exact cap and minimum-allocation definition, rather than assuming that a smaller root pool automatically helps. By (R.persistence), the actual lower vector \((z_s)\) is feasible in (R.score). Thus \[ 0c_0\), the same score, aggregate source-weight bound and coverage at every large real scale with clock \(c\log x\). The order is: canonical constants, certificate, startup, pool and score; then \(c\), a positive terminal ratio, a positive graft tolerance and one finite splice; finally every sufficiently large cutoff. No root or seed precision is selected afresh as a function of that cutoff. The terminal stage and level may vary along the fixed graft schedule, as in (N.sourcefamily). \subsubsection{General residue-complete blocks and their common score}\label{general-residue-complete-blocks-and-their-common-score} Let \(\mathcal F\) be a nonempty finite collection of disjoint root blocks. Fix \(q\ge2\). Each block has one odd root \(M_i(r)\ge16^b\) in every unit residue \(r\bmod3^q\). Put \[ \mathcal U_q=(\mathbb Z/3^q\mathbb Z)^\times,\qquad A_{\mathcal F}(r)=\sum_i\frac1{M_i(r)},\qquad \mathcal V(q,p)=\{v:0\le v_r\le B_q,\ \sum_rv_r\ge K_qp\}, \] \[ W_{\mathcal F}^{\rm joint}(q,p) =\min_{v\in\mathcal V(q,p)}\sum_rv_rA_{\mathcal F}(r),\qquad W_i(q,p)=\min_{v\in\mathcal V(q,p)}\sum_r\frac{v_r}{M_i(r)}. \tag{R.jointscore} \] Here \(0\alpha^{-1}W(q,p)\quad(p>0). \tag{R.blockcomparison} \] Indeed \(M_s<\alpha R_s\) pointwise. Evaluate the strict reciprocal inequality at a minimizer for this block; at least one coordinate has positive mass. The baseline-root objective of that vector is at least \(W(q,p)\). This argument uses the common cap and total mass and does not assume invariance of \(g_N\) under the residue permutation. A proved upper coefficient \(\alpha\le\beta<1\) gives the weaker factor \(\beta^{-1}\). These finite comparisons will be used in Appendix C. \subsubsection{Quantitative score bounds and the fixed-recipe optimum}\label{quantitative-score-bounds-and-the-fixed-recipe-optimum} For \(p>0\), every root in \(\mathcal S_q\) is less than \(4^{H+3^q}/12\). Hence the required total mass gives \[ W(q,p)>8p\,3^q4^{-(H+3^q)}. \tag{R.scorelower} \] For \(q\ge3\) and \(018\), use \(3^q<2^{2q}\) and \(3^{18}-36=387420453\) to obtain \[ 0N\) and any \(03^q-2H+3-\log_2 3-1575N-7324\\ &>3^q-17q-7337\ge12193. \end{split} \tag{R.startupoptimum} \] Here \(1575N\le q\) since \(b>1575\). The last function equals 12193 at \(q=9\) and has positive successive increments \(2\,3^q-17\); all actual levels exceed nine. Earlier startups leave no positive guaranteed budget under this same majorant. At \(N\), (R.residual) is the largest budget it allows, and fixed-level score monotonicity shows that no smaller budget improves the score. Every later startup loses even if allowed budget one. Thus the selected startup and its full residual uniquely optimize this minimum-score recipe. This does not optimize the underlying variation coefficient, reference family, actual marks, or alternative root pools. The rational residual condition and large-level condition are essential: an arbitrary tiny real budget in an earlier small pool need not beat a later pool. The coefficient dependence of the startup, residual, and root score is proved in \hyperref[appendix_provenance]{Appendix B}. \subsubsection{The score used in counting}\label{the-score-used-in-counting} The fixed-depth survival proof, complete variation bound and earliest positive allocation provide the score used below. They use the affine transfer, root-period, capacity and finite-graft estimates proved in the preceding sections. Appendix A compares this score with the central first-crossing benchmark. The reproducibility archive contains the rational survival and rate checks. The all-stage estimates use the primitive coefficient (M.canonicalC); (B.coefficientscope) gives the variant with a general mixing coefficient. We next estimate the reference moments used to convert the score into an ordinary counting bound. \subsection{9. Reference moments and their finite certificates}\label{chapter09} We now bound the concentration of the reference fan by its maximum, variance, and average power of order \(3/2\). The counting section will use these three bounds to turn a guaranteed marked sum into a lower bound on total source weight. All moments concern the auxiliary law (T.law). A product inequality propagates finite moment bounds to every level, so finite certificates suffice for the estimates used at arbitrarily large moduli. \subsubsection{Orientation and a conditional affine mixture}\label{orientation-and-a-conditional-affine-mixture} Let \(T_a(x)=(3x+1)/2^a\), interpreted in the appropriate ternary ring. The composition \(T_{a_n}\cdots T_{a_1}(0)\) is \[ \sum_{i=1}^n3^{n-i}2^{-(a_i+\cdots+a_n)}. \] Reversing a finite iid word turns this into the offset sum in (T.law), without changing its probability \(2^{-\sum a_i}\). Thus the forward random affine construction and the inverse-prefix construction have the same law at every level. The physical histories retain the orientation fixed in (W.inverse). For \(p\ge1\) put \[ M_p(n)=\langle f_n^p\rangle_n,\qquad A_n=\max_{G_n}f_n. \tag{F.norms} \] All averages in this section are over the full group, including zero and the nonunits. We have \(M_1(n)=1\), \(M_p(0)=A_0=1\). \textbf{Theorem 9.1 (moment inheritance).} For integers \(u,v\ge0\) and real \(p\ge1\), \[ M_p(u+v)\le M_p(u)M_p(v),\qquad A_{u+v}\le A_uA_v. \tag{F.inheritance} \] \emph{Proof.} Zero-length blocks give equality. Otherwise condition on the last \(v\) forward valuations, a word \(w\) with probability \(\theta_w\). Its affine action is \(b_w+3^vc_wx\), where \(c_w\) is a ternary unit. For \(y\in G_v\), write the lifts as \(y+3^vz\), \(z\in G_u\). For each word with \(b_w\equiv y\pmod{3^v}\), the equation for the final residue determines the preceding residue as an affine permutation \(x=\psi_w(z)\) of \(G_u\). Independence of the two reference blocks gives the exact identity \[ f_{u+v}(y+3^vz)=3^v \sum_{w:b_w\equiv y\ (3^v)}\theta_w f_u(\psi_w(z)). \tag{F.mixture} \] The nonnegative coefficients sum to \(f_v(y)\). If this sum vanishes, the whole fiber vanishes. Otherwise divide by it and apply Jensen's inequality to the probability mixture. Every \(\psi_w\) preserves the full average, so averaging first over \(z\) and then over \(y\) gives the first assertion. Bounding each \(f_u\) by \(A_u\) instead gives \(f_{u+v}(y+3^vz)\le A_uf_v(y)\), proving the second. Countably many words cause no problem: the functions on each finite group are bounded, and the coefficients have finite sum; Jensen follows by truncation and one remainder atom, or directly for a countable probability distribution. \(\square\) For a probability law on a finite group, its collision probability is the sum of the squared point probabilities: this is the chance that two independent samples agree. In particular the normalized collision energy is \(M_2(n)=3^n\sum_y\mu_n(y)^2\), and is submultiplicative. General \(p\) matters: interpolation between the maximum and this second moment does not supply the smaller direct \(p=3/2\) table below. The general convex-order inequality and the divergence consequences of moment inheritance are collected in \hyperref[appendix_reference_consequences]{Appendix D}. The moment inequality above is all that is needed for the finite certificates and their propagation below. \subsubsection{The whole affine fan and the exact coset split}\label{the-whole-affine-fan-and-the-exact-coset-split} Conditioning on the last forward valuation gives \[ f_{n+1}(y)=3\sum_{\substack{a\ge1\\2^ay\equiv1\ (3)}} 2^{-a}f_n\left((2^ay-1)/3\bmod3^n\right). \tag{F.transfer} \] The divisibility condition and the modulus \(3^{n+1}\) on \(y\) make the argument well defined. Define \[ F_j(x)=4^jx+(4^j-1)/3,\qquad H_m(x)=\sum_{j\ge0}4^{-j}\rho_m(F_j(x)),\qquad \phi_m(x)=(3x+1)/2\pmod{3^{m+1}}. \tag{F.fan} \] Every \(F_j\) permutes \(G_m\); the series converges absolutely. The map \(\phi_m:G_m\to G_{m+1}\) is a bijection onto the \(2\pmod3\) coset: equality of its values is equivalent to equality of its inputs modulo \(3^m\). At its image the allowed exponents in (F.transfer) are \(a=2j+1\), with predecessor \(F_j(x)\). Therefore \[ H_m(x)=\frac49f_{m+1}(\phi_m(x)). \tag{F.fanidentity} \] If \(y\equiv1\pmod3\), shifting its even admissible exponents down by one proves \(f_{n}(2y)=2f_n(y)\) for \(n\ge1\). The nonunit coset is zero. Since multiplication by two bijects the two unit cosets, the \(p\)-power sums on them have ratio \(1:2^p\). Consequently \[ \langle H_m^p\rangle_m= \frac{3(8/9)^p}{1+2^p}M_p(m+1),\qquad \langle H_m\rangle_m=\frac89=:\mu, \quad \langle H_m^2\rangle_m=\frac{64}{135}M_2(m+1). \tag{F.fanmoments} \] These constants use full-group averages, not unit-relative ones. The maximum likewise lies in the \(2\)-coset, since every value on the \(1\)-coset is doubled on its image. Thus \[ \|H_m\|_\infty=\frac49 A_{m+1}. \tag{F.fanmax} \] The word with all \(m+1\) valuations equal to one has probability \(2^{-(m+1)}\) and endpoint \(-1\pmod{3^{m+1}}\). Since \(\phi_m(-1)=-1\), it also proves the coherent lower bound \[ H_m(-1)\ge\frac23(3/2)^m. \tag{F.spike} \] This all-depth family rules out a polynomial fan maximum. It does not rule out a polynomial collision or fractional moment, because its support occupies a decreasing fraction of the residue group. \subsubsection{The finite inequalities used by the density recipe}\label{the-finite-inequalities-used-by-the-density-recipe} The finite certificate consists precisely of \[ A_d\le a_d\quad(0\le d\le16),\qquad M_2(d)\le e_d\quad(0\le d\le16),\qquad M_{3/2}(d)\le b_d\quad(0\le d\le16), \tag{F.finitetarget} \] with the following rational values. The first three maximum and collision values are exact; the subsequent comparisons may be proved strictly. {\def\LTcaptype{none} % do not increment counter \begin{longtable}[]{@{}rrrr@{}} \toprule\noalign{} \(d\) & \(a_d\) & \(e_d\) & \(b_d\) \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot 0 & 1 & 1 & 1 \\ 1 & 2 & 5/3 & 1276143/1000000 \\ 2 & 22/7 & 15/7 & 711317/500000 \\ 3 & 24/5 & 521/200 & 769803/500000 \\ 4 & 73/10 & 3069/1000 & 32779/20000 \\ 5 & 11 & 707/200 & 862279/500000 \\ 6 & 17 & 4001/1000 & 449901/250000 \\ 7 & 25 & 2233/500 & 116627/62500 \\ 8 & 38 & 1233/250 & 962777/500000 \\ 9 & 57 & 5399/1000 & 1979303/1000000 \\ 10 & 85 & 2933/500 & 253519/125000 \\ 11 & 127 & 3167/500 & 259093/125000 \\ 12 & 948/5 & 6803/1000 & 422721/200000 \\ 13 & 1896/5 & 7273/1000 & 134447/62500 \\ 14 & 20856/35 & 7743/1000 & 109287/50000 \\ 15 & 3201/5 & 4107/500 & 2217669/1000000 \\ 16 & 4802/5 & 1737/200 & 561799/250000 \\ \end{longtable} } The horizons fix the finite work needed by this recipe. Depth sixteen supplies the retained second-moment block rate \(e_{16}^{1/16}\); depth twelve supplies the direct fractional rate \(b_{12}^{1/6}\) and the rational inequality \(b_{12}^2<(57/50)^{12}\), together with its remainder bounds. The latter rate is smaller, so the comparison compensates for the fractional branch's extra residual-mass power. The quantitative comparison (Q.paid) checks this on its stated small-score range. These are sufficient certificate choices, not minimal depths for a rate advantage: already \(b_{11}^{32}5/3\) for \(3\le n<16\), and \(D_n\ge1737/200\) for \(n\ge16\). The supremum envelope's restriction \(m\ge2\) matters: \(\|H_1\|_\infty=88/63>(8/9)\beta_s\). The prefactor in \(\mathcal B_m\) is less than one: its square is \((512/81)/(9+4\sqrt2)<1\). Together with (F.fracgates) and \(4\beta_f<5\), this yields the useful but weaker comparison bound \[ \mathcal B_m^2\le5\beta_f^m. \tag{F.fracsimple} \] The density formula uses the exact table expression (F.ceilings), not (F.fracsimple). Jensen and the exact mean also give \(\mathcal B_m^2\ge\mu^3\); this will check the radial counting domain. For the sixteen-step fractional bound set \[ D_{16,n}^{(3/2)}=b_{16}^{\lfloor n/16\rfloor}b_{n\bmod16},\qquad \mathcal B_{16,m}= \frac{3(8/9)^{3/2}}{1+2^{3/2}}D_{16,m+1}^{(3/2)},\qquad \mathcal B_m^{\rm best}=\min\{\mathcal B_m,\mathcal B_{16,m}\}. \tag{F.sixteenceiling} \] Inheritance and the same fan identity prove that each is a valid \(3/2\)-moment ceiling. In particular \(\mathcal B_m^{\rm best}\ge\mu^{3/2}\). Both fractional ceilings are retained: their minimum is valid without requiring compatible extremizers. \subsubsection{Use of the moment estimates}\label{use-of-the-moment-estimates} The finite maximum, second-moment and direct fractional-moment tables supply the capacity inputs used by the density formulas. The two transfer implementations use distinct recurrences with the same reference law and rational table targets. The product inequality extends these finite bounds to every level. The next section applies them to actual source weights and logarithmic-time convergence. \subsection{10. Actual-weight capacity and lower natural density}\label{chapter10} The parameterized construction supplies integer trajectories reaching one at every fixed clock above the reference ratio, with the same root score (R.score). We now count their distinct starting integers. The aggregate source-weight bound controls weighted sums in residue classes. An elementary maximum estimate already gives a common positive density; the certified moment bounds refine it quantitatively. We count each odd source together with its even multiples, using the actual source weights throughout. \subsubsection{Counting notation}\label{counting_notation} The following lookup collects the quantities already defined in the proof. {\def\LTcaptype{none} % do not increment counter \begin{longtable}[]{@{} >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}} >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}} >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedright Proof stage \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright Symbols and meanings \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright Definition \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot Integer trajectories & \(w,d,A,C_w,\omega_w\): inverse word, length, total valuation, affine offset and slope weight & (W.inverse), (W.offset), (W.charge) \\ Reference distribution & \(G_q,\mu_q,f_q,\rho_q\): residue group, probability law, density relative to the uniform measure, and its scaled version & (T.law) \\ Approximation & \(\epsilon_m,C_6\): uniform error bound from level \(m\) to finer levels, and its fixed coefficient & (M.constants), (M.envelope) \\ Positive root mass & \(\Pi,V(N),P\): guaranteed reference survival, total future variation bound, and their positive difference & (R.certificate), (R.residual) \\ Root family & \(R_s,W_*\): known convergent roots and a lower bound for their reciprocal-root weighted marked mass & (R.pool), (R.score) \\ Sources and reference factors & \(a_x,H_m\): aggregate weight at integer \(x\), and the weighted reference fan & (N.fan), (N.everyclock) \\ Moment bounds & \(S_m,V_m,\mathcal B_m\): upper bounds for the fan maximum, variance, and average \(3/2\)-power & (F.ceilings) \\ Fractional refinement & \(\mathcal B_{16,m},\mathcal B_m^{\rm best}\): sixteen-step ceiling and the smaller fractional ceiling & (F.sixteenceiling) \\ Density conversion & \(\mathcal D(u)\): lower-density bound from an all-scale harmonic mass \(u\) & (I.conversion), (D.radial) \\ \end{longtable} } The variance bound \(V_m\) is distinct from the seed variation bound \(V(N)\). In this section \(U\) denotes total source weight, not the odd accelerated map, and \(T\) denotes a residue-capacity coefficient. Fix an arbitrary \(c>c_0=3/\log(4/3)\) throughout this section, and write \[ \mathcal G_c=\{x\in\mathbb N_{>0}:\tau_1(x)\le c\log x\}, \qquad \mu=8/9,\quad L=2\log2,\quad\kappa=16\log2/9. \tag{D.constants} \] The counting density is lower natural density, \(\underline d(\mathcal G_c)=\liminf_{Y\to\infty} \#(\mathcal G_c\cap[1,Y])/Y\). The clock is ordinary unaccelerated Collatz time, and logarithms are natural unless marked \(\log_2\). \subsubsection{Harmonic sum bound for the source weights}\label{harmonic-sum-bound-for-the-source-weights} Fix an odd integer modulus \(Q\), a real \(X>0\), a radius \(R>1\), and a finite set of positive odd integers in \([X,RX)\) carrying weights \(0\le a_x\le1/x\). Put \(U=\sum_xa_x\) and \(w_r=\sum_{x\equiv r\ (Q)}a_x\). Odd representatives of one residue class modulo \(Q\) have spacing \(2Q\). The monotonicity of \(1/t\) and comparison of all but the first summand with their preceding intervals give \[ \sum_{\substack{X\le x16\). For fixed coarse \(m\ge2\), the terminal level \(k=k(X)\) eventually exceeds \(m\), and \(3^k/X\to0\). Every \(F_j\) in (F.fan) permutes \(G_k\). The mixing estimate (T.mix) for \(f\) and the normalization \(\rho=(2/3)f\) yield \[ \langle|H_k-H_m\circ\pi_{k,m}|\rangle_k \le\frac89\epsilon_m. \tag{D.fandiscrepancy} \] Indeed, sum the individual discrepancies after the affine permutations; their factors are \(\sum_{j\ge0}4^{-j}=4/3\). Nonnegativity and absolute convergence justify exchanging sums and averages. Harmonic charging at the fine modulus \(3^k\) then proves \[ \mathscr A_X\le\sum_xa_xH_m(x) +\frac89 T_{X,3^k,R_X}\epsilon_m, \qquad T_{X,3^k,R_X}\longrightarrow L. \tag{D.coarse} \] Thus the limiting error bound is exactly \(\kappa\epsilon_m\). Fixing \(\Lambda>16\) in the final shell declaration does not replace this error bound by \((4/9)\log\Lambda\,\epsilon_m\): the physical family still has its sharper radius tending to sixteen. Conversely, one may not assume a literal radius sixteen at finite cutoffs. \subsubsection{Three capacity bounds for the same physical mass}\label{three-capacity-bounds-for-the-same-physical-mass} For a fixed level \(m\), let \(Q=3^m\), \(T=T_{X,Q,R_X}\). The exact mean of \(H_m\) is \(\mu\), and its variance is at most \(V_m\) from (F.ceilings). Subtract the constant \(U/Q\) from each residue mass; the centered marker sums to zero. Then \[ \sum_r(w_r-U/Q)^2=\sum_rw_r^2-U^2/Q \le(TU-U^2)/Q. \] Cauchy--Schwarz, the maximum bound and weighted Hölder give respectively \[ \begin{split} \sum_xa_xH_m(x)&\le\mu U+\sqrt{V_mU(T-U)},\\ \sum_xa_xH_m(x)&\le S_m U,\\ \sum_xa_xH_m(x)&\le U^{1/3} \left(\sum_xa_xH_m(x)^{3/2}\right)^{2/3} \le U^{1/3}(T\mathcal B_m)^{2/3}. \end{split} \tag{D.capacity} \] The last inequality is (D.harmonic) applied to the nonnegative \(3/2\)-power. All three bounds use the same actual source weights. Their validity depends on the source-weight bound (N.charge) established before the residue masses were formed. \textbf{Proposition 10.1 (general moment-to-mass conversion).} Let \(s>1\) and \(B_{s,m}\ge\langle H_m^s\rangle_m\), with \(m\ge2\) fixed. For a source-family score \(016\). \emph{Proof.} At a finite cutoff, weighted Hölder and (D.harmonic) give \[ \sum_rw_rH_m(r) \le U^{1-1/s}\bigl(T_{X,3^m,R_X}B_{s,m}\bigr)^{1/s}. \] By (D.coarse), the left side is at least \(W-\eta-\kappa\epsilon_m-o(1)\) for a fixed positive graft tolerance \(\eta\). Since \(T_{X,3^m,R_X}\to L\), inversion gives every strict lower bound in (D.generalp) by first choosing a sufficiently small fixed \(\eta\), then a sufficiently large cutoff. No positive lower bound is asserted when \(p_m=0\). Jensen and the mean \(\mu\) give \(B_{s,m}\ge\mu^s>0\). Since \(p_m\le W1\). Suppose \(0\le w_r\le T/Q\) and \(\sum_rw_rh_r\ge p\), where \(00. \] Consequently \(\widetilde v_s(p,T)\le p/\mu\). It strictly exceeds the occupied-only bound \(p^{s/(s-1)}/(TB_s)^{1/(s-1)}\), since the subtracted complementary term is positive. At a finite cutoff use \(T=T_{X,3^m,R_X}\), the actual capacity coefficient, and a guaranteed positive marked mass below \(W-\eta-\kappa\epsilon_m-o(1)\). First fix a sufficiently small graft tolerance \(\eta\), then take the cutoff large. Continuity as \(T\to L\) gives every strict lower bound below the displayed inverse with \(p=p_m=(W-\kappa\epsilon_m)_+\), \(T=L\). Set the inverse to zero when \(p_m=0\). The bound by \(p_m/\mu\) also gives continuity there and keeps it in (D.domain). Thus the same radial counting applies; no limiting capacity is imposed at a finite cutoff. Fix a score \(00),\qquad \widetilde v_m(W)=0\quad(p_m=0),\\ u_{16,m}(W)&=\max\{u_m(W),v_{16,m}(W)\},\\ u_m^{\rm comp}(W)&=\max\{u_{16,m}(W),\widetilde v_m(W)\}. \end{aligned} \tag{D.refinedmass} \] The complementary branch uses the better of the two moment ceilings. All branches concern the same actual source family and fixed score. Their limiting mass bounds are at most \(W/\mu\), as are the original branches by (D.domain). The label \({\rm MV}\) denotes the combination of the centered second-moment and maximum bounds; it does not change the root score or mixing estimate. The added term \(v_m(W)\) is the direct fractional-moment bound. Each expression estimates the same unmarked source mass. The square root and its choice of branch have the following justification. On \([0,L/2]\), the function \(g_m(u)=\mu u+\sqrt{V_mu(L-u)}\) is strictly increasing, with \(g_m(0)=0\) and \(g_m(L/2)\ge\mu L/2>W_{\max}\). Consequently there is a unique inverse for \(0\le p\le W_{\max}\). Squaring on the branch \(p-\mu u\ge0\) gives \[ (\mu^2+V_m)u^2-(2\mu p+V_mL)u+p^2=0. \] The required solution is the smaller root, whose rationalized expression is (D.inverse). It obeys \(\Psi_m(p)\le p/\mu2/3\), \(L^2\mu^3>(16/9)(512/729)>1\). Therefore \(v_m\le W^3\le W\). In particular \[ 0\le u_m(W)\le W/\mu <\tfrac12\log(3/2)1/6\), the latter by integrating \(1/t>2/3\) on \([1,3/2)\). \textbf{Lemma 10.3 (all-scale unmarked mass).} For each fixed \(m\ge2\), each \(u16\), every sufficiently large real \(X\) has a finite family of odd sources in \(\mathcal G_c\cap[X,\Lambda X)\), with \[ a_x\le1/x,\qquad\sum_xa_x\ge u. \tag{D.mass} \] \emph{Proof.} Fix \(m,\Lambda\) and a positive graft tolerance \(\eta\). (N.everyclock) and (D.coarse) give coarse marked mass at least \(W-\eta-\kappa\epsilon_m-o(1)\); both fine and coarse capacity coefficients tend to \(L\), uniformly at large cutoffs. Apply (D.capacity). To justify the limit without postulating one zero-error graft, first allow \(\eta\downarrow0\) in these scalar inequalities. If an asserted branch lower bound failed by a fixed positive amount for every sufficiently small tolerance, one could choose \(\eta_i\to0\) and corresponding sufficiently large bad cutoffs. Their masses have a convergent subsequence, since \(U\le T\to L\). The limit \(U_*\) would satisfy all three inequalities with left side \(p_m\) and capacity \(L\). If \(U_*\ge L/2\), (D.domain) already contradicts the failure; otherwise monotonicity of \(g_m\) and the other two inversions give \(U_*\ge u_m(W)\), again a contradiction. Hence a single sufficiently small fixed tolerance and a single associated finite graft work for the chosen strict \(u\) and all large \(X\). At \(p_m=0\), all the asserted branch bounds are zero and there is nothing positive to prove. \(\square\) The same conclusion (D.mass) holds for every \(0\le uc_0\log2>1\). Thus all the counted towers remain inside the same literal logarithmic-time set. \textbf{Theorem 10.4 (conversion from small harmonic mass to lower density).} Suppose, for every fixed \(R>16\) and every sufficiently large real \(X\), the odd members of a dyadically closed set \(G\) in \([X,RX)\) admit weights \(0\le a_x\le1/x\) totaling at least \(u\). If \(0\le u\le\tfrac12\log(3/2)\), then \[ \underline d(G)\ge \mathcal D(u):=\frac{32}{225}(e^{2u}-1). \tag{D.radial} \] The function \(\mathcal D\), introduced in (I.conversion), converts weighted counts over intervals of sizes comparable to \(X\) into density below a final cutoff. We call it the radial conversion. \emph{Proof.} The zero case is immediate. First consider one exact band \((X_j,16X_j)\), where \(X_j=Y/16^j\), \(j\ge1\) is fixed. Away from its four dyadic boundaries, the tower count on \((2^hX_j,2^{h+1}X_j)\) is \(4j-h\), for \(h=0,1,2,3\). Relax integer selection to occupations \(0\le t_x\le1\) at all ambient odd points, taking \(t_x=xa_x\) on the actual family and zero elsewhere. Their harmonic mass is \(\sum t_x/x\), and their fractional counting cost is \(\sum t_xD_Y(x)\). The actual integer count is at least that cost. The cost per unit harmonic mass is \(xD_Y(x)\). In the first subband it increases as \(4jx\). The minimum in any later subband is at least \(2^h(4j-h)X_j\ge(8j-2)X_j\ge6jX_j\). For fixed \(116\), then choose the hypothesized families in the pairwise disjoint shells \([Y/R^j,Y/R^{j-1})\), \(1\le j\le J_0\). Remove sources below \(Y/16^j\). At most \[ \frac j2\log(R/16)+O(Y^{-1}) \tag{D.stripdebit} \] of harmonic mass is lost, by the same odd census, with constants allowed to depend on the fixed \(j,R\). The remainder lies in the exact band \([Y/16^j,Y/16^{j-1})\). Its mass is at least \(a_j-o(1)\), where \(a_j=(u-\tfrac j2\log(R/16))_+\). For each positive \(a_j\), apply (D.bandcost) with an arbitrarily small fixed downward tolerance; the zero cases contribute zero. The selected odd sources in different shells are distinct and their towers are disjoint. After dividing their total count by \(Y\), first let \(Y\to\infty\), then \(R\downarrow16\) with \(J_0\) fixed, and finally \(J_0\to\infty\). All inequalities concern the same set \(G\), so source families and startup cutoffs may depend on these fixed choices. Since \(\sum_{j\ge1}4j/16^j=64/225\), the result is \[ \underline d(G)\ge\frac{e^{2u}-1}{2} \sum_{j\ge1}\frac{4j}{16^j} =\frac{32}{225}(e^{2u}-1). \] This proves (D.radial). \(\square\) The linear consequence is \(\mathcal D(u)\ge64u/225\), which retains the factor \(32/15\) over the older separate packing-and-doubling coefficient \(2u/15\). The nonlinear expression is stronger still, but its ratio to the linear one tends to one as \(u\downarrow0\). No extra fixed multiplicative improvement is claimed from that small second-order correction. \subsubsection{A common positive density without finite moment certificates}\label{a-common-positive-density-without-finite-moment-certificates} \textbf{Proposition 10.5 (elementary common-density bound).} Suppose (T.mix) has \(\epsilon_m\to0\), and a fixed persistent pool supplies a score \(0c_0:\qquad \underline d(\mathcal G_c)\ge\frac{4W}{25\,2^m}>0. \tag{D.elementarydensity} \] Both \(W\) and \(m\) are fixed before \(c\). This deduction uses no moment certificates at level twelve or sixteen. \emph{Proof.} Such an \(m\) exists because \(\epsilon_m\to0\). The elementary recursion (B.markercap) and the fan definition (F.fan) give \[ \|H_m\|_\infty \le\sum_{j\ge0}4^{-j}\|\rho_m\|_\infty \le\frac89\,2^m. \tag{D.elementarycap} \] For a fixed \(c>c_0\), the source theorem (N.everyclock) and (D.coarse) therefore supply every mass strictly smaller than \[ \frac{W-\kappa\epsilon_m}{(8/9)2^m} >a_m:=\frac{9W}{16\,2^m}. \] Indeed, for a fixed positive graft tolerance \(\eta\), the finite coarse marked mass is at least \(W-\eta-\kappa\epsilon_m-o(1)\). First make \(\eta\) small enough for the desired strict mass, then take all cutoffs sufficiently large. This works for every fixed containing ratio \(\Lambda>16\), using the actual radius tending to sixteen. Since \(m\ge2\), \(a_m0 \qquad(c>c_0). \end{aligned} \tag{D.headline} \] In particular the baseline score \(W_*\) gives \[ \forall c>c_0:\qquad \underline d(\mathcal G_c)\ge c_{\rm FM}(W_*)>0. \tag{D.everyclock} \] All scores and density functions are independent of \(c\). The same implications hold for any fixed integer \(C\ge1\) satisfying (M.primitive), with its corresponding mixing, startup and score recipes. For fixed \(c>c_0\) and any strict lower bound \(d0\) eventually. The maximum branch tends to zero by (F.spike), and the centered inverse tends to zero since \(V_m\to\infty\) and \(\Psi_m\le2W^2/(V_mL)\). Both fractional branches tend to zero because \(b_{12},b_{16}>1\). Finally \(\mathcal B_m^{\rm best}\to\infty\); once it exceeds \(2\mu^{3/2}\), \[ \widetilde v_m(W)\le4\max\{v_m(W),v_{16,m}(W)\}\longrightarrow0. \] This follows by bounding the subtracted complementary term by \(\mu^{3/2}\). Thus each positive mass sequence tends to zero and attains its supremum at a finite level. Inclusion of branches proves the non-strict comparisons. Section 11 proves the strict improvements. The pool and score precede \(c\) by (N.everyclock); none of \(L,\kappa,\epsilon_m\), the moment ceilings or the optimizations depends on \(c\). Only the graft and eventual cutoffs do. The final counting assertion is the definition of lower density. In particular, \(c=10431/1000\) gives the certified numerical specialization. \(\square\) \textbf{Corollary 10.7 (an unspecified vanishing clock loss).} If a constant \(d_0>0\) satisfies the common-clock conclusion above, there exists a nonincreasing \(\eta:[1,\infty)\to(0,\infty)\), with \(\eta(x)\to0\), such that \[ \underline d\{x\ge1:\tau_1(x)\le(c_0+\eta(x))\log x\} \ge d_0. \tag{D.vanishingclock} \] In particular one set of lower density at least \(d_0\) has \(\limsup_{x\to\infty}\tau_1(x)/\log x\le c_0\), with the limit restricted to that set. No decay rate for \(\eta\) is specified. \emph{Proof.} Put \(A_j=\mathcal G_{c_0+1/j}\); these sets decrease with \(j\). Choose strictly increasing integer cutoffs \(X_j\ge j\) such that \(\#(A_j\cap[1,X])\ge(d_0-1/j)X\) for every \(X\ge X_j\). Set \(\eta=1/j\) on \([X_j,X_{j+1})\) and \(\eta=1\) below \(X_1\). If \(X\) is in the \(j\)-th band, every point of \(A_j\) below \(X\) satisfies the new clock bound: its earlier band has allowance at least \(1/j\). The resulting set therefore has at least \((d_0-1/j)X\) points below \(X\). Let \(X\to\infty\). This diagonalization uses the nested fixed-clock sets; the endpoint \(c=c_0\) is discussed in Section 5. The same proof applies to any fixed target with a common density constant throughout \(c>c_0\). \(\square\) \textbf{Theorem 10.8 (finite-pool transfer to a fixed target).} Under the same analytic and moment inputs as (D.everyclock), fix a positive target \(y_\star\) and a nonempty finite pool satisfying (G.marks), (W.antichain), and the finite root-to-target path hypothesis of (N.targetfamily). If \(0c_0:\qquad \underline d(\mathcal G_c(y_\star))\ge c_{\rm comp}(W) \ge c_{\rm FM}(W)>0. \tag{D.finitepool} \] Without the moment-table premise, the same pool gives the elementary bound (D.elementarydensity), with \(\mathcal G_c(y_\star)\) replacing \(\mathcal G_c\). \emph{Proof.} Apply (N.targetfamily). Every step of the harmonic-capacity and coarse-transfer argument uses its source weights, radius and precision; none depends on which fixed target is eventually visited. The target-clock set is dyadically closed because \(\tau_{y_\star}(2^rx)\le r+\tau_{y_\star}(x)\) and \(c\log2>1\). Thus (D.mass), including its refined branches, the general set formulation of (D.radial), and the attainment argument in (D.headline) give the bound. The elementary maximum argument proves the table-free variant. The pool, score and coarse estimates precede \(c\), so one constant works throughout its stated range. \(\square\) \textbf{Corollary 10.9 (odd sources in every large shell).} Fix \(c>c_0\), \(\Lambda>16\), \(m\ge2\), and \(016\) and a strict mass below the limiting bound. \subsubsection{Every prescribed target not divisible by three}\label{fixed_targets} Two complete incoming periods suffice for the general target theorem. \textbf{Theorem 10.10 (every fixed positive unit target).} For every positive integer \(y_\star\), \(3\nmid y_\star\), an effectively defined \(d_{y_\star}>0\) satisfies \[ \forall c>c_0:\qquad \underline d(\mathcal G_c(y_\star))\ge d_{y_\star}. \tag{D.everytarget} \] \emph{Proof.} Use the selected startup \(N\) and \(q=q_N\), as in (R.markapplication). Write \(y_\star=2^v y\), with \(y\) odd. In (W.constructor), choose \(t_0\ge0\) so that \(X_{t_0}(y)\ge16^b\) and \(2t_0+e(y)\ge v+1\). Take the unit roots in each of the disjoint index intervals \([t_0,t_0+3^q)\) and \([t_0+3^q,t_0+2\cdot3^q)\). Each contains one root in every unit residue. All roots have next odd value \(y\). By the cycle-predecessor lemma of Section 1, at most one root in their union is periodic. At least one whole block therefore satisfies (W.antichain). Let \[ M_{\max}=X_{t_0+2\cdot3^q-1}(y),\qquad W_{y_\star}=\min\{W_{\max},K_qP/M_{\max}\}>0,\qquad d_{y_\star}=c_{\rm FM}(W_{y_\star}). \tag{D.targetscore} \] The identity (B.physical), uniform over roots, and full future variation bound (R.latermarks) give \(z_r=(g_N(r)-V(N))_+\) on either block, with \(\sum_rz_r\ge K_qP\). Hence the suitable block has \(\sum_Mz_M/M\ge K_qP/M_{\max}\). Its root indexed by \(t\) reaches \(y_\star\) along the ordinary odd-and-halving segment in \(1+2t+e(y)-v\) steps. These specified finite paths need no information about the orbit after \(y_\star\). Apply (D.finitepool). The score recipe is effective and the same for either candidate block; no cycle-detection procedure is needed to define it or to prove that one block is suitable. All choices are finite and precede \(c\). With the qualitative analytic input, (D.elementarydensity) instead gives \(4W_{y_\star}/(25\,2^m)>0\) at one fixed \(m\) making the mixing error smaller than half the score, without the large moment tables. \(\square\) This proves (I.targets). It does not assert convergence of the target. \textbf{Corollary 10.11 (one-period quantitative target score).} Retain the selected startup \(q=q_N\), \(K=K_q\), \(B=B_q\), \(P=\Pi-V(N)\) and the index \(t_0\) of the preceding proof. Set \[ M_1=X_{t_0+3^q-1}(y),\qquad W_{y_\star}^{\rm one}=\frac{KP-B}{M_1}. \tag{D.oneperiodscore} \] Then \(0c_0:\qquad \underline d(\mathcal G_c(y_\star)) \ge c_{\rm comp}(W_{y_\star}^{\rm one})>0. \tag{D.oneperioddensity} \] Its score strictly improves the two-period value in (D.targetscore): \[ \frac{W_{y_\star}^{\rm one}}{W_{y_\star}} >\tfrac12\,4^{3^q}. \tag{D.oneperiodgain} \] \emph{Proof.} Equations (R.denominator) and (R.conductorguards), with \(b=2^{25}\), \(N\ge16\), and \(3/2>\sqrt2\), give \[ \frac{KP}{B}=P(3/2)^q >2^{q/2-1575N-6305} \ge2^{(b/2-1575)N-6305}>2. \tag{D.singlecap} \] The last exponent is at least \(268403951\). One full unit period has persistent marks totalling at least \(KP\). Remove its periodic root if present. There is at most one, so its removal costs at most \(B\); the remaining pool is nonempty and satisfies (W.antichain). It retains the specified finite paths to the actual target, including its even factor. Root-uniform persistence gives score at least \((KP-B)/M_1\). Apply (D.finitepool). For the domain, put \(T=3^q\). The recurrence \(X_{t+1}(y)=4X_t(y)+1\) gives \(M_1\ge16^b4^{T-1}\), while \(KP4^TM_1. \] The clipping in (D.targetscore) is inactive by the same domain bound. Since \(KP-B>KP/2\), division proves (D.oneperiodgain). The score is a finite rational recipe; no cycle-decision procedure is needed to define it. The two-period construction remains available for the qualitative theorem without the scalar cap comparison. \(\square\) \textbf{Corollary 10.12 (exact obstruction to positive-density hitting).} For every positive integer \(y\) and every fixed \(c>c_0\), \[ \boxed{\underline d(\mathcal G_c(y))>0 \quad\Longleftrightarrow\quad 3\nmid y.} \tag{D.targetcriterion} \] \emph{Proof.} If \(3\nmid y\), apply (D.everytarget). Suppose \(3\mid y\). An odd predecessor of a number divisible by three is impossible, since \(3x+1\equiv1\pmod3\). Its only predecessor is therefore its double, which is again divisible by three. Iterating this observation backwards along any finite path, and conversely taking repeated halvings, gives \[ \begin{aligned} \{x\ge1:\tau_y(x)<\infty\}&=\{2^j y:j\in\mathbb Z_{\ge0}\},\\ \#\{x\le X:\tau_y(x)<\infty\} &=1+\left\lfloor\log_2(X/y)\right\rfloor =O_y(\log X)\qquad(X\ge y). \end{aligned} \tag{D.divisiblepredecessors} \] The full predecessor basin has natural density zero, and so does its subset \(\mathcal G_c(y)\). \(\square\) There can be no positive density constant uniform over all admissible targets: the predecessor basins of the infinitely many distinct \(R_s>1\) with \(3\nmid s\) are pairwise disjoint, and these targets are units modulo three. If any \(J\) such basins each had lower density at least \(d>0\), their union would have lower density at least \(Jd\le1\). Letting the finite number \(J\) increase rules out a common positive \(d\). \textbf{Remark (a score from one persistent root).} At least one root \(M\) from (R.pool) has a fixed score \(W_M>0\) such that, for every \(c>c_0\), \[ \underline d\{x:\tau_M(x)\le c\log x\} \ge c_{\rm FM}(W_M)>0, \tag{D.roottarget} \] where \(\tau_M\) is the least ordinary hitting time of \(M\). \emph{Proof.} By (R.persistence), \(\sum_s z_s\ge K_qP>0\). Choose the first \(s\), in increasing order in the finite set \(\mathcal S_q\), with \(z_s>0\), put \(M=R_s\), and take \(W_M=\min\{z_s/M,W_{\max}\}>0\). These are finite rational operations, although no numerical target is evaluated here. The singleton pool obeys (G.marks) with that score. Every constructed inverse history passes through \(M\), so its bound on time to one also bounds its time to \(M\). The latter logarithmic-time set is dyadically closed. Repeat the proof of (D.everyclock) for this set. This singleton argument selects a positive entry of the persistent vector; it does not assert that every entry is positive. \(\square\) \textbf{Remark (one-block construction for a convergent target).} For every fixed convergent odd unit \(y\), the set \(\mathcal G_c(y)\) has positive lower natural density, with one target-dependent constant for all \(c>c_0\). Indeed choose a barrier exceeding \(16^b\) and every odd state on the finite trajectory of \(y\). A complete period of the roots \(X_t(y)\) above that barrier is residue-complete and a convergent antichain: after its first odd step every later odd state is below the barrier. The persistent vector has positive total mass on this block. Apply (D.finitepool) with target \(y\). \textbf{Corollary 10.13 (qualitative analytic input).} A single positive lower density bound for all \(c>c_0\) follows from the local primitive theorem and the seed, persistent-root, graft and terminal constructions, without the large finite moment certificates. Alternatively, Tao's published reference-law estimates {[}1{]} suffice without specifying a numerical analytic coefficient. \emph{Proof.} For the local route, (E.primitive) supplies (M.primitive), Section 6 gives its mixing envelope, and Sections 7--8 give a fixed positive score. Choose one coarse level at which the error is less than half the score, and apply (D.elementarydensity). This needs no large moment certificate. For the alternative route, {[}1, Proposition 1.17{]} applied at order \(6409\) gives (M.primitive) for an integer coefficient \(C\ge1\), with the Fourier sign changed by conjugation when necessary. Section 6 and the finite startup argument then give a positive score \(W(C)\). Alternatively, {[}1, Proposition 1.14{]} supplies (T.mix) with unhalved error \(K_A m^{-A}\). Take \(A=2314/25\) and an integer \(\widehat C>\max\{K_A,2^{8838}\}\). The envelope in (B.coefficientscope), the survival bound \(\Pi=24/25\), and the root allocation give a positive score. In either case decrease the score into \((0,W_{\max}]\), choose a coarse level with error less than half that score, and apply (D.elementarydensity). In every route the coefficient, pool, score and coarse level are fixed before \(c\). The law and norm are those of (T.law) and (T.mix). \(\square\) The alternative scores depend on their chosen analytic coefficients. The explicit density formulas use \(C_{\rm primitive}\) and the finite tables of Section 9. \textbf{Corollary 10.14 (explicit rational specialization).} For rational \(0 \frac4{1125}W^3(57/50)^{-m_\circ}. \tag{D.explicit} \] \emph{Proof.} The chosen envelope obeys \(\epsilon_{m_\circ}\le C_6m_\circ^{-2314/25}\le2W/5\). Since \(\kappa=16\log2/9<100/81<5/4\), its residual \(p_{m_\circ}\) is strictly greater than \(W/2\). Using \(L^2<2\), (F.fracsimple), and the fractional branch in (D.massrecipe), gives \[ u_{m_\circ}(W)> \frac{W^3}{80(57/50)^{m_\circ}}. \] The inequality \(\mathcal D(u)\ge64u/225\) proves (D.explicit). The logarithmic bounds used here follow from \(\log2<25/36\); then \(L^2<(25/18)^2<2\). For the canonical integer \(C_6\) and rational \(W\), the least integer in the ceiling is determined by \[ m^{2314}(2W)^{25}\ge(5C_6)^{25}. \] Clearing denominators makes this an integer comparison. The recipe terminates, although its integers are not of practical size. \(\square\) For any constant strictly below the displayed density bound, the proof supplies an eventual all-cutoff lower bound with that constant. A lower density liminf alone does not assert exact attainment of the displayed constant at every sufficiently large finite cutoff. Neither the fixed root pool nor the finite coarse precision is chosen anew at each cutoff. The estimate combines the source-family theorem, the reference-moment bounds and the counting argument above. Its analytic input is the primitive Fourier theorem (M.primitive); the survival and moment certificates supply the stated finite inequalities. It remains to quantify the improvement over the formula without fractional moments. \subsection{11. Quantifying the fractional-moment improvement}\label{chapter11} We first compare the twelve-step fractional-moment formula with the optimized maximum and second-moment bounds. Its fractional branch has slower growth in the ternary level, but contains an extra power of the small marked mass; the comparison must account for both effects. We then prove the strict improvements from sixteen-step propagation and complementary capacity, retaining the same mixing envelope (M.envelope). \subsubsection{Two properties of the certified error envelope}\label{two-properties-of-the-certified-error-envelope} For every integer \(n\ge2\), the explicit formula (M.envelope) satisfies \[ \epsilon_n\ge\frac1{300}n^{-95},\qquad \epsilon_m\le\frac nm\epsilon_n \quad(m\ge n\ge2,\ \epsilon_n<2). \tag{Q.error} \] These inequalities describe the envelope (M.envelope). To prove them, use \(\log2<25/36\). Throughout the compact range \(80\le v\le679/5\), one has \(q_v<95\) and \(a(v)>1\), while \(A=2314/25<95\). The latter exponent bound follows already from \(6407-(25/72)(679/5)^2>1\). Every nonconstant candidate in (M.envelope) therefore decreases at least as \(x^{-1}\); in the second term of (M.widthbound), \(q_v>2\) and \(1/\log x\) is also decreasing. If \(\epsilon_n<2\), a nonconstant candidate attains that minimum. It remains available at every larger level, including when crossing \(m_0\), where more choices are added. This proves the second assertion. Compactness and continuity ensure attainment of the width minimum. For the first assertion, the second positive term of each compact-width candidate is at least \[ \frac{32}{95^2}n^{-95}>\frac1{300}n^{-95}, \] using \(\log n\le n\) and \(q_v<95\). The constant two and \(C_6n^{-A}\), with \(C_6\ge1\), also exceed this floor. Taking the minimum proves (Q.error) without estimating the actual reference discrepancy from below. \subsubsection{A bound for the maximum and second-moment formula}\label{a-bound-for-the-maximum-and-second-moment-formula} Fix \(0\kappa\epsilon_m\}. \tag{Q.firstpositive} \] This minimum exists because \(\epsilon_m\to0\). Since \(\kappa>1\), (Q.error) implies \(W>n^{-95}/300\), and hence \[ n>2^{(a-9)/95}>1000(2a+32),\qquad n>32. \tag{Q.largeconductor} \] For the second inequality, at \(a=4096\) the middle expression exceeds \(2^{43}>1000(2\cdot4096+32)\). The logarithmic derivative of its ratio to \(2a+32\) is \((\log2)/95-1/(a+16)>1/190-1/4112>0\) thereafter. The constant \(\kappa>1\) follows from \(\log2>2/3\). For the collision table, exact rational comparisons give \[ e_{16}>(8/7)^{16},\qquad \frac{64}{135}(7/8)^{14}>\frac1{16}. \] Because every remainder \(e_r\ge1\), writing \(m+1=16s+r\), \(0\le r<16\), yields \[ (64/135)D_{m+1}\ge\frac1{16}(8/7)^m. \] At \(m\ge32\) this exceeds \(2\mu^2\), so \[ V_m\ge\frac1{32}(8/7)^m. \tag{Q.variancefloor} \] This is a lower bound for the ceiling \(V_m\). If \(U=\Psi_m(p_m)\), the correct inverse relation in (D.inverse) gives \(p_m=\mu U+\sqrt{V_mU(L-U)}\), with \(U1\). The other benchmark branch obeys \(p_m/S_m\le(3/2)W(2/3)^m\), by (F.spike), since \(S_m\) is a valid upper bound for that maximum. All branches vanish at \(mW(1-n/m)\ge W/101. \tag{Q.residual} \] From (F.fracsimple), \(L^2<2\), and \(\mathcal D(u)\ge64u/225\), \[ c_{\rm FM}(W)>\frac{64}{2250\cdot101^3} W^3(57/50)^{-m}. \tag{Q.newbound} \] The bound \(L^2<2\) follows from \((25/18)^2<2\). Dividing by (Q.oldbound), and writing \(\beta_f=57/50\), gives \[ \begin{split} \frac{c_{\rm FM}(W)}{c_{\rm MV}(W)} &>\frac{W^2}{1280\cdot101^3\beta_f} \left(\frac{8/7}{\beta_f^{101/100}}\right)^n\\ &>2^{n/1000-2a-31}>2. \end{split} \tag{Q.paid} \] The middle inequality is exactly the pair of rational inequalities \[ (8/7)^{1000}>2\beta_f^{1010},\qquad 1280\cdot101^3\beta_f<2^{31}, \tag{Q.ratechecks} \] and the last is (Q.largeconductor). Each inequality is verified by integer cross-multiplication in the finite driver. Unlike an interpolation heuristic, the strict exponential separation compensates for the extra \(W\) and the fixed displacement of the chosen coarse level. The auxiliary first positive level can be eliminated from the comparison. For every \(02^{\,2^{(a-9)/95}/1000-2a-31}>2.} \tag{Q.explicitgain} \] Indeed, substitute \(n>2^{(a-9)/95}\) from (Q.largeconductor) into (Q.paid). The other inequality in (Q.largeconductor) gives \(2^{(a-9)/95}/1000-2a-31>1\). This proves the stated strict factor for the complete density formulas, with no new estimate or choice of roots. Combining (D.everyclock), (Q.paid) and (R.scoredomain) gives \[ \boxed{\forall c>c_0:\quad \underline d\{x:\tau_1(x)\le c\log x\} \ge c_{\rm FM}(W_*)>2c_{\rm MV}(W_*)>0.} \tag{Q.headline} \] The explicit specialization \(c=10431/1000\) is included. With \(c_*=c_{\rm FM}(W_*)\), (D.everyclock) proves (I.main), while (Q.explicitgain) gives the stronger comparison (I.gain). The bound uses (M.primitive), its specified coefficient, the finite certificates and the auxiliary estimates proved above. Both comparisons keep the root score and mixing estimate fixed. \subsubsection{Improvement from the sixteen-step and complementary bounds}\label{improvement-from-the-sixteen-step-and-complementary-bounds} \textbf{Theorem 11.1 (strict refinement at a fixed score).} For \(0c_{{\rm FM},16}(W) >\frac{1369}{625}c_{\rm FM}(W), \tag{Q.sixteengain} \] and \[ \frac{c_{{\rm FM},16}(W)}{c_{\rm FM}(W)} >2^{\,2^{(a-9)/95}/30-10}. \tag{Q.sixteenscoregain} \] All formulas have the same root score and mixing envelope. \emph{Proof.} Let \(m_*\) attain the positive maximum defining \(c_{\rm FM}(W)\). By (Q.paid), its attaining branch is fractional. Write \(n=m_*+1=48t+r\), \(0\le r<48\). Equation (Q.largeconductor) gives \(t\ge5\). Every fractional remainder is at least one, and all rows are at most \(b_{16}\); hence \[ D_n^{(3/2)}\ge b_{12}^{4t},\qquad D_{16,n}^{(3/2)}\le b_{16}^{3t+3}. \] The exact scalar comparisons are \[ b_{12}^4>b_{16}^3,\qquad 25b_{12}^{20}>37b_{16}^{18},\qquad b_{12}^{20}>16b_{16}^{15},\qquad b_{16}^6<256. \tag{Q.sixteenguards} \] Writing \(R=b_{12}^4/b_{16}^3\), the first pair implies \(D_n^{(3/2)}/D_{16,n}^{(3/2)}\ge R^t/b_{16}^3>37/25\). The common residual and fan prefactor cancel, so \(v_{16,m_*}>(37/25)^2v_{m_*}\). Increasing convexity of \(\mathcal D\), with \(\mathcal D(0)=0\), proves the factor \(1369/625\) after optimizing. For the stronger estimate, the last pair gives \(R^5>16\) and \[ (D_n^{(3/2)}/D_{16,n}^{(3/2)})^2 \ge R^{2t}/b_{16}^6 >2^{(8/5)t-8}>2^{n/30-10}, \] since \(t\ge(n-47)/48\) and \(47/30+8<10\). Use \(n>2^{(a-9)/95}\) and the same density conversion. Finally, a maximizing branch of \(c_{{\rm FM},16}\) is also fractional: \(c_{{\rm FM},16}\ge c_{\rm FM}>c_{\rm MV}\) excludes both nonfractional branches. At that level the positive complementary term makes \(\widetilde v_m\) strictly larger than both occupied-only fractional bounds, since it uses \(\mathcal B_m^{\rm best}\). Monotonicity of \(\mathcal D\) proves \(c_{\rm comp}>c_{{\rm FM},16}\). All masses remain in (D.domain). \(\square\) \subsubsection{How a larger root score improves the full formula}\label{how-a-larger-root-score-improves-the-full-formula} \textbf{Theorem 11.2 (cubic score comparison).} For the fixed retained envelope and finite tables, if \(0 (W_2/W_1)^3c_{\rm FM}(W_1). \tag{Q.cubicscore} \] \emph{Proof.} The positive maximum at \(W_1\) is attained. By (Q.paid), no benchmark branch can attain it, so at a maximizing level \(m\), \(c_{\rm FM}(W_1)=\mathcal D(v_m(W_1))\) with \(v_m(W_1)>0\). Put \(a=W_2/W_1>1\), \(\delta=\kappa\epsilon_m\). Then \(W_2-\delta=a(W_1-\delta)+(a-1)\delta\ge a(W_1-\delta)\); hence \(v_m(W_2)\ge a^3v_m(W_1)\). Strict convexity, monotonicity and \(\mathcal D(0)=0\) give \(\mathcal D(a^3u)>a^3\mathcal D(u)\) for \(u>0\). Maximizing at \(W_2\) proves the claim. All intermediate masses are at most \(v_m(W_2)\), already in (D.domain). \(\square\) The small-score hypothesis is used at the smaller score to prove which branch attains the maximum. Merely including a cubic branch would not suffice. The inequality is a lower comparison between certified formulas; it gives neither their exact ratio nor an upper bound on that ratio. The same argument for a globally active order-\(s\) branch gives exponent \(s/(s-1)\), provided its dominance is separately proved. For \(c_{{\rm FM},16}\), such dominance follows from \(c_{{\rm FM},16}\ge c_{\rm FM}>c_{\rm MV}\) at the smaller score. A maximizing branch is one of the two pure fractional branches, whose denominator depends only on its level. The identical proof gives \[ c_{{\rm FM},16}(W_2)> (W_2/W_1)^3c_{{\rm FM},16}(W_1). \tag{Q.sixteencubic} \] The denominator of \(\widetilde v_m\) also depends on the residual; no cubic score comparison for \(c_{\rm comp}\) follows from this argument. The pure-fractional comparisons remain available as lower bounds through the retained formulas. Appendix C supplies the available scores \(W_{\mathcal F}^{\rm joint}>\mathfrak A_bW_*>E_bW_*>0\), all below \(2^{-4096}\). Therefore (D.finitepool), with target one, and (Q.cubicscore) give a strictly stronger common-density bound than the baseline (Q.headline). The explicit comparisons are stated in (C.densitygain). The meaning of \(W_*\) is unchanged. \subsubsection{Finite level search}\label{finite-level-search} Each density formula can be approximated effectively. Once a positive rational incumbent for the mass maximum has been certified, a finite level cutoff follows from the decreasing tail bound, valid for \(m\ge32\), \[ u_m(W)\le 64W^2(7/8)^m+\tfrac32W(2/3)^m +4W^3b_{12}^{-2\lfloor(m+1)/12\rfloor}. \tag{Q.searchtail} \] The first two terms were proved above without the additional assumption \(m\ge n\). For the third, all \(b_r\ge1\) and the squared fan prefactor exceeds \(1/4\): indeed \((512/81)/(9+4\sqrt2)>512/1215>1/4\). Also \(L^2>1\). This gives the stated tail, which tends to zero and can be bounded by exact rational arithmetic when \(W\) is rational. The actual \(W_*\) is rational by (R.greedy) and the finite rational startup recipe. A rational positive incumbent exists, for example by using \(\epsilon_m\le C_6m^{-A}\) at a sufficiently large explicit \(m\) and certified logarithm bounds. Compact-width minimization on a fixed bounded interval uses continuous elementary functions with computable uniform error bounds. The remaining finite maximum can therefore be approximated to any prescribed accuracy, using certified enclosures for its finitely many values. For the refined formulas, a tail valid for \(m\ge32\) is \[ \begin{aligned} u_m^{\rm comp}(W)\le{}&64W^2(7/8)^m+\tfrac32W(2/3)^m\\ &+16W^3\bigl(b_{12}^{-2\lfloor(m+1)/12\rfloor} +b_{16}^{-2\lfloor(m+1)/16\rfloor}\bigr). \end{aligned} \tag{Q.refinedtail} \] Indeed both fractional fan ceilings exceed \(2\mu^{3/2}\) there: the common prefactor exceeds \(1/2\), every remainder is at least one, and \(b_{12}^2,b_{16}^2>4\). Thus \(\widetilde v_m\le4\max\{v_m,v_{16,m}\}\). The argument for the third term of (Q.searchtail) applies to each fractional block length. This proves the displayed tail and finite search for both new optimizers. Their expressions are continuous at zero residual by their common bound \(p_m/\mu\), so equality decisions at a zero residual are unnecessary for approximation to a requested accuracy. \subsubsection{The absolute scale of the density formulas}\label{the-absolute-scale-of-the-density-formulas} \textbf{Proposition 11.3 (score-to-density scale).} Put \(a=\log_2(1/W)\) for \(02^{(a-9)/95}>32\). In (Q.searchtail), use \(b_{12}>2\), \((8/7)^6>2\), \((3/2)^6>2\), and \(\lfloor(m+1)/12\rfloor\ge(m-11)/12\). Its three terms are respectively at most \(64W^2\,2^{-m/6}\), \((3/2)W\,2^{-m/6}\), and \(16W^3\,2^{-m/6}\), since \(2^{11/6}<4\). As \(W<1\), their sum is at most \((163/2)W\,2^{-m/6}\). The recipe vanishes below \(n\); on its radial domain \(\mathcal D(u)\le(128/225)u\). Maximizing and using \((128/225)(163/2)=10432/225<47\) proves the upper bound. At \(m=m_+(W)\), (M.envelope) gives \(\epsilon_m\le C_6m^{-2314/25}\le W/3\). Since \(\kappa<5/4\), the resulting error is less than \(W/2\). The maximum branch has \(S_m\le(8/9)2^m\); its inverse and \(\mathcal D(u)\ge64u/225\) give the lower bound, exactly as in (D.elementarydensity). \(\square\) These bounds concern the chosen formula. The error-envelope floor used for the upper bound is not a lower bound on true mixing error. For fixed \(C_6\) and tables, as \(W\downarrow0\), they imply \[ \frac a{95}-O(1) \le\log_2\log_2(1/c_{\rm FM}(W)) \le\frac{25a}{2314}+O_{C_6}(1). \tag{Q.logscale} \] For fixed \(H,P>0\), (R.scorelower)--(R.scoreupper) give \(W(q,P)=2^{-\Theta(3^q)}\) as \(q\to\infty\). Multiplying by the fixed \(\mathfrak A_b\) retains this score scale; composition with (Q.formulascale) gives \(c_{\rm FM}(\mathfrak A_bW(q,P))=2^{-\,2^{\Theta(3^q)}}\). This describes the formulas at fixed parameters as the residue level grows. The canonical level satisfies the domain bound (C.domainguard). The upper bound in (Q.formulascale) retains its literal twelve-step meaning. For the strongest retained formula, a separate estimate is \[ c_{\rm comp}(W)< 55W\,2^{-\frac18\,2^{(a-9)/95}} \qquad(0(11/10)^{12}\), \(b_{16}^2>(11/10)^{16}\), and \(b_{12}^2,b_{16}^2<6\). For either block length \(d=12,16\), \[ b_d^{-2\lfloor(m+1)/d\rfloor} \le b_d^2(10/11)^{m+1}<6(10/11)^m. \] Hence each pure fractional mass is at most \(24W^3(10/11)^m\), and the complementary one at most \(96W^3(10/11)^m\) for \(m\ge32\). The two nonfractional branches are bounded by \(64W^2(7/8)^m\) and \((3/2)W(2/3)^m\), so their maximum together with the complementary branch is at most \(96W(10/11)^m\), since \(W<1\). All branches vanish below the first positive level \(n\). Use \(\mathcal D(u)\le(128/225)u\), \((128/225)96<55\), \((11/10)^8>2\), and \(n>2^{(a-9)/95}\). This proves (Q.refinedscale). The lower bound in (Q.formulascale) remains valid by inclusion of the old formula. Thus the two-sided logarithmic scale in (Q.logscale) also holds for \(c_{{\rm FM},16}\) and \(c_{\rm comp}\), with their own constants. These upper bounds concern the defined density formulas. \subsubsection{Comparison with Mazur's printed bound}\label{comparison-with-mazurs-printed-bound} With \(L_3(x)=\log_2\log_2\log_2 x\), (E.coefficientcomparison) gives \(L_3(C_{\rm primitive})<176\) and \(L_3(C_{\rm primitive}^{\rm Maz})>2^{172}\). \hyperref[appendix_provenance]{Appendix B} proves that this coefficient decrease strictly improves the baseline density formula, accounting for both startup rounding and score allocation. We compare the actual arithmetic recipes, retaining the root parameters printed in {[}3{]}. The printed primitive coefficient remains \(C_{\rm primitive}^{\rm Maz}\), while ours is the smaller local coefficient in (M.canonicalC). The printed sixth-order adapter is \(C_{\rm Maz}\) from (M.sourceadapter), while our proved adapter is \(C_6=2D+2\mathcal M^{-1},\qquad c_{\rm MV}(W_*)>\frac{22208}{825}c_{\rm Maz},\qquad c_{\rm FM}(W_*)>\frac{44416}{825}c_{\rm Maz}. \tag{Q.mazgain} \] \emph{Proof.} Our construction starts at \(b=2^{25}\), with \(t=\lceil\log_2F_b\rceil\), \(N\le4t/5\), \(q=q_N\), and \(H=2b+1\). Since \(F_{b,{\rm Maz}}>F_b\) and these formulas increase with \(b\), \(t_{\rm Maz}\ge t\) and \(N_{\rm Maz}>N\). The floor recurrence, the summands in (B.conductor), and the terminal quarter are nondecreasing in their initial value. The extra positive summand gives \(q_{\rm Maz}\ge q+1\); also \(H_{\rm Maz}>H\). By (R.scorelower), (R.denominator), and (R.conductorguards), \[ \begin{split} \log_2(W_*\mathcal M) &>3-1575N-6305+q\log_2 3 +2(H_{\rm Maz}-H)+2(3^{q_{\rm Maz}}-3^q)\\ &>4\cdot3^q-q-6302>0. \end{split} \] Here \(q\ge bN\ge1575N\). The final expression is already positive at nine and increases thereafter, whereas our levels exceed nine. This proves the first assertion. Set \(m=m_{\rm Maz}\ge2\). The chosen envelope (M.envelope), with exponent \(2314/25>6\), gives \[ \epsilon_m\le C_6m^{-6}\frac{347}{352\mathcal M}, \] using \(\kappa<5/4\) and \(W_*>\mathcal M^{-1}\). Since \(S_m\le(8/9)2^m\) and \(\mathcal D(u)\ge64u/225\), \[ c_{\rm MV}(W_*)\ge\mathcal D(p_m/S_m) \ge\frac8{25}p_m2^{-m} >\frac{347}{1100\mathcal M2^m} =\frac{22208}{825}c_{\rm Maz}. \] The radial domain is (D.domain). Finally (Q.paid), applied to \(W_*\) using (R.scoredomain), gives the last assertion. \(\square\) Retaining the symbolic score in the same calculation gives a stronger comparison. Put \(R=W_*\mathcal M\) and \(E_q=4\cdot3^q-q-6302\). The proof above gives \(R>2^{E_q}>1\) and, at \(m=m_{\rm Maz}\), \(p_m>(R-5/352)/\mathcal M\). Therefore \[ \frac{c_{\rm MV}(W_*)}{c_{\rm Maz}} >\frac{2048}{75}\left(R-\frac5{352}\right),\qquad \frac{c_{\rm FM}(W_*)}{c_{\rm Maz}} >\frac{4096}{75}\left(R-\frac5{352}\right). \tag{Q.mazsymbolic} \] The first bound uses \((8/25)/(3/256)=2048/75\); the second uses the certified factor greater than two at the same \(W_*\). Substituting \(2^{E_q}\) for \(R\) preserves strict lower bounds. The comparison is symbolic in the score and conductor. Thus the conservative factor over the printed constant is \(44416/825\approx53.8376\). Against the benchmark obtained by doubling that constant through dyadic closure, the factor is \(22208/825\approx26.9188\). The explicit numerical clock specialization here is \(10431/1000=10.431\), compared with \(523/50=10.46\) in {[}3{]}; both count ordinary Collatz steps and use natural logarithms. This printed benchmark is distinct from \(c_{\rm MV}(W)\), which fixes our root family and optimizes the maximum and second-moment estimates. The former comparison measures the change from the stated earlier bound; the latter isolates the direct fractional-moment gain. The benchmark {[}3{]} also includes a single explicit finite-cutoff recipe. Here the proof gives effective eventual thresholds for every strictly smaller density constant, without assembling them into a comparable final cutoff formula. \subsubsection{Further quantitative improvements}\label{further-quantitative-improvements} The convex-order inequality (F.convexfan) and optimal allocation formula (D.hinge) in \hyperref[appendix_reference_consequences]{Appendix D} identify a further route: certify upper-tail profiles, then compare the resulting density bound with the complete \(c_{\rm comp}(W_*)\), keeping the roots and mixing envelope fixed. The challenge is to propagate the finite distributions efficiently while preserving certified upper bounds and to control the remaining level search. A quantitative comparison can use the exact two-sided inverse (D.twosidedcapacity) as well as propagated profile bounds. Other direct moment orders can be used in (D.generalp). Interpolating existing moment ceilings is insufficient on its own. Indeed, suppose \(10\), its full logarithm is \[ \log v_s=\frac{s}{s-1}\log(p/\mu) -\frac{\log L}{s-1}-D_s(H_m/\mu), \] where \(D_s(h)=\log\langle h^s\rangle_m/(s-1)\) for a mean-one density \(h\). The residual and capacity terms remain in this formula alongside the divergence. Adding a certified branch to the maximum can never decrease the bound; a strict gain still needs proof against the whole optimized formula. Polynomial or subexponential collision and fractional-moment bounds remain open. Absolute density also depends on the root score. From (D.domain), \[ c_{\rm FM}(W)\le \frac{32}{225}\left(e^{9W/4}-1\right) \sim\frac8{25}W\quad(W\downarrow0). \tag{Q.scoreceiling} \] This bounds the certified formula, not the true convergence density. Even the exact residue-capacity optimum satisfies \(\mathcal C_m(p_m)\le W/\mu\), by the feasible uniform allocation in (D.hinge). Thus a substantial absolute lower bound requires improving the root score or using further constraints on the actual sources, in addition to improving concentration estimates. A separate analytic route concerns the head localization in Section 6. With the present squared-width loss, its two competing exponents are \[ a(v)=6407-\tfrac12(\log2)v^2,\qquad q_v-1=(\log2)v-1. \tag{Q.headtradeoff} \] The first decreases and the second increases with \(v>0\). Their equality determines the best asymptotic exponent within this specific tradeoff; \(v_*=6749/50\) gives the retained exponent \(2314/25\) with explicit rational slack. Improving the quadratic localization loss, or replacing that argument by a multiscale or exponential-moment estimate, could improve the resulting exponent. Such a change must reprove the head-frequency estimate, rejection bound, level induction and complete envelope with explicit constants. The parameterized clock is established in (N.everyclock)--(D.everyclock), with the full fixed density bound for every \(c>3/\log(4/3)\). The endpoint clock remains unproved. A family of bounds valid for every strictly larger coefficient does not imply the endpoint, and the reference-model mean does not establish deterministic optimality below it. \subsection{12. Sharper persistence and source counting}\label{chapter12} The preceding results use a fixed root score and several reductions from weighted trajectories to distinct integers. This section retains information discarded in those reductions. It gives a smaller seed floor, a score based on the location of surviving root mass, adaptive conditional moment bounds, and additional sources obtained from the stopping construction. The original constants in (I.constants) retain their definitions. The resulting bounds form an explicit family: a seed stage is chosen once, before the clock and the counting cutoff. Optimization over the later reference level is part of each bound. The exact product-profile bound and the bound using conditional moments are both retained; their complete small-score comparison is proved below. These refinements do not change \(c_0\). \subsubsection{A smaller seed floor with its full variation cost}\label{a-smaller-seed-floor-with-its-full-variation-cost} Write \(b_{\rm old}=2^{25}\), and retain \(C_6\), \(t\), and the original startup \(N\) from Section 8. Define \[ \begin{gathered} B_0=100000,\qquad B_{j+1}=B_j+\lfloor B_j/100\rfloor,\qquad w(B)=\lceil(4096/125)B^{3/5}\rceil,\\ \alpha=100999/100000,\quad h=503/500,\quad \sigma=797/2000,\quad \nu=h\sigma,\quad A_h=10923000,\quad \eta=1-A_h^{-1},\\ a_*=2314/25,\qquad E=t-134160956,\\ V_\dagger(n)=2^E\left\{\frac{\nu^n}{1-\nu} -\eta\frac{\sigma^n}{1-\sigma}\right\} \quad(n\ge200),\qquad \Pi_\dagger=1-2^{-2046}. \tag{P.seed} \end{gathered} \] At stage \(j\), retain exactly the words of depth \(B_j\) whose valuation sum differs from \(2B_j\) by at most \(w(B_j)\), with their original weights. \textbf{Lemma 12.1 (paid persistence).} These seeds have total reference survival at least \(\Pi_\dagger\). Their marks are physically realized at every positive odd root at least \(2^{100000}\). For every \(n\ge200\), \(V_\dagger(n)\) bounds their entire subsequent variation, uniformly over these roots. The finite graft and the ordinary-clock source construction apply to every fixed finite admissible root pool above this height. \emph{Proof.} The floor recurrence gives \(B_j\ge100000\alpha^j\). Since \((101/100)^3(7231/100000)B>B/16. \] Its inverse slope is below \(2^{-B/16}\), and its offset is at most \((3/2)^B-1\). At a parent at least \(2^B\) this offset is less than half the parent. As \(B/16-1\ge4\lfloor B/100\rfloor\), induction and the offset composition formula give \[ x_j\ge2^{100000}16^{B_j-100000},\quad \omega_{\rm pre}\le16^{100000-B_j},\quad C_{\rm pre}<\frac{32}{29}(3/2)^{100000}<2^{99999}. \tag{P.geometry} \] The partial offsets are smaller, so the same argument realizes every prefix. Also \(\log_2x_j>100000+(21/4)(B_j-100000)\), using \(\sum_{i4\), the stronger old height premise \(x_j\ge16^{B_j}\) holds from stage 200. The fixed-time geometric tail, with this wider window, gives \[ 1-p_j\le2\exp\{-w(B_j)^2/(4B_j+3w(B_j))\} <2e^{-2048\alpha^{j/5}}. \] Here \(4+3(32769/100000)<5\) and \(4096/125>32\). Since \((1001/1000)^5<\alpha\), summing gives \[ \sum_j(1-p_j) <2e^{-2048}\frac{3048}{2048} <(381/128)2^{-2048}<2^{-2046}. \tag{P.survival} \] Finite products are at least one minus the sum of their deficits. For variation, keep the full exponent \(a_*\) from (M.powers). The degree-468 exponential-series term bounds the window failure by \(S_*B_j^{-a_*}\), where \(S_*=8\cdot468!\cdot576^{468}<2^{8895}1,\qquad 8^{a_*}<2^{278},\\ 2A_h(2^{279}+1)<2^{279}b_{\rm old},\qquad 3^{100000}<2^{158497},\qquad (b_{\rm old}/100000)^{a_*}<2^{777} \end{gathered} \] bound it by \(2^E\). Indeed \(2^{279}b_{\rm old}2^{b_{\rm old}}(C_6+1)/b_{\rm old}^{a_*}\) is the old \(F_b\) times \(2^{-100665798}\); the remaining saving is \(2^{-33495158}\). Each increment is consequently bounded by \(2^E(\nu^j-\eta\sigma^j)\). Summing the positive terms proves (P.seed), with its negative term retained. For the graft, \(\omega_{\rm pre}C_{\rm app}\to0\), so one common late splice has total offset below \(2^{99999}+1<2^{100000}\). The original polynomial tag bound also bounds (P.tags). Transition and continuation errors still tend to zero, while the recovered late height supplies their physical premise. The splice is fixed before the physical cutoff. This verifies the changed interfaces to Sections 4--5. \(\square\) Call \(n\) \emph{eligible} if \[ n\ge200,\qquad V_\dagger(n)<3/100. \tag{P.eligible} \] There are eligible stages because \(\nu<1\). An explicit comparison witness is \(n=N-b_{\rm old}\). In fact \(N>2b_{\rm old}\), since \(V(2b_{\rm old})>1\) by \(t\ge4b_{\rm old}\) and \(\lambda>1/4\). The rational guard \(\nu<\lambda\), and \(2/5<\nu\), give \[ V_\dagger(N-b_{\rm old}) <2^{-134160956}\nu^{-b_{\rm old}}V(N) <2^{-89421713}<3/100. \tag{P.witness} \] For the first comparison, bound the new tail above by \(\nu^{N-b_{\rm old}}/(1-\nu)\), and the old cubic tail below by \((N+1)^3\lambda^N/(1-\lambda)\). Thus this witness does not depend on reconstructing a sequence of intermediate startup improvements. Any earlier eligible stage remains an allowed, separately computable choice. \subsubsection{Locating persistent mass on the root list}\label{locating-persistent-mass-on-the-root-list} Fix an eligible \(n\), and put \[ D=\sum_{j\Pi_\dagger. \tag{P.subdensity} \] This is not a normalized conditional distribution. Fixed total depth and the equality \(q-D=a\) are essential. Let \(V=V_\dagger(n)\) and \(z_i=(g(R_{s_i})-V)_+\). These are persistent physical marks by Lemma 12.1. For \(1\le m\le q\), write \(K_m=2\cdot3^{m-1}\). The identity \(\nu_3(4^u-4^v)=1+\nu_3(u-v)\) shows that \(s\mapsto R_s\bmod3^m\) is a bijection of one period and takes unit indices to unit residues. Hence \(\rho_m(R_{s_i})\) has period \(K_m\) and sum \(K_m\) over each period. Nonnegativity gives a prefix sum at least \(j-K_m\). Reference mixing, in unit normalization, gives total absolute discrepancy at most \(K\epsilon_m\). Its negative part has mass at most \(K\epsilon_m/2\), since both full vectors have total \(K\). Subtracting this loss and the entire selected deficit in (P.subdensity) therefore proves \[ \sum_{i\le j}z_i\ge [(1-V)j-K_m-K(1-p+\epsilon_m/2)]_+. \tag{P.prefix} \] The coefficient has not been hidden in the choice \(m=a\). From the definition of \(E\), \(2^E\ge2^{57264}(C_6+1)\). The first positive term in the tail, \(n\ge200\), and \(h^{200}>3/2\) give \(V>2^{57263}C_6\sigma^n\). As \(a\ge B_n/8\) and \(\alpha^{-a_*}<\sigma\), \[ \epsilon_a\le C_6a^{-a_*}<2^{-56985}V,\qquad K_a/K=3^{-D}<2^{-2048}. \tag{P.prefixerror} \] Define the following exact rational quantities: \[ \gamma=1-V,\quad E_{\rm pre}=K_a+K(2^{-2046}+2^{-56986}V),\quad j_0=\lfloor E_{\rm pre}/\gamma\rfloor+1, \] \[ W_{\rm pre}(n)= \max_{j\in\{j_0,j_0+1\}} \frac{\gamma j-E_{\rm pre}}{R_{s_j}},\qquad s_j=50001+\lfloor(3j-1)/2\rfloor. \tag{P.prefixscore} \] Both candidates lie between 1 and \(K\). Indeed \(E_{\rm pre}/K<11\cdot2^{-2049}\) and \(\gamma>97/100\). For \(\bar j=\lceil K/2^{2044}\rceil\), \[ (\gamma\bar j-E_{\rm pre})>(501/800)\bar j>\bar j/2. \] Before \(j_0\) the positive part vanishes. After \(j_0+1\) the numerator increases by a factor less than two at each step, whereas consecutive unit roots increase by a factor greater than four. Thus (P.prefixscore) is exactly the best single-prefix consequence of (P.prefixerror); it requires two comparisons, rather than enumerating the root period. Summation by parts, or just \(\sum_{i\le j}z_i/R_{s_i}\ge(\sum_{i\le j}z_i)/R_{s_j}\), proves \[ 0K/2^{918}. \] To construct it, remove the union of these \(J\ell\le K/2\) positions in residue coordinates and distribute the required mass with height at most two on the complement. The geometric root tails give the displayed cost. The inherited cap exceeds two. Here \(K>16J^2\), so \(K\le\ell^2\le2^\ell\). The prefix witness above has \(W_{\rm pre}(n)>2^{-6\bar j}\), with \(\bar j\le K/2^{2043}\) and \(H\le\bar j\). Consequently \[ W_{\rm pre}(n)>2^{K/2^{919}}W_{\rm mc}(n). \tag{P.rootgain} \] This compares guaranteed scores for the same finite pool; the scores cannot be added over their overlapping roots. \subsubsection{A radius-four source family and its stopping antichain}\label{a-radius-four-source-family-and-its-stopping-antichain} Retain the terminal parameters \(b,w_b,l_b,h_b,e_b,r_b\) from Section 5. Replace its barrier and shift set by \[ H_v(s)=2b+\lceil(\log_2 3)(s-b)+v\rceil-r_b,\qquad v\in\{j/b:0\le j\le b(2r_b-1)\}. \tag{P.barrier} \] The single ceiling has error in \([0,1)\), also when \(s0\), \(k=\lceil\theta B\rceil\), and the fixed graft constants, their total error is bounded by \[ \frac23P_t(B+1)^2(C_*+1)[b(2r_b-1)+1] \{2C_6/k^6+2e^{-b/64000}+2^{-(e_b+1)}\}=o(1). \tag{P.shiftpayment} \] Its polynomial part is \(O_\theta(P_tB^{-2})\). The original total-step allowance \(481b/100\), fine-precision condition, and \(c_\theta-c_0<3\theta\) remain valid. This proves the source theorem with \(R_X\to4\), the same score, and every fixed \(c>c_0\). We will use one more producer property. At each fixed cutoff the selected odd sources form an antichain for \(U(x)=(3x+1)/2^{\nu_2(3x+1)}\). To see this, parse complete reversed histories at their fixed seed cuts, their prefix-free transition, and their fixed number of prefix-free macro blocks. Comparable central words have the same cuts and the same parent. The deterministic terminal shift at that parent is then the same. Compressed terminal words still obey the first-crossing rule; a proper extension would already have crossed. The complete word family is therefore prefix-free. If \(U^h(x)=z\) for selected sources and \(h>0\), distinct root owners contradict the root antichain; the same owner contradicts prefix-freeness. All roots are visited before one, excluding a cycle exception. This property follows from the actual construction, rather than from the reciprocal-weight inequality alone. For a general fixed target, the same reasoning applies at all sufficiently large cutoffs. If that target lies on a cycle, exclude its finite set of cycle states by increasing the cutoff; if it is not periodic, a path cannot return from the target to a source that already reaches it. This uses no assumption that the target itself reaches one. \subsubsection{Conditional reference laws and a finite convex envelope}\label{conditional-reference-laws-and-a-finite-convex-envelope} Let \(X_n\) be the mean-one law of \(f_n\) on either unit coset divided by its first-digit value \(f_1(a)\). The two cosets have the same normalized law, and \(X_1=1\). In the affine mixture (F.mixture), condition on the coarse residue \(Y\bmod3^v\) and a suffix word of length \(v-1\). Its coefficient is \(3^{v-1}\theta_w f_1(a_w)\); these coefficients sum to \(f_v(Y)\). Each normalized fine fibre has law \(X_u\). Jensen for these actual coefficients gives \[ X_{u+v-1}\preceq_{\rm cx}X_uX_v,\qquad u,v\ge1, \tag{P.shared} \] where the variables on the right are independent comparison variables. Zero coarse density gives zero contributions, and the countably many words can be grouped by their finitely many induced maps before Jensen. Projection gives \(X_j\preceq_{\rm cx}X_n\) for \(j\le n\). Consequently \[ H_m/\mu\preceq_{\rm cx}\prod_{i=1}^{d_m}X_{16}^{(i)}, \qquad \mu=8/9,\quad d_m=\lceil m/15\rceil. \tag{P.product} \] The product is an envelope, not the distribution of independent physical Collatz paths. Here is a compact finite replacement for \(X_{16}\). Use the upward recurrence of Section 9 at scale \(S=2^{48}\) on its \(3^{15}\) entries in the first unit coset. Spread each upper entry to its adjacent endpoints in the grid \(\{0\}\cup\{2^{-24}(17/16)^k:0\le k\le378\}\), preserving its mean. For a bin with one entry retain that entry itself. In particular the two singleton values are \[ 40016666875724526/S,\qquad135156244001121987/S. \] Bin counts and scaled sums determine these endpoint probabilities exactly. Remove the excess mean from the smallest positive atoms, transferring the corresponding probability to zero and splitting at the threshold. Denote the resulting 146-point probability law by \(X^\sharp\), including its zero atom. For clarity, this repair has a proof. If \(U\ge X_{16}\) is the upper array and \(V\) its mean-preserving endpoint spread, put \(X'=(X_{16}/U)V\), with zero at \(U=0\). Then \(X_{16}\preceq_{\rm cx}X'\le V\). For any convex \(\Phi\), the slopes \((\Phi(v)-\Phi(0))/v\) increase with \(v>0\). The chord bound shows that, among variables of mean one bounded by \(V\), retaining mean at the largest atoms maximizes every convex test. This is exactly the stated repair. Singleton contractions preserve it: the removed excess is below both changed bins. Thus \[ X_{16}\preceq_{\rm cx}X^\sharp,\quad EX^\sharp=1,\quad E(X^\sharp)^{33/32}<10227/10000,\quad E\sqrt{X^\sharp}<24/25. \tag{P.envelope} \] The last inequality also follows from (P.product) and \(X_2\) uniform on \(\{8/7,11/7,2/7\}\), whose square-root mean is less than \(143/150\). Directed rational root enclosures certify the displayed finite moment; floating-point values are not certificate premises. A stronger conditional bound retains two digits. For \(u\ge h\) and \(f_h(a)>0\), let \(X_{u\mid h,a}\) have values \(f_u(a+3^hz)/f_h(a)\) under uniform fine lifts. In the suffix mixture of length \(v-h\), use coefficients \(\beta_w=3^{v-h}\theta_w f_h(a_w)\). Their sum is exactly \(f_v(Y)\). Therefore \[ M_p(u+v-h)\le M_p(v)\max_a E X_{u\mid h,a}^{\,p}, \qquad p\ge1,\quad u,v\ge h. \tag{P.conditional} \] The maximum cannot be replaced by a uniform average of the fibres; their moments need not agree. At \(h=2\), the states \(1,4,7\bmod9\) have normalizers \(8/7,11/7,2/7\); coset doubling accounts for the other unit states. Split the same depth-sixteen histogram by these states, normalize each, and apply the mean-one repair separately. Exact rational certificates give \[ \max_a E X_{16\mid2,a}^{33/32}639/1000\) give \[ q_{\rm ad}(m,r)\ge r e^{-(309/500)L_m},\quad t_*(m,r)<1/32,\quad \log\frac{q_{\rm ad}(m,r)}{r^{33}B^{-32L_m}} >(21/1000)L_m+32a. \] The square-root and residual penalties are retained in the query. Its leading exponent per reference level is \(h_{\rm ad}/14\), about \(0.0440164\), compared with \(32\log B/14\), about \(0.0457113\), for the fixed-order query. The following recipe keeps both orders. \subsubsection{A closed conditional matrix with its tilt cost}\label{a-closed-conditional-matrix-with-its-tilt-cost} The conditional maximum can be sharpened by retaining the state vector. Fix \(v=16\) and \(A=\{1,4,7\}\bmod9\). Let \(q_a\) equal \(f_2\) on the pair \(\{a,2a\bmod9\}\) and vanish elsewhere, and let \(\beta_a\) be its image under \(v-2\) original positive linear reference transfers. These fields sum to \(f_v\). For \(b\in A\), define \[ \mathsf T_v(t)_{ba}= \frac1{3^{v-2}f_2(b)} \sum_{\substack{Y\in G_v\\Y\equiv b\ (9)}} \beta_a(Y)\left(\frac{f_v(Y)}{f_2(b)}\right)^t, \qquad t\ge0. \tag{P.statematrix} \] Zero density contributes zero. If \(e_u(a,1+t)=E X_{u\mid2,a}^{1+t}\), conditional Jensen on each \(b\)-fibre gives the componentwise inequality \[ \boldsymbol e_{u+v-2}(1+t) \le\mathsf T_v(t)\boldsymbol e_u(1+t),\quad u\ge2. \tag{P.stateclosure} \] The suffix length is fixed, so one matrix can be iterated. At \(t=0\), every row is \(\pi=(8/21,11/21,2/21)\), for every \(v\ge3\). Indeed the depth-one projection of \(q_a\) is \(\pi_a f_1\). Commutation of the linear transfer with uniform projection therefore gives the depth-two projection \(\pi_a f_2\) after the suffix transfers. The identity \(L\phi(2y)=2L\phi(y)\) for \(y\equiv1\pmod3\) follows by shifting the geometric summation index; its omitted term has the wrong integrality residue. It identifies the paired conditional moments. Thus the rank-one order-one matrix and the closure follow from the actual transfer, rather than from a numerical transition model. The three repaired state laws have size-biased entropy upper bounds \[ h_1=h_7=423112/10^6,\quad h_4=616230/10^6,\quad h_{\rm st}=\sum_a\pi_a h_a=220193/420000. \] Set \(d_a=h_a-h_{\rm st}\) and \(w_a(t)=1+td_a\). Then \(\pi\cdot d=0\) and \(|d_a|<1/5\). For an actual conditional law \(X=f_{16}/f_2(b)\), the positive transfer bound gives \(X<2^{18}\). Under its size bias put \(Z=\log X\). Convex-order domination supplies \(E_QZ\le h_b<1\). The inequalities \(x|\log x|\le1/e\) and \(x(\log x)^2\le4/e^2\) on \(00\). For each positive upper tail let \(A_i=E[X1_{X\ge x_i}]\) and \(P_i=\Pr(X\ge x_i)\), and set \[ B_X(t)=\max_i A_i^{1+t}/P_i^t,\qquad c_t=t^t/(1+t)^{1+t}. \] Writing the hinge as the maximum of its tail lines proves \[ \sup_{s>0}s^t E(X-s)_+=c_tB_X(t),\qquad B_X(t)0}\) has supremum \(c_t\) and integral \(1/[t(t+1)]\). Its lattice sum is at most \(2c_t+1/[\Delta t(t+1)]\), by comparison with integrals on either side of the maximum. The resulting hinge coefficient is \[ \begin{split} R_d(t)&=\frac4{\alpha_td} +\frac2{\sqrt{\alpha_td}}\left(2+ \frac1{c_t\Delta t(t+1)}\right),\\ C_d(t)&=\min\{B_X(t)M(t)^{d-1},\,M(t)^dR_d(t)\},\\ q_t(d,r)&=r^{1+1/t}C_d(t)^{-1/t},\qquad q_{\rm fac}(d,r)=\sup_{t>0}q_t(d,r). \end{split} \tag{P.factored} \] Set both queries to zero when \(r=0\). These are lower bounds for the product allocation profile. They retain the ordinary moment query since \(B_X(t)0\), define \[ u_{\rm ov,m}=\max\left\{u_m^{\rm comp}(W), \frac{p_{16,m}^3} {L^2[B'_m-(\mu-p_{16,m}/L)^{3/2}]^2}\right\}, \tag{P.complement} \] and set it to zero otherwise. It is a valid harmonic-mass lower bound also for the new smaller shell. For \(m\ge60\), its explicit decreasing bound is \[ U_m(W)=64W^2(7/8)^m+(3/2)W(2/3)^m +16W^3\gamma_{15}^{-2(\lfloor m/15\rfloor-1)}. \tag{P.oldtail} \] For the last term, \(B'_m\ge J_f\gamma_{15}^{\lfloor m/15\rfloor-1}>2\mu^{3/2}\); the first two terms are the maximum and second-moment branches. The shared complementary branch dominates the older fractional branches. \subsubsection{A single density recipe retaining the completed refinements}\label{a-single-density-recipe-retaining-the-completed-refinements} The harmonic capacity of the new shell is \(T=\log2\), rather than \(2\log2\). At fixed \(m\ge2\) define \[\begin{gathered} p_m=(W-\mu T\epsilon_m)_+,\quad r_m=p_m/(\mu T),\quad d_m=\lceil m/15\rceil,\\ q_{\rm con,m}=r_m^{33}B^{-32L_m},\\ q_{\rm ad,m}=q_{\rm ad}(m,r_m),\quad q_{\rm st,m}=q_{\rm st}(m,r_m), \end{gathered}\] \[ u_m=\max\{u_{\rm ov,m},\ Tq_{\rm fac}(d_m,r_m),\ Tq_{\rm con,m},\ Tq_{\rm ad,m},\ Tq_{\rm st,m}\}. \tag{P.mass} \] The actual finite census coefficient is \(3^m/X+\tfrac12\log R_X\), before taking the cutoff limit. Thus \(p_m\) includes the changed mixing loss; it is not obtained by replacing a finite capacity by its limit prematurely. For \(W\le2^{-4096}\), all the displayed masses are at most \(W/\mu<\log(9/8)\). At cutoff \(Y\), put \(X_j=Y/4^j\). An odd source at \(x=X_jt\), \(1\le t<4\), has at least \[ d_j(t)= \begin{cases}2j,&1\le t<2,\\2j-1,&2\le t<4\end{cases} \] dyadic multiples below \(Y\). The minimal relaxed cost of harmonic mass \(u\le\log(9/8)\) in one band is \[ J^{\rm seq}_1(u)=2(e^u-1),\qquad J^{\rm seq}_j(u)=j(e^{2u}-1)\ (j\ge2). \] At \(j=1\), the bottoms of the two half-bands have the same cost per harmonic mass and must be filled together. For \(j\ge2\), filling the lower half alone suffices on this mass range. Summation gives \[ \mathcal D_4(u)=\sum_{j\ge1}4^{-j}J^{\rm seq}_j(u) =\tfrac12(e^u-1)+\tfrac7{36}(e^{2u}-1), \quad \tfrac89u\le\mathcal D_4(u)\le2u. \tag{P.radial} \] This proves a factor at least \(6575/2176>3\) over the old radius-sixteen conversion at the same positive mass. The capacity change also improves every profile branch: if \(r_i=p_i/(\mu T_i)\), then \(r_4\ge2r_{16}\), and convexity of a profile \(P\), with \(P(0)=0\), gives \(T_4P(r_4)\ge T_{16}P(r_{16})\). For \(t>0\), define the radial power mean \[ \mathcal M_{-t,j} =\left\{\frac1{2T}\int_1^4[d_j(s)s]^{-t}\frac{ds}{s} \right\}^{-1/t}. \tag{P.powermean} \] It is at most its geometric-mean limit \(2\sqrt{2j(2j-1)}\le4j\). The complete band bound is \[\begin{gathered} F_{j,m}=\max\left\{\begin{gathered} J^{\rm seq}_j(u_m),\\ M_\infty^{-d_m}J^{\rm seq}_j(Tr_m),\\ T\sup_{t>0}\mathcal M_{-t,j}q_t(d_m,r_m),\\ T\mathcal M_{-1/32,j}q_{\rm con,m},\\ T\mathcal M_{-t_*(m,r_m),j}q_{\rm ad,m},\\ T\mathcal M_{-t_{\rm st}(m,r_m),j}q_{\rm st,m} \end{gathered}\right\},\\ C_m=\sum_{j\ge1}4^{-j}F_{j,m}. \end{gathered}\tag{P.bandrecipe}\] All terms are zero at \(r_m=0\). This shorter maximum retains the earlier moment, adaptive-order, factored-hinge and singleton-refinement bounds: their moment queries are bounded by \(q_{\rm fac}\), their caps by \(M_\infty\), and their sequential terms by \(u_m\). More explicitly, the two chosen atoms and their size-biased probabilities are unchanged by the singleton refinement, while the maximum decreases. Thus its tilted-mixture probability increases, and both hinge coefficients decrease. The inherited entropy query is bounded by the \(t=1/32\) term, since \(E(X^\sharp)^{33/32}<10227/10000<8423/8192\) and \(r_m\ge2r_{16,m}\). These observations also cover the previous adaptive and fixed-order branches. No competing branch is discarded. Here is the direct source-to-count justification. Put \(o_x=xa_x\le1\). For \(\lambda,\zeta\ge0\), \[ o_xd_j(x/X_j)\ge o_x\frac{\lambda H_m(x)+\zeta}{x/X_j} -\left(\frac{\lambda H_m(x)+\zeta}{x/X_j} -d_j(x/X_j)\right)_+. \] Sum the last term over all odd integers in the band. Each coarse residue class has spacing \(2\cdot3^m\). Integral comparison on the two monotone half-bands has error at most \(2\cdot3^m\sup\psi\), where \(\psi\) is that positive-part function. This is \(o(X_j)\) at fixed \(m,\lambda,\zeta\). Convex-order domination bounds its averaged penalty; inserting the hinge bound from (P.factored) gives the third term of (P.bandrecipe). Hölder with (P.conditionalfan) gives the fourth; (P.adaptivemoment) gives the fifth and (P.statemoment) the sixth. At zero order the power mean is interpreted by its geometric-mean limit. A mean-one law bounded by \(M_\infty^{d_m}\) is convex-dominated by the law taking that value with probability \(M_\infty^{-d_m}\) and zero otherwise; its band allocation is the second term. The first term is (P.radial). These arguments concern actual source occupations and assume no favourable distribution of physical sources. The stronger exact product-profile alternative remains available. For \(Y_m=\prod_{i=1}^{d_m}X_{16}^{(i)}\), define \[ P_{Y_m}(r)=\min_{0\le a\le1,\ E[aY_m]\ge r}E[a],\quad u^{\rm ex}_m=\max\{u_{\rm ov,m},TP_{Y_m}(r_m)\}, \] \[\begin{gathered} \mathcal J_j(p,u;Y_m)= \sup_{\lambda,\zeta\ge0}\left\{\lambda p+\zeta u -\frac12E\int_1^4 \left(\frac{\lambda\mu Y_m+\zeta}{s}-d_j(s)\right)_+ds\right\}, \\ \mathcal C_{\rm ex}(W)= \max_{m\ge2}\sum_{j\ge1}4^{-j}\mathcal J_j(p_m,u^{\rm ex}_m;Y_m). \end{gathered}\tag{P.exactprofile}\] The same census proves this bound. Equivalently, \(\mathcal J_j\) minimizes the band cost over occupations with marked mass at least \(p\) and harmonic mass at least \(u\). Finite support and weak-* compactness give primal attainment; a strictly feasible constant occupation gives duality for our interior mass range. The dual also attains its maximum: its objective is at most \((3j-1)-\lambda(\mu T-p)-\zeta(T-u)\). Its separate residue-then-radius relaxation is \(J^{\rm seq}_j(\max\{u,TP_{Y_m}(r)\})\). Equality occurs exactly when that harmonic mass already supplies \(p\) at the mean marker \(\mu\). Thus retaining residue and radius can strictly improve the separate relaxations. The exact-profile and conditional branches use different information. The small-score comparison (P.adaptivegain) below pays for both complete alternatives, without assuming an ordering of their individual bands. \subsubsection{Additional sources with paid overlap and the same clock}\label{additional-sources-with-paid-overlap-and-the-same-clock} Fix finitely many bands and trim their actual sources to \[ A_j=S_{X_j}\cap[X_j,4X_j-1). \] The removed harmonic mass is \(\tfrac12\log(R_{X_j}/4)+O(1/X_j)=o(1)\). Trim after the fine-to-coarse comparison: at fixed \(m\), the removed coarse marked mass is at most \(\max H_m\) times this loss. Each \(A_j\) remains an odd-orbit antichain. For \(x\in A_j\), \(j\ge3\), choose \((e,r)=(2,1)\) if \(x<2X_j\), and \((e,r)=(1,2)\) otherwise. There is a unique \(k\in\{0,1,2\}\) with \(x+k\equiv r\pmod3\). Set \[ y=4^kx+(4^k-1)/3,\quad z=(2^ey-1)/3,\quad b=j-k, \qquad f(x)=\begin{cases}y,&y\notin A_b,\\z,&y\in A_b.\end{cases} \tag{P.fresh} \] Both \(y,z\) are odd integers in the same band \([X_b,4X_b)\); the unit-width trim pays the additive upper-end offsets. The residue choice gives integrality of \(z\), and \[ U(y)=U(x),\qquad U(z)=y,\qquad \nu_2(3z+1)=e. \] If \(y\in A_b\), its predecessor \(z\) cannot be in \(A_b\) by the antichain property. Hence \(f(x)\) is outside every selected \(A_i\). For one base band the map is injective. Different \(k\) have disjoint target bands. Within fixed \(k\), each output type determines its input. For \(k\ge1\), \(y\)-outputs have valuation at least three whereas \(z\)-outputs have valuation one or two; they cannot collide. For \(k=0\), \(y=x\) is already selected and only the \(z\)-case occurs. Use base bands \(j=3,6,9,\ldots\); their target triples are disjoint. All fresh odd cores and their dyadic towers are therefore distinct from one another and from the original towers. Choose \(c_01\). The same upper inequalities hold for a fixed target once \(x\) exceeds it; no target-to-one orbit assumption is introduced. At level \(m\), \(\#A_j\ge X_j(u_m-o(1))\). Each fresh output from base band \(j\) has at least \(2j-5\) dyadic multiples below \(Y\). The additional density contribution is consequently \[ u_m\sum_{n\ge1}(6n-5)4^{-3n}=\frac{23}{1323}u_m. \] This is an integer injection, not a new reference mark. Define \[ \mathcal C_1(W)=\max_{m\ge2}C_m,\qquad \mathcal C_2(W)=\max_{m\ge2}\{C_m+(23/1323)u_m\}, \qquad \mathcal C_{\rm rev}(W)= \max\{\mathcal C_{\rm ex}(W),\mathcal C_2(W)\}. \tag{P.finalrecipe} \] \textbf{Theorem 12.2 (refined quantitative convergence).} For every eligible seed stage \(n\), and every \(c>c_0\), \[ \underline d(\mathcal G_c)\ge \max\{c_{\rm comp}(W_{\mathcal F}^{\rm joint}), \mathcal C_{\rm rev}(W_{\rm pre}(n))\}>0. \tag{P.density} \] For any admissible fixed-target pool realized by the same source construction and supplying a score \(W\) in the same small-score domain, the counting conclusion \(\underline d(\mathcal G_c(y))\ge\mathcal C_{\rm rev}(W)\) also holds. The explicit canonical prefix score above is for target one. \emph{Proof.} Lemma 12.1 and (P.prefixscore) supply persistent root marks. The fixed-pool graft and (P.shell) supply the actual source families. Use the census and mass bounds for finitely many trimmed bands, then let the physical cutoff tend to infinity. Take arbitrarily close fixed dual or moment witnesses before that limit. Send the fixed graft tolerance to zero, and then increase the finite band collection. All counts are nonnegative; distinct odd cores have disjoint dyadic towers. This proves both alternatives in (P.finalrecipe). The earlier density theorem remains valid independently, giving the maximum in (P.density). The pool, score and reference level precede the choice of clock. \(\square\) \subsubsection{Comparisons, approximation, and scope}\label{comparisons-approximation-and-scope} Since \(F_{j,m}\le8ju_m\), one has \(C_m\le(32/9)u_m\). At an attaining level of the complete \(\mathcal C_1\) formula, \[ \mathcal C_2(W)\ge\frac{4727}{4704}\mathcal C_1(W). \tag{P.freshgain} \] Thus \(23/4704\), approximately \(0.48895\%\), is the last counting improvement only. It is not the combined improvement of this section, nor a measurement of the density of convergent integers. For comparison with the previous complete factored formula, omit only the conditional queries and their joint terms in (P.mass)--(P.bandrecipe), calling the result \(\mathcal C_0\). For \(0(508/263)\mathcal C_0(W). \tag{P.conditionalgain} \] Here is a direct comparison paying for every branch. Let \(A=\log_2(1/W)\), and let \(m\) attain \(\mathcal C_0(W)>0\). Then \(m>2^{(A-9)/95}>2^{43}\) by the lower error envelope. Set \(d=\lceil m/15\rceil\), \(k=\lceil1001m/1000\rceil\), and \(P=P_{\prod_{i=1}^dX^{\sharp(i)}}(r_m)\). Under the size-biased single-factor law, \(E\log X^\sharp>277/400\) and \(|\log X^\sharp|<8\); these are rationally enclosed properties of the finite certificate. Chebyshev supplies a product size-biased event of mass at least \(3/4\) on which the product exceeds \(\exp(d(277/400)-16\sqrt d)\). A fraction of that event carrying mass \(r_m<1/25\) is feasible for the profile, so \[ P\le r_m\exp\{-d(277/400)+16\sqrt d\}. \] Every band of \(\mathcal C_0\) is at most \(J^{\rm seq}_j(u_{\rm ov,m})+8jTP\). The error scaling gives \(p_k\ge W/1001\), \(r_k\ge W/1001\), whereas \(r_m<2W\). Using \(\log B<1/50\), \(\log1001<7\), and the initial and remainder factor in \(L_k\), one obtains \[ \log(q_{\rm con,k}/P)>m/5000-32A-234>4. \] The last inequality follows at \(A=4096\) from \(m>2^{43}\) and then from monotonicity of \(2^{(A-9)/95}/(32A+238)\). To cover the complementary branch directly, (P.oldtail) and \(7/4<\gamma_{15}<2\) give \(u_{\rm ov,m}<322e^{-m/15}\). The same conditional query obeys \[ \log\!\left(\frac{Tq_{\rm con,k}}{u_{\rm ov,m}}\right) >m/50-33A-240>4 \] whenever \(u_{\rm ov,m}>0\); a zero value needs no division. Thus \(q_{\rm con,k}>16P\) and \(Tq_{\rm con,k}>16u_{\rm ov,m}\). It follows that \(\mathcal C_0(W)<(25/72)Tq_{\rm con,k}\), whereas \(\mathcal C_1(W)\ge(8/9)Tq_{\rm con,k}\). Their ratio exceeds \(64/25>508/263\), proving the stated conservative comparison without assuming a winning old moment order. The adaptive orders also give strict comparisons with complete formulas. Let \(\mathcal C_{28}\) be \(\mathcal C_2\) with both adaptive queries and their joint terms omitted, and let \(\mathcal C_{29}\) omit only the state-vector query and its joint term. Write \(\mathcal B_{28}=\max\{\mathcal C_{\rm ex},\mathcal C_{28}\}\). For \(0e^{R_A/1000}\mathcal B_{28}(W), \tag{P.adaptivegain} \] \[ \mathcal C_2(W)>e^{R_A/250}\mathcal C_{29}(W) >e^{R_A/200}\mathcal B_{28}(W). \tag{P.stateweightedgain} \] These compare complete lower-bound formulas at the same score. We first bound the actual product-profile alternative in the required direction. The depth-sixteen original-law entropy certificate gives \[ E_Q\log X_{16}=D_1(16)-(2/3)\log2>69/100. \] Moreover \(X_{16}\le2^{16}\), so its size-biased log variance is below \(16^2+1=257\): the positive log is below sixteen, and the negative second moment is at most \(4/e^2<1\). Chebyshev and \(257/34^2<1/4\) therefore give \[ P^{\rm actual}_m:=P_{Y_m}(r_m) \le r_m e^{-(69/100)d_m+34\sqrt{d_m}}. \tag{P.actualprofileupper} \] A fractional occupation on the size-biased event proves this bound; its probability is at least \(3/4>r_m\). In each exact joint band, an occupation depending only on the product has cost \((3j-1)P^{\rm actual}_m\). Taking its pointwise maximum with the sequential occupation supplying \(u_{\rm ov,m}\) preserves both mass constraints. Since \(\sum_{j\ge1}4^{-j}(3j-1)=1\), the complete exact level is at most \[ \mathcal D_4(u_{\rm ov,m})+P^{\rm actual}_m \le2u_{\rm ov,m}+P^{\rm actual}_m. \] Let \(P^\sharp_m\) be the repaired product profile used above. Thus both old complete alternatives at level \(m\) are at most four times \[ \max\{u_{\rm ov,m},TP^\sharp_m,TP^{\rm actual}_m,Tq_{\rm con,m}\}, \tag{P.oldrecipeupper} \] using \(T>1/2\) and \(4727/1323<4\). Choose a level \(m\) attaining \(\mathcal B_{28}>0\), and put \(k=\lceil1001m/1000\rceil\). The same residual calculation gives \(m>R_A>2^{43}\), \(r_k\ge W/1001\), \(r_m<2W\), and \(L_k\le1001m/14000+2\). The ratio \(R_A/(A+7)\) increases for \(A\ge4096\); its base value gives \(\log(1/r_k)m/800,\\ [(69/1500)-g]m-34\sqrt m-10&>m/800,\\ [(639/14000)-g]m+32A\log2-42&>m/800,\\ [(1/15)-g]m-A-16&>m/800. \end{aligned} \] The profile rate gaps exceed \(19/10000\) and \(18/10000\); \(34\sqrt m2^{43}>(34\cdot80000)^2\). The fixed-order gap exceeds \(7/5000\). For the last inequality use \((1/15)-g>1/50\), \((15/800)2^{43}>4096+16\), and the increasing ratio \(R_A/(A+16)\). Zero denominators need no division. The new level is at least \((1199/1323)Tq_{\rm ad,k}\). The prefactor \(1199/(4\cdot1323)>1/5\), with \(\log5<214\cdot10^8(4096+7)\) gives \(\log(1/r_k)1/5\) and using \(m/1000>2\) gives \(\mathcal C_2>e^{R_A/250}\mathcal C_{29}\). Combining the two exponents proves (P.stateweightedgain). All these consumers scale at least linearly with \(W\). At a fixed level, increasing \(W\) by a factor \(s\ge1\) increases each positive residual by at least \(s\). The queries \(q_t/r\), \(q_{\rm con}/r\), \(q_{\rm ad}/r\) and \(q_{\rm st}/r\) increase with \(r\); both chosen adaptive orders decrease, so their radial power means increase; sequential and cap terms are increasing convex functions vanishing at zero. The exact joint dual is convex and increasing in its two mass constraints. Maxima, sums and an old attaining level preserve the comparison. Thus (P.rootgain) reaches the complete consumer, for example \[ \mathcal C_1(W_{\rm pre}(n))> 2^{\lfloor K/2^{919}\rfloor}\mathcal C_1(W_{\rm mc}(n)). \tag{P.rootdensitygain} \] At the explicit witness (P.witness), the new conductor is at least \(100000b_{\rm old}>1800\) below the old conductor: the new block recurrence is smaller at every common stage. The geometric ceiling used above, applied to the old complete pool, and the elementary new baseline bound \(W_{\rm base}>2^{-4\cdot3^q}\) then give \(W_{\rm base}>2^{3^q}W_{\mathcal F}^{\rm joint}\). Also \(W_{\rm pre}(n)>W_{\rm mc}(n)>W_{\rm base}\). For every valid score, (P.mass) retains \(u_m^{\rm comp}(W)\); (P.radial) therefore gives \(\mathcal C_1(W)>3c_{\rm comp}(W)\). Together with the score comparison this proves that the revision improves the previous complete bound for an explicit eligible choice, rather than only at an isolated reference level. No percentage for the combined canonical improvement is evaluated. For completeness, the maxima and suprema define effective bounds. The ordinary product profile has \(P(r)\le r\rho^{d_m}/(1-\rho^{d_m})\), \(\rho=24/25\): use a fractional allocation on the event that the product is at least one. Together with (P.oldtail) and \(q_{\rm con,m}\le[W/(\mu T)]^{33}B^{-32L_m}\) and \(q_{\rm ad,m}\le[W/(\mu T)]e^{-h_{\rm ad}L_m}\), together with \(q_{\rm st,m}\le[W/(\mu T)]e^{-h_{\rm st}L_m-67/100}\), supply a decreasing coarse-level tail tending to zero. For the exact joint formula, combine that feasible product allocation with the sequential occupation and cap their sum at one. Its complete band sum is at most \(r_m\rho^{d_m}/(1-\rho^{d_m})+2u_m^{\rm ex}\), which also tends to zero. Positive incumbents prove finite attainment. The band tails are explicit: \[ \sum_{j>J}4^{-j}F_{j,m} \le\tfrac89(3J+4)4^{-J}u_m,\qquad \sum_{n>L}(6n-5)64^{-n} =64^{-L-1}\left\{\frac{6L+1}{1-1/64} +\frac{6/64}{(1-1/64)^2}\right\}. \tag{P.tails} \] Finite-law logarithms, powers and moments admit directed rational enclosures. Infinite-order and zero-order limits of (P.factored) are handled separately; interior orders can be approximated on compact intervals with their numerical error included in the full logarithm of the bound. These facts give terminating certified approximation, not a feasible expansion of the canonical product or root arrays. The Fourier coefficient and clock threshold have not improved here. The absolute bound remains extremely small. Subexponential growth of higher reference moments, general mixing for weighted physical source sets, and efficient evaluation of the entire canonical concentration profile are not premises of Theorem 12.2. The new seed, conditional certificates, root-prefix and fresh-source arguments are paper proofs with finite computational checks; the selected eighteen-statement Lean release does not yet formalize them. \subsection{Appendix A. Comparison of fixed-depth and first-crossing seeds}\label{appendix_fixed_budget} This appendix compares the fixed-depth seed with a central first-crossing benchmark. The constants of both constructions are specified below. Throughout this appendix \(b=2^{25}\), the sequence \(b_j\) and the root index origin \(H=2b+1\) are fixed. The following comparison is uniform in a fixed integer primitive coefficient \(C\ge1\). Put \(D=C20^{6409}\); use the same \(C\) in both seed recipes. The canonical instance takes \(C=C_{\rm primitive}\). The comparison with {[}3{]} takes \(C=C_{\rm primitive}^{\rm Maz}\). Each comparison uses one common value of \(C\). In this appendix \(W_*\) denotes the fixed-depth score at that coefficient; it agrees with (R.score) in the canonical instance. \subsubsection{The two seed recipes}\label{the-two-seed-recipes} Use \(+\) for the fixed-depth seed and its local adapter, and \(-\) for the first-crossing benchmark: \[ C_6^+=2D+2,\qquad C_6^-=2D+2+2^{481}=:C_{\rm fc}(C), \] \[ q^+(n)=\sum_{j0\), \[ \frac{q^+(n)}{q^-(n)} \le\frac{1+29/1600}{8/5-4/(5b)+29/1600}<\frac{63}{100}; \] the last inequality is rational at \(b=2^{25}\). This is a reduction greater than 37 percent at a common stage. Moreover \(b_n=b+S/100+O(n)\) and \(n=o(b_n)\), by geometric growth, so the same-stage ratio tends to \(401/641\). Write \(N^+=N\) for the fixed-depth startup. The tail at 16 exceeds one after multiplication by \(2^t\), so \(N^-\ge17\). For \(n\ge1\), \[ T_3(n-1,\lambda) =\lambda^{-1}\sum_{k\ge n}k^3\lambda^k <\lambda^{-1}T_3(n,\lambda). \] Together with \(\lambda^{-1}<5/2\) and (Q.oldsurvival), this gives \[ V(N^--1)<(5/2)\Pi^-<5/8<24/25,\qquad N^+2^{12193}W(q^+(N^-),P^-)>2^{12193}W_{\rm fc}. \tag{Q.seedscoregain} \] The corresponding fixed-budget comparison follows from the same allocation estimates. In the following denominator, \(a,d\) are the rate integers of (R.variationdata). The positive rational \(\Pi^--V(N^-)\) has denominator dividing \(2^{1024}d^{N^-}(d-a)^4\), and so exceeds \(2^{-1575N^--7324}\). The bounds \(q^-(n)\ge bn\), \(H\le8q^-(n)+7\) and \(q^-(n+1)\ge q^-(n)+1\) also hold. Apply the proof of (R.startupoptimum) with \(q^-\), \(N^-\) and that denominator bound. Since \(N^-2^{12193}W_{{\rm fixed,fc}},\qquad W_*>2^{24386}W_{{\rm fixed,fc}}. \tag{Q.historicalscoregain} \] \subsubsection{Passing the gains to the complete density formulas}\label{passing-the-gains-to-the-complete-density-formulas} Here \(L=2\log2\) is the density interval length, distinct from \(L_t\). For fixed \(m\), the function used in (D.inverse) is increasing and concave on \([0,L/2]\), and its inverse is increasing and convex. Consequently the maximum branch, the centered second-moment branch and the direct fractional-moment branch are convex in \(W\); the last is a positive multiple of a cubed positive part. Their maximum, its composition with the increasing convex \(\mathcal D\), and the supremum over levels remain convex. Both density formulas vanish at zero. For either formula \(c\), \[ c(aW)\ge a\,c(W) \quad(a\ge1,\ 0\le W\le aW\le W_{\max}). \tag{Q.scoregain} \] Let \(c_{{\rm FM},\pm}\) and \(c_{{\rm MV},\pm}\) denote the complete formulas with adapter \(C_6^\pm\); all reference tables and the compact-width terms \(E_v\) are identical. Thus \(\epsilon_m^+\le\epsilon_m^-\) and each \(+\) formula dominates its \(-\) counterpart at fixed score. Within this comparison, formulas without subscripts are the \(+\) formulas at the chosen common \(C\); at the canonical instance these are the main text's local formulas. All scores above lie in \((0,2^{-4096})\), by (R.scoreupper) and their large levels. The quantitative comparison (Q.paid) holds for either adapter: its proof uses the same compact envelope and a coefficient at least one. Applying (Q.scoregain) with the actual score ratios, (Q.seedscoregain) and (Q.historicalscoregain) now gives \[ c_{\rm FM}(W_*)> 2^{12193}c_{{\rm FM},-}(W_{\rm fc}),\qquad c_{\rm FM}(W_*)> 2^{24387}c_{{\rm MV},-}(W_{{\rm fixed,fc}})>0. \tag{Q.retainedgain} \] The second inequality includes the certified fractional-moment factor greater than two. These inequalities hold separately at each common coefficient \(C\). For the coefficient in {[}3{]}, use \(C=C_{\rm primitive}^{\rm Maz}\) throughout this appendix, then apply (D.coefficientgain) to its left-hand current-seed formula. Thus the main text's local formula also retains the displayed strict gains over those specified first-crossing bounds. The rounding identity (Q.roundinginvariance) is used at a common primitive coefficient. \subsection{Appendix B. Dependence on the analytic coefficient}\label{appendix_provenance} Fix the reference law, seed schedule, and finite moment tables. This appendix compares the resulting startup, root score, and density bound as the valid primitive-decay coefficient varies. \subsubsection{Dependence on the analytic coefficient}\label{dependence-on-the-analytic-coefficient} Write \(t(C),V_C,N(C),P(C),W(C)\) for the fixed-depth startup and allocation of Section 8 with integer primitive coefficient \(C\ge1\) and adapter \(2C20^{6409}+2\). The seed, its survival probability, the rate \(\lambda\), and the level sequence \(q(n)\) are fixed. The coefficient guards in Sections 6--8 hold for every such \(C\). If \(C_1\le C_2\), then \[ t(C_1)\le t(C_2),\quad V_{C_1}(n)\le V_{C_2}(n),\quad N(C_1)\le N(C_2),\quad W(C_1)\ge W(C_2). \tag{R.coefficientmonotonicity} \] Moreover \(t(C_1)W(C_2)\). \emph{Proof.} The first three comparisons follow from the increasing adapter, the ceiling logarithm and the definition of the first positive residual. If the startup indices agree, the smaller variation tail gives a larger residual at that same level; use the strict increase in (R.greedy). If \(N(C_1)2^{12193}W(C_2)\). If the rounded exponents differ but startup indices agree, the residual comparison is strict. \(\square\) The arithmetic bounds (E.coefficientcomparison) give \(C_{\rm primitive}^{\rm Maz}>4C_{\rm primitive}\). To see that the factor four is allowed, (E.local-size) implies \(L_3(4C_{\rm primitive})<177<2^{172}\). Consequently the ratio of the corresponding \(F_b\)'s exceeds two, so \(t(C_{\rm primitive}^{\rm Maz})\ge t(C_{\rm primitive})+1\). Thus \[ W_* = W(C_{\rm primitive})> W(C_{\rm primitive}^{\rm Maz})>0. \tag{R.localcoefficientgain} \] The comparison includes both the rounded startup and the resulting allocation residual. \subsubsection{Density bounds}\label{density-bounds} \textbf{Corollary B.1 (coefficient improvement through the complete formula).} For the same reference tables, let \(c_{{\rm FM},C}\) denote the density formula with coefficient \(C\). If \(C_1\le C_2\) are valid integer primitive coefficients, every term of the mixing envelope for \(C_1\) is no larger than the corresponding term for \(C_2\). The maximum and moment branches therefore give \[ c_{{\rm FM},C_1}(W)\ge c_{{\rm FM},C_2}(W) \quad(0\le W\le W_{\max}). \] The score scaling (Q.scoregain), whose proof uses only the defining functions, then implies, with \(W_i=W(C_i)\), \[ c_{{\rm FM},C_1}(W_1) \ge \frac{W_1}{W_2}\,c_{{\rm FM},C_2}(W_2). \tag{D.coefficientgain} \] Both scores are in the required small-score domain by (R.scoredomain). For \(C_1=C_{\rm primitive}\), \(C_2=C_{\rm primitive}^{\rm Maz}\), (R.localcoefficientgain) makes this a strict improvement of the baseline density formula. The larger coefficient is also a valid primitive bound by (E.primitive) and (E.coefficientcomparison). The other retained branches remain valid with the new envelope, and every fixed-target source construction still fixes its new startup, pool and score before choosing the clock. \subsection{Appendix C. Proofs and certificates for the finite-root refinement}\label{appendix_roots} This appendix proves the finite-root refinement (I.rootdensity). The finite-pool theorem (D.finitepool) and the allocation comparison (R.blockcomparison) reduce that task to residue coverage, separation of root trajectories, and a lower bound for the combined score. We verify these properties for valuation redistributions of certified trajectories and evaluate their parameter counts exactly. Section C.1 fixes the notation. Sections C.2--C.5 supply the constructions and counting lemmas; Section C.6 assembles the score comparison. Section C.7 gives the canonical finite certificates. The final two sections explain a strict shared-allocation gain and the limitations of increasing the number of blocks at fixed height. The baseline score \(W_*\) and the density formulas in the main text retain their definitions. \subsubsection{C.1. Setup and notation}\label{c.1.-setup-and-notation} In this appendix, a \emph{seed} is a certified finite trajectory used to construct additional roots. It is distinct from the fixed-depth word family used for the initial marks in Sections 4 and 7. The constructor (W.constructor) supplies each incoming block. We redistribute valuations along certified finite trajectories while keeping complete residue coverage and a finite root-to-one clock. Throughout this appendix, \(b\ge2^{25}\), \(H=2b+1\), and the residue period and unit index set are those of the root-pool section. All blocks use the same analytic inputs. Let a seed have a certified first-hitting trajectory \[ y_0\xrightarrow{a_1}y_1\xrightarrow{a_2}\cdots\xrightarrow{a_D}1, \qquad 3y_{i-1}+1=2^{a_i}y_i, \] and let \(A_j=\sum_{i\le j}a_i\), \(A_0=0\). Fix \(e=e(y_0)\), \(d\ge0\), and \[ 2^ey_0\equiv4^d\pmod9,\qquad \alpha_0=2^ey_0/4^d>3/4. \tag{C.seed} \] The finite base-height condition and the certified trajectory-height condition must also hold: \[ (2^ey_0\,4^{H-d}-1)/3\ge16^b, \qquad \max_j y_j<16^b, \qquad d\le H. \tag{C.baseheight} \] The base conditions imply \(T_0=2b-d+e-1\ge0\). Its only possible negative case is \(e=1,d=H\), when the base root \((2y_0-1)/3{\raggedright\arraybackslash}p{(\linewidth - 2\tabcolsep) * \real{0.5000}} >{\raggedright\arraybackslash}p{(\linewidth - 2\tabcolsep) * \real{0.5000}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedright Notation \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright Role in this appendix \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot \(y_0,\ldots,y_D=1;\ a_i,A_j\) & Certified seed trajectory, its valuations and their partial sums. \\ \(b,H=2b+1;\ e,d\) & Fixed height parameter, initial root index, and seed-dependent parameters in (C.seed). \\ \(r,\ell_i,S_j\) & Prefix length, added valuations and their partial sums in the general construction below. \\ \(Y_j,M_s\) & Modified trajectory states and the incoming root with index \(s\). \\ \(L_j^{(r)},\beta_r,\Gamma_r\) & Leading height coefficients defined in (C.leading); \(L_j^{(r)}\) is unrelated to the counting constant \(L\). \\ \(\chi_r,T_r\) & Height correction and remaining increment budget, defined in (C.heightdebit) and (C.budget). \\ \(N_r,Q_r,\Lambda_{y,r}\) & Rectangular count, quotient-simplex count and selected layer count, defined in (C.rectcount), (C.quotientcount) and (C.selectedcount). \\ \(E_b,\mathfrak A_b\) & Rectangular benchmark and canonical guaranteed score multiplier. \\ \end{longtable} } \subsubsection{C.2. Rectangular redistribution and retained layers}\label{c.2.-rectangular-redistribution-and-retained-layers} The rectangular rule takes \[ t_i=3^{i+1}k_i,\quad h=\sum_it_i\le T_0:=2b-d+e-1, \] \[ Y_r=y_r,\quad Y_{i-1}=\frac{2^{a_i+2t_i}Y_i-1}{3},\quad M_s=\frac{2^{2(s-d-h)+e}Y_0-1}{3}. \tag{C.rectangular} \] A sufficient rectangular condition is \(\Gamma_r<1\). It implies \(\beta_r<1\). \textbf{Proof of its arithmetic interface.} The valuation identity \(\nu_3(4^t-1)=1+\nu_3(t)\) gives \(4^{t_i}\equiv1\pmod{3^{i+2}}\). Backwards induction gives \(Y_j\equiv y_j\pmod{3^{j+2}}\). In particular \(Y_0\equiv y_0\pmod9\), and \(h\) is a multiple of nine, so the root's compensated affine coefficient is one modulo nine. The original unit index set is therefore unchanged. The geometric and clock arguments are the special case \(\ell_i=2t_i\) of the height and clock lemmas below. Every new root is at least \(16^b\), all later odd states are below \(16^b\), and its clock is \(2s+e+1+\tau_1(y_0)-2d\). A parameter vector is recovered from its actual valuations. Across different seeds, distinct full first-hitting depths are a sufficient distinctness certificate. \(\square\) \textbf{Theorem C.1 (exact count recurrence).} Set \[ N_r(T)=\#\{k\in\mathbb Z_{\ge0}^r:\sum_{i=1}^r3^{i+1}k_i\le T\}, \] with \(N_r(T)=0\) for \(T<0\) and \(N_0(T)=1\) for \(T\ge0\). Then \[ \boxed{N_r(T)=N_{r-1}(T)+N_r(T-3^{r+1}).} \tag{C.rectcount} \] The number with last nonzero coordinate exactly \(r\) is \[ \Delta_r(T)=N_r(T-3^{r+1}). \tag{C.rectlayer} \] \textbf{Proof.} Partition by \(k_r=0\) or \(k_r>0\). In the second part subtract one from \(k_r\); this is a bijection to the indicated lower-budget set. \(\square\) For \(T\ge0\), the unit-cube bounds give \[ \frac{T^r}{r!\,3^{r(r+3)/2}}\le N_r(T) \le\frac{(T+\sum_{i=1}^r3^{i+1})^r}{r!\,3^{r(r+3)/2}}. \tag{C.volume} \] Flooring points of the real simplex proves the lower bound via unit cubes; its cube cover lies inside the enlarged simplex for the upper bound. The resulting asymptotic is for fixed \(r\), not uniformly growing dimension. \textbf{Theorem C.2 (retention of sharper earlier layers).} Unrolling the original prefix gives \[ L_0^{(r)}=y_0+\sum_{i=1}^r\frac{2^{A_{i-1}}}{3^i}, \qquad \beta_{r+1}-\beta_r=\frac{2^{e+A_r}}{4^d3^{r+1}}>0. \tag{C.betastep} \] Thus replacing each retained block ceiling by the final \(\beta_r\) can weaken a coarse certificate. Instead define \[ G_R(T)=\alpha_0^{-1}+\sum_{r=1}^R\frac{N_r(T-3^{r+1})}{\beta_r}. \tag{C.rectscore} \] Then \[ \boxed{G_R(T)-G_{R-1}(T)=N_R(T-3^{R+1})/\beta_R.} \tag{C.rectincrement} \] The right side is positive exactly when that new rectangular layer is nonempty. \textbf{Proof.} The zero vector gives the base target block and the factor \(\alpha_0^{-1}\). A nonzero vector has a unique last nonzero coordinate \(r\); its later coordinates vanish, so the target is exactly its \(r\)-prefix construction, with coefficient less than \(\beta_r\). Apply (R.blockcomparison) separately to those \(\Delta_r\) blocks and retain their budgets. \(\square\) Equation (C.rectincrement) gives the exact increment of the guaranteed multiplier. \paragraph{Exact evaluation without canonical root enumeration}\label{exact-evaluation-without-canonical-root-enumeration} The same counts admit the following exact radix evaluation. Let \[ H_{r,p}(n)=[z^n](1-z)^{-(p+1)}\prod_{j=0}^{r-1}(1-z^{3^j})^{-1}. \] Then \(N_r(T)=H_{r,0}(\lfloor T/9\rfloor)\), \(H_{0,p}(n)=\binom{n+p}{p}\), and all values with \(n<0\) vanish. For \(n=3m+a\), \(a\in\{0,1,2\}\), define \[ c_j(p,a)=\sum_{v=0}^j(-1)^v\binom jv \binom{-3(v+1)+a+p+1}{p+1}. \] Using generalized binomial coefficients for negative upper arguments, \[ H_{r,p}(3m+a)=\sum_{j=0}^{p+1}c_j(p,a)H_{r-1,j}(m). \tag{C.radix} \] To prove it, expand the polynomial \(\binom{3u+a+p+1}{p+1}\) in the basis \(\binom{u+j}{j}\), whose coefficients are the displayed backward differences. Apply the generating-function identity for division of the geometric weights by three. This is a signed-integer recurrence; no decimal values decide the counts. The canonical rectangular prefixes \((7,15,14)\) give \[ E_b=1+G_{101,7}(2b-4)+G_{193,15}(2b-4)+G_{12289,14}(2b-7). \tag{C.E} \] It remains a separately named benchmark in the canonical comparison below. \subsubsection{C.3. General increments, heights, residues and clocks}\label{c.3.-general-increments-heights-residues-and-clocks} Now allow \textbf{arbitrary nonnegative integer increments} \(\ell_1,\ldots,\ell_r\), not necessarily even. Write \[ S=\sum_{i=1}^r\ell_i,\qquad S_j=\sum_{i=1}^j\ell_i. \] Define, whenever the divisions are integral, \[ Y_r=y_r,\qquad Y_{i-1}=\frac{2^{a_i+\ell_i}Y_i-1}{3}, \tag{C.redistribution} \] and let the incoming root exponent be \[ a_{\rm in}(s)=2(s-d)+e-S,\qquad M_s=\frac{2^{a_{\rm in}(s)}Y_0-1}{3}. \tag{C.incoming} \] Here \(e\) is the original seed parameter. It need not equal \(e(Y_0)\). Correct incoming parity will follow from the compensated congruence; one must not substitute a nonintegral half-shift into the one-step constructor formula. Assume the backward values are positive odd units and satisfy \[ 2^eY_0\equiv4^d2^S\pmod9. \tag{C.compensated} \] The residue-completion theorem gives a complete deterministic way to meet these conditions. \textbf{Lemma C.3 (geometry and exact affine coefficient).} For \(0\le j1\) for \(j16^5\), while all its subsequent odd states are below \(16^5\). Its ordinary first hitting time is 41. This is an arithmetic example only; the density theorem still retains its canonical analytic range for \(b\). Here \(e(Y_0)=2\), although the seed parameter is \(e=1\). The compensated incoming exponent has automatically changed parity. Keeping \(e(Y_0)=e(y_0)\) would wrongly exclude this valid family. \paragraph{Relation to the rectangular family}\label{relation-to-the-rectangular-family} Every rectangular vector satisfies the completed family's conditions: it is integral, all intermediate states are units, and its affine coefficient is one modulo nine. Theorem C.6 therefore recovers it uniquely. The completed family is strictly larger; the example above contains an odd increment and cannot be a rectangular vector. The original recurrence (C.rectcount) remains the exact count of its original rectangular set. It must not be presented as the exact count of the larger residue-completed set. \subsubsection{C.5. Counting completed layers}\label{c.5.-counting-completed-layers} \textbf{Lemma C.7 (quotient-simplex count).} For \(r\ge1\), define \[ Q_r(n)=\#\{k\in\mathbb Z_{\ge0}^r:3k_1+k_2+\cdots+k_r\le n\}, \qquad Q_r(n)=0\ (n<0). \] Then \[ \boxed{ Q_r(n)=\sum_{v=0}^{\lfloor n/3\rfloor} \binom{n-3v+r-1}{r-1}. } \tag{C.quotientcount} \] This counts all choices of the remaining \(r-1\) coordinates and unused budget after choosing \(k_1=v\). For an efficient exact evaluation, write \(n=3m+a\), \(a\in\{0,1,2\}\), and \[ P_{r,a}(u)=\binom{3u+a+r-1}{r-1},\qquad d_j=\sum_{v=0}^j(-1)^{j-v}\binom jvP_{r,a}(v). \] Then \[ \boxed{ Q_r(3m+a)=\sum_{j=0}^{r-1}d_j\binom{m+1}{j+1}. } \tag{C.quotienteval} \] \textbf{Proof.} Reindex the sum in (C.quotientcount) as \(\sum_{u=0}^mP_{r,a}(u)\). The polynomial has degree \(r-1\), so its finite Newton expansion is \(P(u)=\sum_{j=0}^{r-1}d_j\binom uj\). Summing and using \(\sum_{u=0}^m\binom uj=\binom{m+1}{j+1}\) proves (C.quotienteval). All arithmetic is integer arithmetic. \(\square\) Thus the dimension-43 count is evaluated from a degree-42 polynomial using exact binomial operations. \textbf{Lemma C.8 (disjoint last-nonzero layers with explicit correction budgets).} A nonzero increment vector has a unique last nonzero coordinate \(r\). Later coordinates are zero, so it is the \(r\)-prefix construction. At its last internal step, \(Y_r=y_r\). The two residue completions for \(c_r\) are \[ c_r=0\quad\text{or}\quad c_r=\sigma_r:=\begin{cases} 2,&y_{r-1}\equiv1\pmod3,\\ 4,&y_{r-1}\equiv2\pmod3. \end{cases} \] The first choice preserves the original residue of \(y_{r-1}\), and the other switches it. For \(r\ge2\), each of the earlier \(r-2\) internal residues is freely chosen from two units. The first correction is at most 17, and those earlier internal corrections are at most five each. Consequently a fixed last correction \(c\in\{0,\sigma_r\}\) has total correction at most \[ \delta_{r,c}=17+5(r-2)+c=5r+7+c. \tag{C.correctionbudget} \] To ensure \(\ell_r>0\), use three disjoint classes: {\def\LTcaptype{none} % do not increment counter \begin{longtable}[]{@{} >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.1667}} >{\raggedleft\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2222}} >{\raggedleft\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2222}} >{\raggedleft\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2222}} >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.1667}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedright Class \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft Last residue correction \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft Last quotient \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft Earlier branch choices \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright Guaranteed free-quotient budget \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot A & \(0\) & \(k_r\ge1\) & \(2^{r-2}\) & subtract \(\delta_{r,0}+6\) from \(2T\) \\ B & \(\sigma_r\) & \(k_r\ge1\) & \(2^{r-2}\) & subtract \(\delta_{r,\sigma_r}+6\) from \(2T\) \\ C & \(\sigma_r\) & \(k_r=0\) & \(2^{r-2}\) & subtract \(\delta_{r,\sigma_r}\) from \(2T\) \\ \end{longtable} } Subtract one from \(k_r\) in A and B. In C omit the last quotient. The exact number in the selected parameter classes is \[ \boxed{ \begin{aligned} \Lambda_r(T)=2^{r-2}\bigg[& Q_r\!\left(\left\lfloor\frac{2T-\delta_{r,0}-6}{6}\right\rfloor\right) +Q_r\!\left(\left\lfloor\frac{2T-\delta_{r,\sigma_r}-6}{6}\right\rfloor\right)\\ &+Q_{r-1}\!\left(\left\lfloor\frac{2T-\delta_{r,\sigma_r}}6\right\rfloor\right) \bigg] \quad(r\ge2), \end{aligned}} \tag{C.selectedcount} \] where \(\Lambda_r(T)=0\) for \(T<0\), and \[ \Lambda_1(T)=\max(0,\lfloor T/9\rfloor). \] \textbf{Proof.} The last-step residues follow by multiplying the original numerator residue, either four or seven modulo nine, by powers of two. The alternate valid unit output occurs at exponent two or four respectively. At this last step, \(c_r\) is independent of its own quotient \(k_r\), because \(2^6\equiv1\pmod9\). Earlier corrections may depend on downstream quotients through the computed states; the uniform bounds cover that dependence. For each fixed earlier branch string, (C.totalincrement) and the correction budget ensure \(S\le2T\). The quotient counts are precisely (C.quotientcount). Theorem C.6 is injective in all the choices, and the three last-quotient/residue classes are disjoint. Therefore multiplication by \(2^{r-2}\) and addition are exact counts of the chosen parameter classes. At \(r=1\), the nonzero shifts are exactly \(18k_1\le2T\), \(k_1\ge1\). \(\square\) These are exact counts of \textbf{explicitly selected safe parameter sets}. They are lower bounds on the count of all admissible completed vectors, because an individual vector can have a smaller correction than the common budget used above. Neither rounding nor a distributional assumption enters the selected counts. \paragraph{The larger finite domain}\label{the-larger-finite-domain} For every prefix \(r\), let \(\mathscr L_r\) consist of all nonnegative vectors with last nonzero coordinate \(r\), satisfying the physical unit conditions, (C.compensated), and \(S\le2T_r\). This is a finite, deterministically decidable set. Theorem C.6 enumerates it by bounded quotient vectors and unit branches, followed by its literal total-budget test. It contains the selected classes of Lemma C.8. Padded zero suffixes identify these as disjoint layers of the full original certificate. The single all-zero vector is added once per seed, not once for every prefix. For a depth-\(D\) seed, all these choices inject into the unrestricted depth-\(D\) completion parameters with \(18k_1+6\sum_{i\ge2}k_i\le2T_0\). Thus the number of its blocks, including its zero vector, is at most \[ 2^{D-1}Q_D(\lfloor T_0/3\rfloor). \tag{C.blockcount} \] This upper bound deliberately ignores some completion and height costs. It is used only for the domain condition. \paragraph{Structural count improvement at fixed dimension}\label{structural-count-improvement-at-fixed-dimension} For fixed \(r\), let \(\mathcal N_r(T)\) count all completed vectors with \(S\le2T\), before imposing a prefix-specific height correction. For \(r\ge2\), the last residue correction is at most four, the first is at most seventeen, and the other corrections total at most \(5(r-2)\). Thus \[ 2^{r-1}Q_r\!\left(\left\lfloor\frac{2T-(5r+11)}6\right\rfloor\right) \le \mathcal N_r(T) \le2^{r-1}Q_r(\lfloor T/3\rfloor). \] The same quotient-simplex argument as for (C.volume), now with weights \((3,1,\ldots,1)\), gives \(Q_r(n)=n^r/(3r!)+O_r(n^{r-1})\). Therefore \[ \mathcal N_r(T)= \frac{2^{r-1}}{3^{r+1}r!}T^r+O_r(T^{r-1}). \] For \(r=1\) and \(T\ge0\), the exact count is \(1+\lfloor T/9\rfloor\), with the same leading coefficient. The selected last-layer formula (C.selectedcount) has this same degree-\(r\) leading coefficient: the first two terms supply it, while the third and the shorter layers have lower degree. Relative to the rectangular lattice's degree-\(r\) coefficient in (C.volume), the ratio is \[ 2^{r-1}3^{(r-1)(r+2)/2}. \] This asymptotic comparison holds with \(r\) fixed. The canonical finite result uses the explicit counts and inequalities below. \paragraph{Retained-layer multiplier for the completed construction}\label{retained-layer-multiplier-for-the-completed-construction} Define \[ \widehat G_R =\alpha_0^{-1}+\sum_{r=1}^R\frac{\Lambda_r(T_r)}{\beta_r}. \tag{C.layerscore} \] Then \[ \widehat G_R-\widehat G_{R-1}=\Lambda_R(T_R)/\beta_R\ge0. \] Every nonempty selected new layer gives a strictly positive increment. Crucially, a later \(\beta_R\), height correction, or shorter budget never replaces the earlier layers' retained certificates. This is the analogue of (C.rectincrement), with both residue corrections and height bounds included. Using the exact count of \(\mathscr L_r\), or finer per-vector coefficients from (C.exactalpha), can improve this convenient explicit multiplier further. The selected-set formula is not asserted to equal the exact shared score. \subsubsection{C.6. Finite-family score comparison}\label{c.6.-finite-family-score-comparison} \textbf{Theorem C.9 (residue-completed finite-family score comparison).} Retain the analytic and counting inputs of the main text. Fix a nonempty finite collection of certified convergent odd unit seeds \(y\), greater than one, each with parameters \(e=e(y),d\) satisfying (C.seed)--(C.baseheight). For each seed choose a terminal prefix length \(1\le R\le D\), where \(D\) is its certified first-hitting depth, and assume \(\beta_R<1\). Earlier leading coefficients are smaller by (C.betastep). Form \(\chi_r,T_r\) separately for every \(1\le r\le R\). Use the zero vector once and any or all of the finite completed layers \(\mathscr L_r\). In particular one may use the full layer sets. Require that generated targets from different seeds be distinct; pairwise distinct full accelerated first-hitting depths suffice. Retain the original block \(\{R_s:s\in\mathcal S_q\}\). The finite score comparisons below hold for every \(q\ge2\). For the persistent-mark and density conclusions, take \(q=q_N\) and \(p=P\), as in (R.markapplication). The resulting root family is a finite convergent antichain, partitioned into complete unit-residue blocks; at \(q=q_N\) these blocks carry the common persistent marks (G.marks). With the selected classes included, set \[ \mathfrak A=1+\sum_{\text{seeds }y}\widehat G_{R_y}. \] Then \[ \boxed{ W_{\mathcal F}^{\rm joint}(q,P)>\mathfrak A\,W(q,P)>0. } \tag{C.master} \] All roots are at least \(16^b\), every later odd state from a new root is below \(16^b\), the largest root is the largest baseline root, and every root clock is given by (C.clock). Let \[ J_{\rm up}=1+\sum_y2^{R_y-1}Q_{R_y}(\lfloor T_{0,y}/3\rfloor). \] The following bound holds: \[ W_{\mathcal F}^{\rm joint} <\frac43J_{\rm up}16^{-b}. \tag{C.domain} \] At \(q=q_N\), whenever the shared score is within \(W_{\max}\), (D.finitepool) gives the corresponding every-clock density bound. If the right side of (C.domain) is at most \(2^{-4096}\), all positive scores in (C.master) are also in the cubic comparison range. \textbf{Proof.} The geometric and residue lemmas prove each block's admissibility. A target's modified first-hitting depth is unchanged, so the stated cross-seed condition proves target distinctness. Within a seed, actual valuations recover the increment vector and hence its unique layer. A root's next odd value identifies its block. The height barrier proves no root subsequently visits any selected root. Therefore there is one aggregate \(1/x\) bound on each source weight. Every block uses the same capacity \(B_q\) and mass requirement \(K_qP\). At \(q=q_N\), the mark identity, uniform over roots, and full variation bound supply these as actual persistent marks in every complete block. For the zero vector use the factor \(\alpha_0^{-1}\). In layer \(r\), (C.affine) gives \(M_s<\beta_rR_s\), so (R.blockcomparison) supplies a strict block factor \(\beta_r^{-1}\). Count the disjoint selected classes by (C.selectedcount), sum their budgets, retain the baseline block, and apply the shared-score dominance (R.jointdominance). This proves (C.master). For (C.domain), the constant vector \(v_r=P\le1\) is feasible in (R.jointscore). Each block obeys \(M_{s+1}=4M_s+1\) on the full sequence of consecutive indices and starts at least at \(16^b\). Its finite unit-subset reciprocal sum is strictly less than \(\sum_{j\ge0}4^{-j}16^{-b}=(4/3)16^{-b}\). Bound the number of blocks by (C.blockcount) and include the baseline block. This proves an upper bound on the finite allocation minimum, with no sum over infinitely many physical histories. The remaining density conclusion is (D.finitepool). \(\square\) For a seed used only through a proper prefix, (C.blockcount) and (C.domain) use that prefix length and pad shorter layers only to it. Distinctness still uses the full first-hitting depth of its certified original trajectory. The canonical application below uses full prefixes, so these lengths coincide there. \paragraph{Strongest finite certificate retained by the theorem}\label{strongest-finite-certificate-retained-by-the-theorem} For the explicitly defined finite family, (R.jointscore) retains the full residue coupling and exact root denominators. It dominates the sum of individual block minima, which in turn dominates the exact coefficient sum (R.blockcomparison) and the selected-count multiplier (C.master). The resulting certificate hierarchy is \[ \begin{gathered}\text{shared residue allocation}\ \ge\ \text{separate block allocations}\\ \ >\ \text{exact affine-coefficient comparison}\ \ge\ \text{certified cell/layer comparisons}.\end{gathered} \] The rectangular \(E_b\) certificate is retained as a benchmark. In the canonical completed family its rectangular blocks are contained in the full completed layers: their original prefixes have zero height correction, their budgets are unchanged, and the completion theorem recovers their vectors. The larger family is not obtained by changing the definition of \(E_b\). \subsubsection{C.7. Canonical certificates and density bounds}\label{c.7.-canonical-certificates-and-density-bounds} Here \(q=q_N\), \(p=P\), and \(W_{\mathcal F}^{\rm joint}=W_{\mathcal F}^{\rm joint}(q_N,P)\). The three certificates used here have the following exact data: {\def\LTcaptype{none} % do not increment counter \begin{longtable}[]{@{} >{\raggedleft\arraybackslash}p{(\linewidth - 12\tabcolsep) * \real{0.1481}} >{\raggedleft\arraybackslash}p{(\linewidth - 12\tabcolsep) * \real{0.1481}} >{\raggedleft\arraybackslash}p{(\linewidth - 12\tabcolsep) * \real{0.1481}} >{\raggedleft\arraybackslash}p{(\linewidth - 12\tabcolsep) * \real{0.1481}} >{\raggedleft\arraybackslash}p{(\linewidth - 12\tabcolsep) * \real{0.1481}} >{\raggedleft\arraybackslash}p{(\linewidth - 12\tabcolsep) * \real{0.1481}} >{\raggedright\arraybackslash}p{(\linewidth - 12\tabcolsep) * \real{0.1111}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedleft Seed \(y\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft \(d\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft \(e\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft Full odd depth \(D\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft Ordinary time \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft Maximum odd state \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright \(\alpha_0\) \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot 101 & 4 & 1 & 7 & 25 & 101 & \(\tfrac{101}{128}\) \\ 193 & 5 & 2 & 43 & 119 & 3077 & \(\tfrac{193}{256}\) \\ 12289 & 8 & 2 & 14 & 50 & 12289 & \(\tfrac{12289}{16384}\) \\ \end{longtable} } Their complete valuation words, in forward order and ending at the first hit of one, are: \begin{verbatim} 101: 4, 1, 3, 1, 2, 3, 4 193: 2, 2, 3, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 2, 2, 4, 3, 1, 1, 5, 4 12289: 2, 2, 2, 2, 2, 4, 1, 5, 3, 3, 1, 2, 3, 4 \end{verbatim} Apply \(3y+1\), check the displayed exact valuation, and divide by its power of two at every step. This reconstructs each finite certificate; no assumption about an uncertified trajectory is needed. The full-prefix leading data are {\def\LTcaptype{none} % do not increment counter \begin{longtable}[]{@{} >{\raggedleft\arraybackslash}p{(\linewidth - 10\tabcolsep) * \real{0.1818}} >{\raggedleft\arraybackslash}p{(\linewidth - 10\tabcolsep) * \real{0.1818}} >{\raggedright\arraybackslash}p{(\linewidth - 10\tabcolsep) * \real{0.1364}} >{\raggedright\arraybackslash}p{(\linewidth - 10\tabcolsep) * \real{0.1364}} >{\raggedleft\arraybackslash}p{(\linewidth - 10\tabcolsep) * \real{0.1818}} >{\raggedleft\arraybackslash}p{(\linewidth - 10\tabcolsep) * \real{0.1818}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedleft Seed \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft \(A_D\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright \(\beta_D\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright \(\Gamma_D\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft \(\chi_D\) \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft Final \(T_D\) at \(b=2^{25}\) \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot 101 & 18 & \(\tfrac{2048}{2187}\) & \(\tfrac{2048}{2187}\) & 0 & 67,108,860 \\ 193 & 76 & \(\tfrac{2^{68}}{3^{43}}\) & \(\tfrac{262144}{19683}\) & 2 & 67,108,858 \\ 12289 & 36 & \(\tfrac{4194304}{4782969}\) & \(\tfrac{4194304}{4782969}\) & 0 & 67,108,857 \\ \end{longtable} } All full-prefix \(\beta_D\) are less than one, and hence so are the earlier \(\beta_r\). For 193, \(\Gamma_{43}\) lies strictly between four and sixteen, so the least height correction is exactly two. Its earlier layers retain the smaller correction appropriate to each prefix. The base minimum-root excesses over \(16^b\) are, respectively, \[ \frac{5\,16^b-32}{96},\qquad \frac{16^b-64}{192},\qquad \frac{16^b-4096}{12288}. \] They are positive for \(b\ge4\), and the maximum certified odd state of each target is below \(16^4\). These verify the finite base conditions. The analytic application continues to use \(b=2^{25}\). \paragraph{Exact score multiplier}\label{exact-score-multiplier} For every seed and prefix, compute \(\beta_{y,r},\Gamma_{y,r},\chi_{y,r}\) from (C.leading) and (C.heightdebit), and \(\Lambda_{y,r}\) from (C.selectedcount). Define \[ \boxed{ \mathfrak A_b =1+\sum_{(y,d,D)=(101,4,7),(193,5,43),(12289,8,14)} \left[ \frac1{\alpha_y} +\sum_{r=1}^D \frac{\Lambda_{y,r}(2b-d+e(y)-1-\chi_{y,r})}{\beta_{y,r}} \right]. } \tag{C.factor} \] This is an exact rational definition involving 64 prefix layers. Each integer count is evaluated by the finite-difference formula (C.quotienteval). It does not require a root list, a residue table at the canonical \(q_N\), or an optimized value of \(c_{\rm FM}\). At \(b=2^{25}\), \[ \begin{aligned} \mathfrak A_b &\approx2.9273409933096237\times10^{275},\\ \mathfrak A_b^3 &\approx2.5085337209674080\times10^{826},\\ (\mathfrak A_b/E_b)^3 &\approx2.0173520932097131\times10^{697}. \end{aligned} \] The following strict inequalities \[ \mathfrak A_b^3>\frac{250}{100}10^{826},\qquad (\mathfrak A_b/E_b)^3>\frac{201}{100}10^{697} \tag{C.numericguards} \] are checked by integer cross-multiplication of exact fractions. Decimal displays play no role in the inequalities. \paragraph{Benchmark root families}\label{benchmark-root-families} {\def\LTcaptype{none} % do not increment counter \begin{longtable}[]{@{} >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3000}} >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3000}} >{\raggedleft\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.4000}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedright Certificate \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright Fixed meaning \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft Guaranteed score multiplier over \(W_*\) \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot \(C_b\) & One-coordinate period-nine lifts & \(\begin{gathered}57101167060327/1953295\\\approx2.9233253\times10^7\end{gathered}\) \\ \(D_b\) & Rectangular multivariate volume estimate, prefixes \((7,13,13)\) & \(\begin{gathered}5.324667695451\ldots\\\times10^{42}\end{gathered}\) \\ \(E_b\) & Exact rectangular counts with retained layers, prefixes \((7,15,14)\) & \(\begin{gathered}1.0753406914595744\ldots\\\times10^{43}\end{gathered}\) \\ \(\mathfrak A_b\) & Residue-completed increments, height corrections, full prefixes \((7,43,14)\) & \(\begin{gathered}2.9273409933096237\ldots\\\times10^{275}\end{gathered}\) \\ \end{longtable} } For clarity, the one-coordinate and rectangular-volume benchmarks are the literal quantities \[ \begin{aligned} C_b={}&1+\left(1+\left\lfloor\frac{2b-4}{9}\right\rfloor\right) \left(\frac{24}{19}+\frac{192}{145}\right) +\left(1+\left\lfloor\frac{2b-7}{9}\right\rfloor\right)\frac{12288}{9217},\\ D_b={}&1+\frac{(2b-4)^7}{7!\,2^{11}3^{28}} +\frac{(2b-4)^{13}}{13!\,1269760\,3^{91}} +\frac{(2b-7)^{13}}{13!\,1310720\,3^{91}}. \end{aligned} \tag{C.earlier} \] The affine comparison and the rectangular simplex bound give these weaker benchmarks by retaining their indicated blocks and leading ceilings. The definition of \(E_b\) in (C.E) retains each earlier layer's sharper coefficient. The full completed family contains those rectangular blocks: their original prefixes have zero height correction and unchanged budgets. None of these symbols is redefined to mean the larger completed family. The exact counts and rational evaluations accompany the computational evidence. \paragraph{Domain and root clocks}\label{domain-and-root-clocks} The canonical block-count upper bound is \[ J_{\rm up}=1+ 2^6Q_7(\lfloor67108860/3\rfloor)+ 2^{42}Q_{43}(\lfloor67108860/3\rfloor)+ 2^{13}Q_{14}(\lfloor67108857/3\rfloor). \] The exact scalar check is \[ 4J_{\rm up}<3\cdot2^{916}. \tag{C.countguard} \] Equations (C.master)--(C.domain) therefore give \[ 0<\mathfrak A_bW_*c_0\), \[ \begin{aligned} \underline d(\mathcal G_c) &\ge c_{\rm FM}(W_{\mathcal F}^{\rm joint}) >c_{\rm FM}(\mathfrak A_bW_*) >2.50\times10^{826}\,c_{\rm FM}(W_*),\\ c_{\rm FM}(\mathfrak A_bW_*) &>2.01\times10^{697}\,c_{\rm FM}(E_bW_*). \end{aligned} \tag{C.densitygain} \] These compare certified lower-bound formulas at the same analytic envelope and moment tables. They do not compare actual densities or evaluate the absolute optimized constant. The second line follows from the literal score pair, not division of independent lower bounds. The primary quantitative improvement of this appendix is the score guarantee (C.master). Within these three fixed prefixes and incoming budgets, replacing every selected count and every reciprocal coefficient by its allowed upper bound changes the additive comparison factor by at most \[ \frac{1+(4/3)(J_{\rm up}-1)}{\mathfrak A_b} =1.1989376587603659\ldots<1.199. \tag{C.localceiling} \] Every block coefficient exceeds \(3/4\), so each contributes less than \(4/3\), and (C.blockcount) bounds the number. This bounds that additive coefficient comparison only. It does not bound the shared allocation minimum, other seeds, or a changed height geometry. The combined refined bound is \(\underline d(\mathcal G_c)\ge c_{\rm comp}(W_{\mathcal F}^{\rm joint})\) for every \(c>c_0\), by (D.finitepool). Equation (Q.sixteengain) makes it strictly larger than the pure-fractional bound displayed here. The cubic multipliers in (C.densitygain) retain their named \(c_{\rm FM}\) meaning; (Q.sixteencubic) supplies the corresponding comparison for \(c_{{\rm FM},16}\), not for \(c_{\rm comp}\). \subsubsection{C.8. A shared-allocation example: the 101 block}\label{c.8.-a-shared-allocation-example-the-101-block} There is a strict gain beyond summing separate minima even before the valuation extension. For the base block \[ M_s=\frac{101}{128}R_s-\frac9{128}, \qquad \phi(r)=128^{-1}(101r-9)\pmod{3^q}, \] the affine permutation sends the baseline root residue to the new one. Writing \(\alpha=101/128\), the valuation identity gives \[ \phi^k(r)-r=(\alpha^k-1)(r+1/3),\qquad \nu_3(\phi^k(r)-r)=2+\nu_3(k) \quad(k\ge1). \tag{C.101orbits} \] Indeed \(\nu_3(\alpha-1)=3\); cubing a unit congruent to one modulo three raises the valuation of its difference from one by one, and an exponent prime to three preserves it. Also \(\nu_3(r+1/3)=-1\) for every integer \(r\). Thus, for \(q\ge3\), every unit orbit has length \(3^{q-2}\). For \(q\ge4\), \(0W(q,p)+W_{101}(q,p) >(1+128/101)W(q,p). \tag{C.strict101} \] No invariance of \(g_N\) under \(\phi\) is used. The strict conclusion excludes \(p=0\); at \(q=1,2\) the affine map is the identity. This argument supplies strictness without a numerical lower bound on the extra shared-allocation gap. \subsubsection{C.9. Limits at fixed height}\label{c.9.-limits-at-fixed-height} The completed construction remains in a specific height geometry. Put \(X=16^b\), with the canonical \(b=2^{25}\). Its affine blocks have \[ M_s=4^{s-H}n+\frac{4^{s-H}-1}{3},\qquad X\le n\le R_H=\frac{4X-1}{3}. \] To preserve exactly the baseline unit index set, one requires \(n\equiv2\pmod3\), since \(H\equiv2\pmod3\). For an odd \(n\) in this interval, the next odd value is below \(X\) exactly when \(\nu_2(3n+1)\ge2\): valuation one gives a value exceeding \(X\), while valuation at least two gives a value at most \(X\), with equality impossible for an odd output. Hence \(n\equiv1\pmod4\). As \(X\equiv4\pmod{36}\), the exact list satisfying these structural conditions is \[ n=X+1+12j,\quad 0\le j\le\frac{X-4}{36}, \qquad J_{\rm struct}=\frac{X+32}{36}. \tag{C.structuralcount} \] This counts candidates. Convergence and the remaining trajectory-height conditions still require certificates. Distinct seed descriptions cannot be counted twice if they give the same \(n\). There is a separate upper bound on the score obtainable within this geometry. For \(J\ge1\) distinct such complete blocks, put \(K=K_q=2\cdot3^{q-1}\) and \(\ell=\lfloor K/(2J)\rfloor\). Remove the \(\ell\) smallest-index roots from every block and exclude all their residue classes. At least \(K/2\) residues remain, and every root at a remaining residue is at least \(X4^\ell\). Assign total mass \(KP\) uniformly to the remaining residues. Its entries are at most \(2P\le2\le B_q\), so it is feasible for (R.jointscore). Therefore \[ W_{\mathcal F}^{\rm joint} \le \frac{KPJ}{X}\,4^{-\ell}. \tag{C.geometryceiling} \] At the canonical level, \(N\ge17\): \(V_N\le1\) cannot hold at \(N=16\), since \(t\ge4b\), \(\lambda>1/4\), and \(V_{16}>2^{4b-32}>1\). Consequently \(q\ge17b\ge16b\), \(K\ge2^q\ge X^4\), and \(J\le J_{\rm struct}\le X\). It follows that \[ \ell\ge K/(4X)\ge2^{12b-2},\qquad K\le\ell^2\le2^\ell,\qquad W_{\mathcal F}^{\rm joint} \le2^{-\ell}\le2^{-\,2^{402653182}}. \tag{C.fixedceiling} \] The floor costs at most a factor two, and \(\ell\ge\sqrt K\), since \(K\ge X^4\ge16X^2\). The integer inequality \(\ell^2\le2^\ell\) holds for \(\ell\ge4\). This bounds the stated mean-and-capacity allocation certificate. Since \(c_{\rm comp}(W)0\), divide its coefficients by their sum \(f_v(y)\) and apply Jensen to the convex function \(z\mapsto\Phi(f_v(y)z)\). Averaging over the affine permutations of \(G_u\) gives \[ 3^{-u}\sum_{z\in G_u}\Phi(f_{u+v}(y+3^vz)) \le3^{-u}\sum_{x\in G_u}\Phi(f_v(y)f_u(x)). \] If \(f_v(y)=0\), both sides equal \(\Phi(0)\). Average over \(y\). The countable mixture can be grouped by the finitely many values of \(f_u\), so ordinary finite Jensen suffices. A zero-length block gives equality. \(\square\) Both compared distributions have mean one. Thus (F.convex) is a convex-order bound: every convex test has no larger average for the combined law than for the product of the two independent finite-table variables. Moment inheritance is its power-function specialization. The positive-part tests used in (D.hinge) will retain more of the upper tail than one power moment. The same mixture gives the version needed for the whole fan: \[ \langle\Phi(H_{u+v})\rangle_{u+v} \le\frac1{3^{u+v}}\sum_{x\in G_u}\sum_{y\in G_v} \Phi(f_u(x)H_v(y)). \tag{F.convexfan} \] To prove this, split \(f_{u+v+1}\) into a prefix of length \(u\) and a suffix of length \(v+1\). In the fiberwise proof of (F.convex), restrict the suffix residue to the \(2\bmod3\) coset and replace the test by \(z\mapsto\Phi(4z/9)\). The bijections \(\phi_{u+v}\) and \(\phi_v\) turn the two sides into (F.convexfan). The suffix coset has \(3^v\) elements, giving the displayed normalization. This also covers \(v=0\), for which the suffix has length one and \(H_0=8/9\). Both sides have mean \(8/9\). Repeated application therefore bounds convex tests by product distributions; it need not recover the exact distribution of \(H_{u+v}\). \subsubsection{D.2. The full concentration profile and optimal residue allocation}\label{appendix_profiles} The moment-to-mass estimate (D.generalp) is one way to use residue capacity. The following finite optimization records the strongest bound obtained from the whole marker distribution and that capacity alone. \textbf{Proposition D.2 (optimal capacity bound).} Fix \(m\ge2\), put \(Q=3^m\), and let \(0\le p\le L\mu\). Define \[ \mathcal C_m(p)= \min\left\{\sum_{r\in G_m}w_r: 0\le w_r\le L/Q,\quad \sum_{r\in G_m}w_rH_m(r)\ge p\right\}. \tag{D.allocation} \] Then \[ \mathcal C_m(p)= \sup_{t>0} \frac{\bigl[p-L\langle(H_m-t)_+\rangle_m\bigr]_+}{t}, \qquad 0\le\mathcal C_m(p)\le p/\mu. \tag{D.hinge} \] \emph{Proof.} Put \(h_r=H_m(r)\) and \(c=L/Q\). For every admissible allocation and \(t>0\), \[ p\le\sum_rw_rh_r \le t\sum_rw_r+c\sum_r(h_r-t)_+. \] This proves the lower bound by the displayed supremum. To attain it when \(p>0\), order the entries as \(h_1\ge\cdots\ge h_Q\) and let \(j\) be the least index with \(c\sum_{i\le j}h_i\ge p\). Feasibility follows from \(c\sum_i h_i=L\mu\), and \(h_j>0\). Fill the first \(j-1\) entries to capacity, allocate \((p-c\sum_{iL\mu\), no allocation is feasible, and the hinge supremum is infinite as \(t\downarrow0\). Optimality here concerns the residue-capacity constraints, not realization of an optimizing allocation by integer orbits. For actual finite source families the coefficient is \(T_X=T_{X,3^m,R_X}\), not \(L\). If \(p_X\ge0\) is a guaranteed coarse marked mass at that cutoff, the correct finite inequality is \[ U_X\ge \frac{\bigl[p_X-T_X\langle(H_m-t)_+\rangle_m\bigr]_+}{t}. \tag{D.finitehinge} \] For fixed \(m\), any strict lower bound below \(\mathcal C_m(p_m)\) can be witnessed by one fixed threshold \(t>0\). First choose that threshold, then a sufficiently small fixed graft tolerance, then a cutoff beyond which \(T_X\) and the coarse error bound have their required accuracy. This proves the analogue of (D.mass) with \(\mathcal C_m(p_m)\) in place of \(u_m(W)\). Its radial domain follows from \(\mathcal C_m(p_m)\le W/\mu\). Consequently (D.radial) also gives the valid fixed-level bound \[ \underline d(\mathcal G_c)\ge\mathcal D(\mathcal C_m(p_m)). \tag{D.profiledensity} \] Only a fixed threshold is needed before the cutoff limit. The functions \(z\mapsto(z-t)_+\) are increasing and convex. Thus (F.convexfan), and finite pointwise upper tables with directed rounding, can supply upper bounds for their averages. General convex tests need not be increasing and cannot automatically use such upper tables. An efficient representation of these propagated profiles would permit quantitative comparison with the explicit bound \(c_{\rm comp}\) and the exact two-sided inverse in (D.twosidedcapacity). \subsubsection{D.3. Rényi divergence rates and bounded relative entropy}\label{appendix_entropy} For \(s>1\) define the Rényi divergence from the uniform law, with relative entropy at order one, by \[ D_s(n)=\frac{\log M_s(n)}{s-1},\qquad D_1(n)=\langle f_n\log f_n\rangle_n, \quad 0\log0:=0. \] These are normalized using the mean-one density \(f_n\). \textbf{Proposition D.3 (rates justified by inheritance).} For every \(s\ge1\), \[ r_s:=\lim_{n\to\infty}\frac{D_s(n)}n =\inf_{n\ge1}\frac{D_s(n)}n\ge0. \tag{F.divergencerate} \] The rates are nondecreasing in \(s\), and \(\lim_{s\downarrow1}r_s=r_1\). For a fixed \(s>1\), if a block length \(\ell\ge1\) has certificates \(M_s(j)\le B_j\) for \(0\le j\le\ell\), with \(B_0=1\), then \[ D_s(n)\le n\bar r_s+O_{s,\ell}(1),\qquad \bar r_s=\frac{\log B_\ell}{(s-1)\ell}\ge r_s. \tag{F.blockrate} \] The implicit constant here also depends on the fixed finite certificates. \emph{Proof.} For \(s>1\), (F.inheritance) makes \(D_s\) subadditive; Jensen gives nonnegativity. Finite-level differentiation at order one gives \(D_s(n)\to D_1(n)\), including zero entries by continuity. Taking this limit in subadditivity proves it also for \(D_1\). For any nonnegative subadditive sequence \(a_n\), write \(n=k\ell+j\), \(0\le j<\ell\). Then \(a_n\le ka_\ell+a_j\), so \(\limsup a_n/n\le a_\ell/\ell\). Taking the infimum over \(\ell\) and using the opposite bound from the definition of the infimum proves (F.divergencerate). The same division into blocks using \(\log B_j/(s-1)\) proves (F.blockrate). For a finite mean-one density, Hölder makes \(s\mapsto\log\langle f^s\rangle\) convex. Its value at one is zero, so its secant slope from one, \(D_s\), is nondecreasing. Thus \(r_s\) is nondecreasing, and the two infima commute: \[ \inf_{s>1}r_s =\inf_{s>1}\inf_{n\ge1}\frac{D_s(n)}n =\inf_{n\ge1}\inf_{s>1}\frac{D_s(n)}n=r_1. \] This proves the order-one limit. \(\square\) For a fixed \(s>1\), subadditivity gives \(D_s(n)=nr_s+o(n)\); by itself it does not upgrade that remainder to \(O(1)\) at the true rate. The block rate \(\bar r_s\) is a certified upper bound. At order one, projectivity and the summable mixing envelope give more. \textbf{Proposition D.4 (bounded order-one divergence).} For every \(n\ge2\), \[ 0\le D_1(n)-D_1(n-1)\le\min\{\log3,\epsilon_{n-1}\}. \tag{F.entropyincrement} \] Consequently \(D_1(n)\) increases to a finite limit \(D_1^\infty\), and \[ \begin{split} \sup_{n\ge0}D_1(n)=D_1^\infty &\le\frac23\log2+ \sum_{j\ge1}\min\{\log3,C_6j^{-2314/25}\}<\infty,\\ 0\le D_1^\infty-D_1(n)&\le\sum_{j\ge n}\min\{\log3,\epsilon_j\} \qquad(n\ge1),\\ r_1&=0,\qquad\lim_{s\downarrow1}r_s=0. \end{split} \tag{F.entropybound} \] \emph{Proof.} Fix \(n\ge2\) and put \(g=f_{n-1}\circ\pi_{n,n-1}\). By the projectivity following (T.law), \(g\) is the average of \(f_n\) on each three-point fiber. Nonnegativity gives \(f_n=0\) on a zero-\(g\) fiber, and \(0\le f_n/g\le3\) on every other fiber. For \(0\le u\le3\), define \(\phi(u)=u\log u-u+1\), with \(0\log0=0\). Convexity and its minimum at one give \(\phi(u)\ge0\). If \(u\le1\), then \(u\log u\le0\), so \(\phi(u)\le1-u\). If \(1\le u\le3\), its chord from one to three gives \[ \phi(u)\le\frac{3\log3-2}{2}(u-1)\le u-1. \] Here \(\log3=(\log_2 3)\log2<(8/5)(7/10)=28/25<4/3\), using the elementary logarithmic inequalities already established in the construction. Thus \(0\le\phi(u)\le|u-1|\) on the whole interval. On each nonzero fiber, the sum of \(f_n-g\) is zero and \(\log g\) is constant. These facts cancel the linear and \(\log g\) terms in the entropy difference, giving \[ D_1(n)-D_1(n-1) =\langle g\phi(f_n/g)\rangle_n \le\langle|f_n-g|\rangle_n\le\epsilon_{n-1}. \] The integrand is defined as zero on zero-\(g\) fibers. Its nonnegativity proves the lower bound; the last inequality is precisely the unhalved bound (T.mix). The same fiber cancellation also writes the increment as \(\langle f_n\log(f_n/g)\rangle_n\). Since \(f_n/g\le3\) wherever \(f_n>0\) and \(\langle f_n\rangle_n=1\), this is at most \(\log3\), proving (F.entropyincrement). At level zero \(f_0=1\), while the even and odd geometric letters give \(f_1(0)=0,f_1(1)=1,f_1(2)=2\). Hence \(D_1(0)=0\) and \(D_1(1)=(2/3)\log2\). Summing (F.entropyincrement) and using (M.envelope) proves boundedness, since \(2314/25>1\). Summing from \(n+1\) onward gives the stated tail. Boundedness gives \(r_1=0\), and (F.divergencerate) gives the right-hand limit in the order \(s\). \(\square\) For scale, the following values are certified to round to four decimal places; the complete record covers every \(0\le n\le16\). {\def\LTcaptype{none} % do not increment counter \begin{longtable}[]{@{}lrrrrrr@{}} \toprule\noalign{} \(n\) & 1 & 2 & 4 & 8 & 12 & 16 \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot \(D_1(n)\) & 0.4621 & 0.6304 & 0.8240 & 1.0087 & 1.1008 & 1.1544 \\ \(D_1(n)/n\) & 0.4621 & 0.3152 & 0.2060 & 0.1261 & 0.0917 & 0.0721 \\ \end{longtable} } These are finite-level values, not an estimate of \(D_1^\infty\). The comparison \(\bar r_{3/2}=(\log b_{12})/6\approx0.1247\) is a certified block-rate upper bound from (F.blockrate), not an identified value of \(r_{3/2}\). The proof identifies \(r_1=0\). The computation uses positive upper arrays as in \hyperref[chapter09]{Section 9}, rather than treating their rounded entries as the exact law. For an upper density \(h\ge f_n\), put \(\eta=\langle h\rangle_n-1\), \(B=\max h\), and \(T=\langle h\log h\rangle_n\). For \(0<\eta<1\), \[ T-\eta(\log B+1)\le D_1(n)\le T-\eta\log\eta. \tag{F.entropyrounding} \] Indeed, with \(d=h-f_n\ge0\), the identity \(h\log h-f_n\log f_n=d\log h+f_n\log(h/f_n)\) and \(\log(1+x)\le x\) bound this difference above by \(d(\log B+1)\) and below by \(d\log d\). Zero entries are interpreted by continuity. Averaging and convexity of \(d\log d\) prove the two bounds; at \(\eta=0\), \(h=f_n\). Rational logarithm enclosures and this correction give intervals of width less than \(1.3\times10^{-7}\) for the displayed entropies. The archive contains the rational interval endpoints determining the displayed rounding. The tail estimate concerns convergence as the level \(n\) increases. It gives no quantitative rate as \(s\downarrow1\), no conclusion \(r_s=0\) at a fixed \(s>1\), and no polynomial moment bound at orders \(3/2\) or \(2\). Nor does it improve the optimized density constant without a further quantitative capacity comparison. For the fan, whose mean is \(\mu=8/9\), divergence is computed from \(H_m/\mu\). The capacity branches retain their residual-mass powers and capacity coefficient. \subsection{Appendix E. Primitive decay from the original pair process}\label{appendix_primitive_local} We prove an explicit primitive Fourier-decay estimate for the reference law. The argument uses the pairing and separated-triangle method of Tao {[}1, Section 7{]} and its quantitative development in {[}3{]}. Working directly with the original pair process gives an exact binomial first-passage law. Its exponential moment yields a linear vertical window and the degree-four recurrence used to specify the coefficient. \subsubsection{E.1. Primitive Fourier decay and its coefficient}\label{e.1.-primitive-fourier-decay-and-its-coefficient} Let independent positive integers \(K_i\) satisfy \(\Pr(K_i=a)=2^{-a}\), and put \[ X_n=\sum_{i=1}^n3^{i-1}2^{-(K_1+\cdots+K_i)}\pmod{3^n}, \qquad F_n(\xi)=\mathbb E e^{2\pi i\xi X_n/3^n}. \tag{E.law} \] This is (T.law), with the unnormalized positive Fourier convention of Section 6. All inverse powers of two are taken in the indicated residue ring. For a positive integer \(B\), define \[ \epsilon=2^{-78},\quad d=\epsilon^2/2,\quad \eta=d/(32B),\quad \delta=(16\cdot10^B)^{-1}, \] \[ K=16(B+1)/\epsilon^2,\quad R=4(K+16(B+1))+1, \quad A=64/\delta,\quad H=2^{17}A^2/\delta. \] The symbols \(K,R,A,H\) in this appendix are local parameters; in particular \(K\) is not a valuation letter \(K_i\). They are positive integers. Set \[ u(p)=H(p+1)^4,\qquad v_0=K,\qquad v_{i+1}=v_i+u(v_i)+K+2\quad(0\le i0\), the integer order \(B=\lceil b\rceil\) supplies the same assertion with \(n^{-b}\). We prove the theorem below. \subsubsection{E.2. Pairing and the discounted process}\label{e.2.-pairing-and-the-discounted-process} Fix \(n,\xi\) as in the theorem and let \(N=\lfloor n/2\rfloor\). The case \(n=1\) follows from \(|F_n|\le1\); assume \(n\ge2\). Put \(B_j=K_{2j-1}+K_{2j}\) and \(S_j=B_1+\cdots+B_j\), with \(S_0=0\). Then \[ \Pr(B_j=b)=(b-1)2^{-b}\quad(b\ge2). \] Conditional on a pair sum \(b\), its first letter is uniform on \(1,\ldots,b-1\). The contribution of that pair, after \(j\) preceding pairs and total valuation \(l\), is \[ 3^{2j}2^{-l}\bigl(2^{-a}+3\,2^{-b}\bigr). \] The conditional character is a unit-modulus factor times \((b-1)^{-1}\sum_{a=1}^{b-1}e_{3^n}(\xi3^{2j}2^{-(l+a)})\). Its modulus is at most one. When \(b=3\), that modulus is \(\cos(\pi\theta(j,l))\), where \[ \theta(j,l)=\operatorname{bal}\left( \frac{\xi3^{2j}2^{-(l+2)}}{3^n}\right)\in(-1/2,1/2] \quad(0\le js\}\) and \(H_s=S_{J_s}-s\). Realize each geometric letter as the waiting time for one success of fair coin tosses. If \(T_s\) counts the successes in the first \(s\) tosses, then \[ T_s\sim\operatorname{Bin}(s,1/2),\qquad J_s=\lfloor T_s/2\rfloor+1. \tag{E.passage} \] Conditional on \(T_s\), the overshoot waits for two more successes when \(T_s\) is even and one when it is odd. Future tosses are independent. Consequently, for integers \(j,h\ge1\), with out-of-range binomial coefficients zero, \[ \Pr(J_s=j,H_s=h)=2^{-s-h} \left[\binom{s}{2j-2}(h-1)+\binom{s}{2j-1}\right]. \tag{E.joint} \] In particular \[ J_s\le s/2+1,\quad \mathbb EH_s\le4,\quad \Pr(H_s>h)\le(h+1)2^{-h}. \tag{E.overshoot} \] For \(s\ge1,a>0\), the variance \(s/4\) of \(T_s\) gives \[ \Pr(|J_s-s/4|>a\sqrt{s}+1)\le(16a^2)^{-1},\qquad \sup_j\Pr(J_s=j)\le4/\sqrt{s}. \tag{E.passage-bounds} \] For the atom bound, each atom is the sum of two binomial atoms. The central estimate \(\binom{2k}{k}/4^k\le1/\sqrt{k+1}\) follows by induction: its squared step is \((2k+1)^2(k+2)\le4(k+1)^3\), with difference \(3k+2\). The odd case follows from the neighboring even case, so one binomial atom is at most \(\sqrt{2/(s+1)}\), proving the displayed bound. For \(s\ge1\) and \(t\ge0\), we also use \[ \Pr(T_s-s/2\ge t)\le e^{-2t^2/s}. \tag{E.binomial-tail} \] Indeed its centered exponential moment is \(\cosh(a/2)^s\le e^{sa^2/8}\), by integrating \(\tanh x\le x\); exponential Markov with \(a=4t/s\) gives the claim. Start at \((j,l)\) in a triangle with top \(l_0\), and set \(s=l_0-l\). Except with probability \(e^{-8}\), \(J_s\le s/4+\sqrt{s}+1\); for \(s=0\) this holds deterministically. The top right corner is at least \(s\log2/\log9\) columns to the right of \(j\). Since \(\log2/\log9>5/16\) and \(\sqrt{s}\le s/16+4\), the exit's horizontal overshoot past the last top-row lattice point is less than six. Also \(H_s\le8\) except with probability \(9/256\). On their intersection the exit lies outside the triangle within distance ten, and is before \(N\). Its collar makes it white. Thus \[ \Pr\bigl((j+J_s,l_0+H_s)\text{ is white}\bigr) \ge1-e^{-8}-9/256>15/16. \tag{E.white-exit} \] \subsubsection{E.5. Long crossings and the degree-four threshold}\label{e.5.-long-crossings-and-the-degree-four-threshold} Observe the process \(p\ge0\) steps after that exit. Its position is \((j+J_s+p,l_0+V_p)\), where \(V_p=H_s+B_{J_s+1}+\cdots+B_{J_s+p}\). For \(1Y)\le e^{-(A/4-2)(p+1)} \le e^{-A(p+1)/8} \le\frac{128}{A^2(p+1)^2}\le\frac4{A(p+1)^2}. \tag{E.vertical-tail} \] Together with (E.passage-bounds), this puts the point in the rectangle \[ |x-(j+s/4+p)|\le W:=A(p+1)\sqrt{s}+1, \qquad l_01/16\) for its right end. Consider a different triangle of size \(t'\ge u=u(p)\) meeting the rectangle at a lattice point \((x,y)\), with left column \(a\). Then \(x-Yl_0\). Moving from \((x,y)\) to \((a,y)\) decreases the triangle weight, so \(b'\le y\le l_0+Y\). At the common integer height \(l_*=l_0+\lceil u/(2\log2)\rceil\), each such triangle has a horizontal lattice interval starting at \(a\), of length at least \(u/12\). Its top, at \(b'+t'/\log2\), is above \(l_*\). Since \(u\ge4Y\log2\) and \(u\ge64\), the real section length is \[ \frac{(l_*-b')\log2}{\log9} \ge\frac{u/2-Y\log2}{\log9} \ge\frac{u}{4\log9}>\frac{u}{10}. \] Taking an integer part loses less than one, while \(u/10-1\ge u/12\). These lattice intervals are disjoint, so their left columns are separated by at least \(u/32\). All such left columns lie in an interval of length at most \(2W+Y+2\). Each makes at most \(Y+1\) columns of the rectangle bad, because \(x-Y\sigma_i:l+S_t>l_i\},\qquad \sigma_{i+1}=\inf\{t\ge\tau_i:z_t\text{ is black}\}, \] with \(\inf\varnothing=\infty\). In particular, a black exit is the next entry at that same time. The top \(l_i\) stays fixed during its crossing. The gaps before the first entry and between each exit and the next entry are white, and the exit times are strictly increasing. By (E.white-exit) and the strong Markov property, the conditional expectation of \(e^{-\mathbf1_{\{\text{white exit}\}}}\) at each entry is at most \(1-(15/16)(1-e^{-1})\le3/4\). Iterating with the indicator that the next entry exists gives \[ \Pr(R\text{ entries before }P,\ \#\text{white states before }P\eta m\), the recipe gives \(\sqrt{s}\ge Z\). The triangle fits before \(N\), so \(s\log2\le m\log9\) and \(s<16m/5\). Also \(P\le m/20\). Thus \(J_s+P>9m/10\) implies \(J_s>17m/20\), and (E.passage) gives \(T_s-s/2>m/10-2\ge m/20\). The binomial tail implies \[ \Pr(J_s+P>9m/10)\le e^{-m/640}. \tag{E.terminal-tail} \] On the complementary event all sampled states are before \(N\), and the remaining distance is at least \(m/10\). Discard factors before \(J_s\), use (E.postexit), and then the later-layer bound. On the exceptional event use one. Since \(D\ge1\), \[ \frac{U(j,l)}{Dm^{-B}} \le10^B\mathbb E[\text{post-exit auxiliary product}] +m^Be^{-m/640}\le\frac14+\frac14<1. \tag{E.long} \] For the second term, \(\log m\le\sqrt m\) and \(m\ge(2560(B+1))^2\) give \(m^Be^{-m/640}\le e^{-m/1280}\le1/4\). On the good event, every factor of the auxiliary product is before column \(N\) and agrees with the original product. On the exceptional event, the bound uses \(U\le1\) and its probability in (E.terminal-tail). This proves \(D_m=D_{m-1}\) for \(m\ge M\). For \(m2^{4E_0}\), \(p_0\ge2\), and \(p_{i+1}\ge p_i^3\), so \(p_r\ge2^{3^r}\). The coefficient is at least \(p_r\) through its \(P_{\max}^{10}\) term. With \(L_3(x)=\log_2\log_2\log_2 x\), we obtain \[ L_3(C_{6409}^{\rm loc})<176, \qquad L_3(C_{\rm primitive}^{\rm Maz})>2^{172}, \qquad C_{6409}^{\rm loc}A_*+2\), both exceed \(A_*\) precisely on the intersection of \([80,679/5]\) with \(((A_*+1)/\log2,\allowbreak\sqrt{2(\beta-2-A_*)/\log2})\). For \(\beta\le A_*+2\), the intersection is empty. At width 80 the smallest integral \(\beta\) making both exceed six is \[ \beta\ge2227. \tag{E.floor} \] At \(\beta=2227\), the smaller power is about 6.929. The order-92 seed estimate and geometric level contraction use the stronger choice \(\beta=6409\): the rational width \(6749/50\) gives both powers strictly above \(2314/25\), as in (M.exponentguards). Thus the entire Section 6 envelope and order-92 seed estimate use the new coefficient without a change of exponent. Lower-bound guards and quantitative sensitivity have different roles. For any valid primitive coefficient \(C\ge1\), the adapter \(C_6(C)=2C20^{6409}+2\) satisfies \(C_6(C)>2^{27700}\), enough for (B.coefficientscope), \(S_{92}{\raggedright\arraybackslash}p{(\linewidth - 2\tabcolsep) * \real{0.5000}} >{\raggedright\arraybackslash}p{(\linewidth - 2\tabcolsep) * \real{0.5000}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedright Result family \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright Formalized conclusion \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot Collatz convergence to one & Common-clock positive lower density, the rational clock specialization, and selected depth-eleven second-moment and fractional bounds. \\ Fixed-target hitting & Positive lower density for every positive target not divisible by three, the exact target criterion, and corresponding depth-eleven bounds. \\ Corollaries & A positive individual-root consequence and vanishing-clock-loss statements for convergence to one and for each admissible target. \\ Concentration profiles & Exact finite fan allocation and its hinge dual, each real-moment lower bound, and profile-density bounds at an existential positive paid level. \\ Reference entropy & Increment bounds, a finite monotone limit with an explicit tail, and zero normalized growth, including the positive-index normalized infimum. \\ \end{longtable} } The primitive Fourier-decay argument of Appendix E is proved within the formal development, including its all-order coefficient recipe and the order-6409 specialization used in Section 6. The selected proof dependencies do not assume Mazur's numerical primitive-decay theorem. Adapted probability and word constructions retain their original attribution and license notices. The coefficient-size estimate, its comparison with the historical coefficient, and the resulting optimized-density comparisons remain paper proofs. The other divergence-rate results, the Rényi-order right limit, complementary capacity, deeper tables, full optimization, and amplified root families are outside the selected Lean release. The additional results in Section 12 are also outside that release: the coupled seed schedule, persistent-root prefix score, radius-four terminal family, conditional depth-sixteen certificates, factored and joint counting formulas, and fresh-source injection. Their exact finite checks support the written proofs; they do not constitute Lean or registry verification of the stronger constants. \subsubsection{Source record and Palomar registration}\label{source-record-and-palomar-registration} The companion development defines the ordinary Collatz map, logarithmic hitting sets, and lower natural density directly. Its theorem map records the correspondence with this paper, and its dependencies are pinned to specified Lean and Mathlib versions. The eighteen selected statements have passed local compilation and axiom audits. The formal artifact and any \href{https://palomar-registry.org/}{Palomar registry} record concern the selected statements and their pinned source revision. This revised source draft makes no registry-verification claim for the new Section 12 bounds. The scope table above describes the companion development; the manuscript proofs stand independently of registration. \subsubsection{Computational data and reproducibility}\label{computational-data-and-reproducibility} The finite certificates are specified by the recurrences, pseudocode, and integer inequalities in Section 9 and the appendices. The reference-density tables use exact integer arithmetic, upward rounding, and explicit positive-tail bounds. The depth-sixteen cyclic calculation covers all \(3^{16}=43{,}046{,}721\) residues. Fractional-moment bounds use integer square roots, and entropy values use rational logarithm enclosures. Section 9 specifies the transfer recurrences and certificate inequalities. The supplementary computations also include scalar comparisons, root-trajectory certificates, and residue-counting calculations used in the explicit bounds; these broader paper computations should be distinguished from the selected formal proof package. Section 12 additionally uses two supplementary exact histogram records: the \href{anc/profile_envelope_depth16_2026_09_27.json}{unconditional depth-sixteen envelope} (SHA-256 \texttt{\seqsplit{39440f290eba119bfb0549588bf77e598c48257b66460b41cdb7ce210efe939d}}) and the \href{anc/conditional_profile_depth16_2026_09_28.json}{three conditional histograms} (SHA-256 \texttt{\seqsplit{0622b7e5390741e0c3ec287768d7cc6f947951a0bd9472260012bb018be5aa4e}}). Counts and scaled sums reconstruct the mean-one laws and their rational moment enclosures. The native recurrence agrees with the retained full-array hashes at every depth through sixteen. This is a reproducible finite calculation, not a second independent derivation of that recurrence or a formal proof of the complete optimization. \subsubsection{Use of AI-assisted tools}\label{use-of-ai-assisted-tools} AI-assisted tools were used in manuscript preparation, Lean proof development, code preparation, and editorial revision. \subsection{References}\label{references} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \item Terence Tao, \emph{Almost all orbits of the Collatz map attain almost bounded values}, Forum of Mathematics, Pi 10 (2022), e12. \href{https://arxiv.org/abs/1909.03562v7}{Corrected author preprint, v7}, 16 July 2026. \item Terence Tao, \emph{Equidistribution of Syracuse random variables and density of Collatz preimages}, 25 January 2020. \href{https://terrytao.wordpress.com/2020/01/25/equidistribution-of-syracuse-random-variables-and-density-of-collatz-preimages/}{Author's exposition}. \item L. Mazur, \emph{Explicit Positive-Density Collatz Convergence in Logarithmic Time}, manuscript v2.1, 6 September 2026. \href{https://www.proofatlas.ai/formalizations/positive-density-log-time-collatz/}{Manuscript and supplementary material}. \item L. Mazur, \emph{Positive Lower Density of Collatz Predecessors}, manuscript v1.0, 6 September 2026. \href{https://www.proofatlas.ai/formalizations/positive-lower-density-collatz-predecessors/}{Manuscript and theorem statements}. \item Alex Kontorovich and Jeffrey C. Lagarias, \emph{Stochastic Models for the \(3x+1\) and \(5x+1\) Problems}, in Jeffrey C. Lagarias (ed.), \emph{The Ultimate Challenge: The \(3x+1\) Problem}, American Mathematical Society, 2010, pp.~131--188. \href{https://arxiv.org/abs/0910.1944}{Author preprint}. \item Ilia Krasikov and Jeffrey C. Lagarias, \emph{Bounds for the \(3x+1\) Problem using Difference Inequalities}, Acta Arithmetica 109 (2003), 237--258. \href{https://arxiv.org/abs/math/0205002}{Author preprint}; \href{https://doi.org/10.4064/aa109-3-4}{published article}. \item Idris Ali Shaik, \emph{Polylogarithmic Descent for Almost All Collatz Orbits in Natural Density}, preprint, 6 September 2026. \href{https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7290240}{SSRN 7290240}, \href{https://doi.org/10.2139/ssrn.7290240}{doi:10.2139/ssrn.7290240}. \item Idris Ali Shaik, \emph{Quantitative Endpoint Transport for the Collatz Map with a Natural-Density Descent Application}, preprint, 6 September 2026. \href{https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7341398}{SSRN 7341398}, \href{https://doi.org/10.2139/ssrn.7341398}{doi:10.2139/ssrn.7341398}. \item Riho Terras, \emph{A stopping time problem on the positive integers}, Acta Arithmetica 30 (1976), no. 3, 241--252. \href{https://doi.org/10.4064/aa-30-3-241-252}{doi:10.4064/aa-30-3-241-252}. \item C. J. Everett, \emph{Iteration of the number-theoretic function \(f(2n)=n\), \(f(2n+1)=3n+2\)}, Advances in Mathematics 25 (1977), no. 1, 42--45. \href{https://doi.org/10.1016/0001-8708(77)90087-1}{doi:10.1016/0001-8708(77)90087-1}. \item Ivan Korec, \emph{A density estimate for the \(3x+1\) problem}, Mathematica Slovaca 44 (1994), no. 1, 85--89. \href{https://dml.cz/handle/10338.dmlcz/133225}{DML-CZ article record}. \item Jeffrey C. Lagarias, \emph{The \(3x+1\) problem and its generalizations}, American Mathematical Monthly 92 (1985), no. 1, 3--23. \href{https://www.cecm.sfu.ca/organics/papers/lagarias/}{Author's article}. \item Manuel Inselmann, \emph{An approximation of the Collatz map and a lower bound for the average total stopping time}, preprint (2024), arXiv:2402.03276. \href{https://arxiv.org/abs/2402.03276}{Author preprint}. \end{enumerate} \end{document}