% Hexagon source adaptation of the published Zenodo edition. % Changes are limited to packaging, bibliography rendering, navigation and archive links. % Mathematical formulas and their order are preserved; see the local integrity record. \PassOptionsToPackage{expansion=false,nopatch=footnote}{microtype} \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern,microtype} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage[a4paper,margin=25mm,headheight=15pt]{geometry} \usepackage{booktabs,array,longtable,enumitem} \usepackage{tikz,pgfplots,fancyhdr,xurl} \pgfplotsset{compat=1.18} \usepackage[colorlinks=true,allcolors=blue!45!black]{hyperref} \hypersetup{pdftitle={Arithmetic structure in a complete atlas of power-residue runs below 2 to the power 32},pdfauthor={Brice Pouly}} \makeatletter \setlength{\@fptop}{0pt} \setlength{\@fpsep}{20pt} \setlength{\@fpbot}{0pt plus 1fil} \makeatother \pagestyle{fancy}\fancyhf{} \fancyhead[L]{\small Arithmetic structure in power-residue runs} \fancyhead[R]{} \fancyfoot[C]{\thepage}\renewcommand{\headrulewidth}{0.3pt} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition}\newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newcommand{\Fp}{\mathbb F_p} \newcommand{\Q}{\mathbb Q} \newcommand{\Ftwo}{\mathbb F_2} \newcommand{\Z}{\mathbb Z} \newcommand{\chr}{\chi_{p,r}} \newcommand{\Span}{\operatorname{span}} \newcommand{\Gal}{\operatorname{Gal}} \newcommand{\Stab}{\operatorname{Stab}} \newcommand{\Nrm}{\operatorname{N}} \newcommand{\Prob}{\operatorname{Pr}} \newcommand{\EE}{\mathbb E} \newcolumntype{P}[1]{>{\raggedright\arraybackslash}p{#1}} \title{Arithmetic structure in a complete atlas\\ of power-residue runs below $2^{32}$\\[3pt] \large Rabung profiles, Kummer densities, norm dependencies, and local codes} \author{Brice Pouly\thanks{Prepared with artificial-intelligence assistance. The scope of the assistance and the status of the manuscript are stated in Section~\ref{sec:reproduction}.}} \date{12 October 2026} \hypersetup{pdfsubject={Publication edition v1.0; 12 October 2026}} \usepackage{bookmark} \begin{document} \maketitle \begin{abstract} We study the arithmetic behind long runs in a complete census of power-residue colourings below $2^{32}$, covering every admissible order $2\le r\le9$. The dataset contains $203\,280\,221$ primes and $677\,591\,249$ profiles, with exact quadratic four-window counts at every odd prime. A bounded study of its deep tail identifies algebraic windows with fewer independent character conditions than their lengths suggest. We certify their power-class groups and conjugate-root overlaps, obtaining exact Kummer densities for fixed templates, including eight additional order-four models. In a quadratic family, eleven constant positions force twenty-six neutral positions at odd character orders. In two further families, norm identities generate all remaining higher-root relations. A length-$53$ cubic realizes a three-wise independent binary code whose monochromatic probability is $4096$ times that of independent signs; its local run law is computed exactly. A separate stratified million-profile ternary sample is analysed with the same finite catalogue filter applied to the data and to an independent-colour comparator. Finite census statements, selected explanations and asymptotic template laws concern distinct populations. No universal law of global maxima or finite-range error estimate for the template densities is asserted. \end{abstract} \noindent\textbf{Keywords.} Multiplicative characters; residue runs; van der Waerden numbers; Rabung construction; Kummer theory; Chebotarev density; binary codes; exact computation. \section{Introduction and scope} A long monochromatic residue run may resemble a rare string of independent colours. Some extreme runs instead lie on algebraic windows whose character values satisfy many exact relations. One window of length $53$ gives a sharp contrast: every projection onto three positions is uniform, yet its monochromatic probability is $2^{-40}$ rather than the independent-sign value $2^{-52}$. Section~\ref{sec:code} proves this statement for all prime/rational-root pairs of one fixed cubic template in the limit over primes. A prime with three rational roots contributes three pairs. It is not a finite-range frequency estimate for the canonical witnesses selected from the atlas. Three questions organize the paper: what occurs in the complete finite data; which exact identities explain selected exceptional windows; and what the resulting template laws and finite diagnostics allow us to conclude about run statistics. Rabung's criterion~\cite{rabung} and earlier large searches, including Monroe's~\cite{monroe}, supply the construction context. The atlas covers all primes below $2^{32}$ and every admissible order from $2$ to $9$. The scanner method is developed in~\cite{gpumethod}; its frozen implementation and data commitments are described in Section~\ref{sec:reproduction}. The classification of its $11\,557$ deep-tail profiles is selective: $23$ have no retained model after the higher-degree extensions. A template suggested by a selected witness can subsequently have a proved density over varying primes, but that selection is not an independent test of the density. Certified Kummer groups, norm-generated root dependencies and the exact local code law provide the mathematical explanations. The $496$ order/length pairs and the bounded residual qualification are machine-readable. A separate million-profile ternary supplement supplies finite continuation diagnostics with a same-filter comparator. The short quadratic elliptic formula is background to the census. Some ingredients are classical: the study of consecutive power residues~\cite{lehmer}, Kummer theory~\cite{milneFT}, Chebotarev's theorem~\cite{milneANT}, and the MacWilliams identity~\cite{macwilliams}. Character-pattern counts have established connections with curves and their Jacobians~\cite{patterns}. Prescribed residue patterns on finite sets of rational integers were studied by Wright~\cite{wright07,wright08} and by Karthick Babu and collaborators~\cite{babu24,babu25}. The Kummer--Chebotarev framework is classical; our certifications concern the relations of selected algebraic-root windows and their interaction across conjugate roots. Dual distance, orthogonal arrays and limited independence are standard coding theory~\cite{delsarte,cameron}. Section~\ref{sec:code} gives a specific arithmetic realization and its exact local run distribution, not a new general independence principle. No worldwide priority claim is made for the examples. \paragraph{Relation to succinct certificates.} The published certificates paper~\cite{boundsv2} proves the equivalence of the implemented boundary test with Rabung's criterion and owns the recurrence-closed comparison of lower bounds. Here the subject is the structure visible in a complete finite population: its distributions, selected algebraic windows and their exact relations. Pointwise construction certificates, complete census coverage and limiting prime densities have different verification requirements. We retain a short Rabung specialization for readability and identify the frozen scanner and finite audit artifacts in the reproduction section. \section{The atlas and the Rabung contract}\label{sec:atlas} \paragraph{Four populations used below.} \begin{center}\small \begin{tabular}{@{}P{6.0cm}P{8.4cm}@{}}\toprule Population & Scope of the conclusion \\ \midrule All primes below $2^{32}$ and admissible orders $2$--$9$ & Complete finite census in that domain. \\ \addlinespace Canonical witnesses selected in the deep tail & Descriptive classification under stated thresholds and catalogues. \\ \addlinespace Prime/rational-root pairs of a fixed template & Asymptotic laws from its certified power-class groups and Chebotarev. \\ \addlinespace One million ternary profiles stratified on $[2^{33},2^{34})$ & Finite comparisons of four equally sized lists, rather than a natural prime average over the band. \\ \bottomrule \end{tabular} \end{center} These populations have different weights and selection rules. In particular, neither the selected witnesses nor the equally weighted ternary strata are substituted for the prime/root population of a template law. \subsection{Definitions and conventions} For a prime $p$ and an order $r\mid p-1$, let \[ \chr(x)=x^{(p-1)/r}\in\mu_r(\Fp),\qquad x\in\Fp^\times. \] Its $r$ values are the colours. The neutral colour is $1$. Relabelling the non-neutral colours does not change any constant-run length. All maxima below concern the ordinary word at positions $1,\ldots,p-1$: the word is not joined across zero or interpreted as a circular word. Define \[ M_r(p)=\max\{m:\chr(w)=\cdots=\chr(w+m-1),\ 1\le w\le p-m\}, \] and let $L_r(p)$ be the largest $\ell$ for which $\chr(1)=\cdots=\chr(\ell)=1$. The canonical witness $w_r(p)$ is the least starting position attaining $M_r(p)$. Put $\sigma=\chr(-1)$. As $\sigma^2=1$, it is $1$ or $-1$, with $\sigma=1$ for odd $r$. The atlas also records \begin{equation}\label{eq:K} K_r(p)= \begin{cases} \max(M_r(p)+1,\,2L_r(p)+2),&\sigma=1,\\ \max(M_r(p)+1,\,L_r(p)+2),&\sigma=-1. \end{cases} \end{equation} We use \emph{colour-first} notation: $W(r,k)$ is the least $n$ such that every $r$-colouring of $[n]$ contains a monochromatic progression of length $k$, as in the published certificates paper~\cite{boundsv2}. Historical files retain their original schema: in particular, \texttt{W\_length\_first\_lower\_bound} in the catalogue stores the weak integer bound $W(r,k)\ge(k-1)p+2$. Its name is not a convention for the revised article. A strict bound $W(r,k)>B$ means $W(r,k)\ge B+1$. \subsection{Internal and boundary conditions} For $p>k$, the atlas calls $M_r(p)k$ and $K_r(p)\le k$, the power-residue colouring extends to an $r$-colouring of $[0,(k-1)p]$ with no monochromatic $k$-term arithmetic progression, by colouring its $k$ multiples of $p$ non-constantly. Consequently $W(r,k)>(k-1)p+1$. \end{proposition} \begin{proof} Consider a progression of common difference $d$. If $p\mid d$, the only possible $k$-term progression in this interval is precisely the $k$ multiples of $p$, which have been coloured non-constantly. Otherwise multiply its residues by $d^{-1}$. If none is zero, a constant progression becomes a constant ordinary consecutive segment of length $k$, contradicting $M27\,333\,622\,969. \] At $(3,17)$ the prime is $2\,072\,886\,931$, giving $W(3,17)>33\,166\,190\,897$. Relative to the fixed direct-construction entries in Table~2 of Monroe's version~7~\cite{monroev7}, these primes increase by $18.82\%$ and $113.84\%$. They are the two improved entries among the $29$ fixed comparison points. This is a comparison with that identified historical table, not a claim of current world records. The $(2,25)$ certificate is already in the first published edition~\cite{bounds}, which credits Monroe's phase-2 project. It is not a new construction here. The $(3,17)$ value increases that edition's published strict bound $31\,385\,622\,833$ by $5.67\%$. Separate full ordinary CPU scans are retained in \path{reproducibility/bounds_update/results/}, with their source identities and commands. They also check the larger primes $5\,899\,373\,413$, $14\,413\,721\,599$ and $17\,177\,303\,971$ at lengths $18$, $19$ and $20$. Their consequences exceed the corresponding bounded-atlas rows and retain their own pointwise checks; they are not added to the census population. The current pointwise bounds and recurrence-closed comparisons are reported in the second edition~\cite{boundsv2}. Large $k$ values in Table~\ref{tab:targets} also approach the upper boundary of the search domain, which censors any conclusion about the largest possible construction prime. \subsection{Prime size, refinement, and apparent dependence} Maxima for different orders are correlated when all prime sizes are pooled. For $(3,6)$ the raw correlation is $0.376413$; after restricting to $p\ge2^{28}$ it is $0.203170$. Pooling deviations from the means in the fifteen remaining bands of width $2^{28}$ gives $0.006293$. Corresponding within-band values are $0.002830$ for $(2,3)$, $0.004642$ for $(2,4)$, $0.003503$ for $(3,9)$, and $0.009308$ for $(4,8)$. These transformations describe size adjustment and are not proofs of independence. There is also a deterministic relation. If $r\mid s\mid p-1$, then $\chi_{p,r}=\chi_{p,s}^{s/r}$, so a constant fine-colour run is constant after projection. Thus \[ M_s(p)\le M_r(p),\qquad L_s(p)\le L_r(p),\qquad K_s(p)\le K_r(p). \] The last inequality follows from the two boundary cases in \eqref{eq:K}; $\sigma_r=-1$ implies $\sigma_s=-1$. These inequalities hold throughout the common admissible populations. Small correlations in the centre do not remove dependence among extreme nested events. For a fine run already selected, a local independent-colour comparator must condition on the fine boundary. If the fine order is $s$, the coarse order is $r\mid s$, and the neighbouring fine colour differs from the run colour, its probability of having the same coarse colour is $(s-r)/(r(s-1))$. Subsequent unconditioned coarse matches have probability $1/r$. The earlier exploratory study uses the equivalent parameterization with $s/r$ fine colours per coarse colour. Such conditioned comparisons are more appropriate than multiplying two marginal rare-tail probabilities; they remain local heuristic models. \subsection{Four-term counts and an elliptic trace}\label{sec:A4} Let $A_4(p)$ count ordinary starts $1\le x\le p-4$ at which all four quadratic characters of $x,x+1,x+2,x+3$ are equal. Both constant signs are included. For $p>3$ write $\chi=\chi_{p,2}$ and define \[ B_p=4\chi(2)+2\chi(6)+2\chi(3)+2\chi(-1)+2\chi(-2). \] For the fixed elliptic curve \[ E_9:y^2=x(x-1)(x-9),\qquad t_p=p+1-\#E_9(\Fp), \] the exact identity used by the CPU calculation is \begin{equation}\label{eq:A4} A_4(p)=\frac{p-11-B_p-t_p}{8}. \end{equation} Here is a derivation. Away from the four zeros, the indicator of four equal signs is \[ \frac18\left(1+\sum_{0\le i3$. Isogenous elliptic curves over $\Fp$ have the same Frobenius trace. Thus its full quartic character sum is $-t_p-1$: the affine count is $p$ plus that sum, and the two infinity points give the projective count $p+1-t_p$. At each excluded zero the quartic term vanishes; the total constant contribution is $4$, and the remaining pair contributions sum to $B_p$. Subtracting these four positions gives \eqref{eq:A4}. The census sum is \[ \sum_{\substack{p<2^{32}\\p\text{ odd}}} A_4(p) =53\,206\,216\,723\,199\,456. \] All integer Hasse bounds are satisfied. Since $|B_p|\le12$, \[ \left|A_4(p)-\frac p8\right|\le\frac{\sqrt p}{4}+\frac{23}{8}. \] The $j$-invariant of $E_9$ is $1\,556\,068/81$. Its non-integrality excludes complex multiplication. With $Z_p=t_p/(2\sqrt p)$, the fixed Sato--Tate comparator is $(2/\pi)\sqrt{1-Z^2}\,dZ$ on $[-1,1]$~\cite{sutherland}. Across $203\,280\,219$ primes $p>3$, the empirical mean is $1.09435\times10^{-5}$ and the second moment is $0.2500016788$, compared with limiting values $0$ and $1/4$. Figure~\ref{fig:atlas} uses this fixed density, without fitting a parameter. \begin{figure}[htbp]\centering \begin{tikzpicture} \begin{semilogyaxis}[width=0.48\textwidth,height=55mm,xlabel={$M$},ylabel={Prime fraction},xmin=0,xmax=102,ymin=2e-9,ymax=1,legend style={font=\scriptsize,draw=none},legend columns=2,legend pos=north east,tick label style={font=\scriptsize}] \addplot+[no marks,line width=0.8pt,color=blue!65!black] coordinates {(1,4.9193177772e-09) (2,9.83863555441e-09) (3,9.83863555441e-09) (4,3.93545422176e-08) (5,2.95159066632e-08) (6,1.32821579985e-07) (7,1.47579533316e-07) (8,3.44352244404e-07) (9,6.14914722151e-07) (10,1.24950671541e-06) (11,2.25304754196e-06) (12,4.55036894391e-06) (13,7.43308916136e-06) (14,1.50481930805e-05) (15,2.68447171102e-05) (16,5.31089547227e-05) (17,9.70925749687e-05) (18,0.000190003729827) (19,0.000351613157443) (20,0.000673395571886) (21,0.00127596772573) (22,0.0024677659243) (23,0.00471604172801) (24,0.00905208583501) (25,0.0173928825933) 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marks,thick,color=orange!70!black]{2/pi*sqrt(max(0,1-x^2))};\addlegendentry{Fixed Sato--Tate} \end{axis}\end{tikzpicture} \caption{Marginal maxima for selected orders and normalized elliptic traces in the full atlas. The trace comparator is fixed and has no fitted parameter.}\label{fig:atlas} \end{figure} The raw correlation between $A_4$ and $M_2$ is $0.531581$. Replacing the quadruplet count by its normalized elliptic trace gives $-0.002831$, or $-0.003086$ within the fifteen large-prime bands. This removes the dominant prime-size trend; it does not prove independence. There is a more basic obstruction to recovering a global maximum from a short-pattern count. If $h_\ell$ counts maximal constant runs of length $\ell$, then \[ A_k=\sum_{\ell\ge k}(\ell-k+1)h_\ell. \] A single weighted sum such as $A_4$ does not determine the farthest nonzero entry of $h_\ell$. \section{The fixed million ternary supplement}\label{sec:ternary} \subsection{A separate population and three different counts} The repository also contains a fixed sample of $1\,000\,000$ ternary profiles on $[2^{33},2^{34})$. The separately frozen catalogue collector in \path{src/ternary_catalogue/} produced these profiles and fixed-threshold counts; its historical qualification has domain $p<2^{34}$ and is documented in \path{data/ternary-million/README.md}. It has a different domain and contract from the worker described for $p<2^{33}$. This supplement does not extend the complete census below $2^{32}$. There are four strata of $250\,000$ primes each: the two bands $[2^{33},3\cdot2^{32})$ and $[3\cdot2^{32},2^{34})$, each divided according to whether two is a cube. The equal sample weights differ from the asymptotic natural relative cube-class density $1/3$. No average with these four equal weights is presented as a uniform prime average over the whole band. For a prime $p$ in the supplement put $h=(p-1)/2$. Reflection $x\mapsto p-x$ preserves the cubic character. An ordinary canonical run is a maximal constant interval strictly inside this half-period; it excludes the initial run and the merged central run. Let $N_{p,k}$ count these ordinary runs of length at least $k$, and let $C_p$ be the full length of the central run. Then the predecessor start count on the unfolded interval is \begin{equation}\label{eq:ternary-starts} Z_{p,k}=2N_{p,k}+\mathbf1_{L_p\ge k}+\mathbf1_{C_p\ge k}. \end{equation} Here the predecessor ranges over $1,\ldots,p-k-1$. The initial run at one has no predecessor, whereas its reflected copy does; a central run has one start. Overlapping $k$-windows inside a longer run are a third observable and must not be substituted for $N_{p,k}$. The stored thresholds are 18 through 27. The supplement contains $33\,257\,058$ canonical maximal runs of length at least 18, and $1\,235\,473$ retained canonical positions at length at least 21. Exactly $1\,234\,808$ of the latter are ordinary. The difference is accounted for by the prefix and centre conventions. Full-million positions of lengths 18 through 20 were not retained. The stored central field is zero below length 18; it does not assert that the central run is absent. At threshold 21, $299\,150$ profiles satisfy the interior criterion $M<21$, while $293\,872$ satisfy both Rabung criteria $K\le21$. For these primes $K=\max(M+1,2L+2)$, consistent with Section~\ref{sec:atlas}. Supplement membership includes earlier diagnostic primes and is not an independent validation of the analyses subsequently made on it. The fixed selection and its scientific commitments are documented in \path{data/ternary-million/}. \subsection{Intensity, continuation, and a fixed comparator} For an independent word of $h$ uniform $r$-colours, the expected number of strictly interior maximal runs of length at least $k$ is, for $k\le h-2$, \begin{equation}\label{eq:iid-ordinary} \mu_{h,k}=\frac{(r-1)(h-k-1)-1}{r^k}+r^{1-h}. \end{equation} For proof, an interval of exact length $\ell$ has probability $(r-1)^2/r^{\ell+1}$ of being constant and different from its two neighbours. There are $h-\ell-1$ interior placements; summing over $k\le\ell\le h-2$ gives \eqref{eq:iid-ordinary}. The formula retains the finite-word endpoints. The terminal term is bounded explicitly below; equal population sizes alone do not cancel it when their word lengths differ. Let $S_c$ and $S_n$ be the two sets of $500\,000$ profiles for which \emph{two} is respectively a cube and a noncube. For any per-prime statistic $U_p$, write $\delta U=\sum_{p\in S_c}U_p-\sum_{p\in S_n}U_p$. Define the count contrast and its iid-centred version by \begin{equation}\label{eq:ternary-contrasts} \delta N_k=\delta(N_{p,k}),\qquad \Delta_k=\delta(N_{p,k}-\mu_{h,k}). \end{equation} For $a20996$ these intervals avoid the word endpoints. Their total length is at most 518112, and there are 2510 rational cuts. Thus the number of intervals plus cuts is at most 6542. In general, if total marked length is at most $B$ and this number is at most $J$, coverage of an exact $\ell$-interval is bounded by $B+(\ell-1)J$. Summing gives the uniform bound \[ \EE_{\rm iid}T_{p,k}\le \frac{(r-1)(B+(k-1)J)+J}{r^k}. \] Using the same word lengths for all sampled primes therefore gives: \begin{center} \begin{tabular}{r rr} \toprule $k$&Observed marked runs&Upper bound for iid expected total\\ \midrule 21&2300&124.703820\\ 24&682&4.757640\\ 27&228&0.181357\\ \bottomrule \end{tabular} \end{center} The decimal upper bounds are rounded upwards. They compare counts with bounded expectations; they are not $p$-values. The continuation correction admits a sharper signed bound. Adding one letter to the length of a test interval increases its covered placements by a number between zero and $J$, apart from the final finite-word cutoff. Summing these increments gives, per profile, \[ -\frac{(r-1)(b-a)}{r^b} \le \EE_{\rm iid}T_{p,b} -r^{a-b}\EE_{\rm iid}T_{p,a} \le\frac{J(r-1)(b-a)}{r^b}. \] The same bounds applied to the two equal-sized cube/noncube populations give an absolute iid contrast below 0.002575 for 24 to 27. In this calculation the background is $\varepsilon^T_{a,b}=\delta(\EE_{\rm iid}T_{p,b} -3^{a-b}\EE_{\rm iid}T_{p,a})$, and the fully centred catalogue fraction is \[ \frac{D^T_{a,b}-\varepsilon^T_{a,b}} {D^N_{a,b}-\varepsilon^N_{a,b}}. \] For 24 to 27, $|\varepsilon^T_{24,27}|\le727000000/282429536481+3^{-100}$; the rational part is approximately 0.002574094 and the final term bounds the finite-word remainder. Retaining also the bound on $\varepsilon^N$, the marked catalogue accounts for a fraction between 90.389665\% and 90.392038\% of the centred initial contrast $217-\varepsilon^N_{24,27}$; the centred residual lies in $[20.849277,20.854427]$ cumulative occurrences. These are outward-rounded rigorous enclosures for the fixed comparator, not bounds for the error of a universal maximum law. \section{Structural classification of extreme witnesses}\label{sec:classification} \subsection{Boundary structure and rational anchors} The neutral prefix is constrained by multiplicativity. In particular, $L_r(p)+1$ must be prime: if it were composite, all its prime factors would already be neutral, giving a contradiction. A related identity describes half-word suffixes. Let $h=(p-1)/2$, and extend $\chi$ by $\chi(0)=0$ when needed. Let $q$ be the least odd prime whose character is not $1$. Then \[ \chi(h-j)=\sigma\chi(2)^{-1}\chi(2j+1). \] Every positive odd integer below $q$ is neutral, so the suffix of $1,\ldots,h$ has length $(q-1)/2$. If $\sigma=1$, reflection joins the two half-word suffixes into a central run of length $q-1$. If $\chi(2)=1$, the ordinary neutral prefix ends at $q-1$; otherwise it has length $1$. This explains long prefix and centre events without a random-sign hypothesis. The atlas has $L_8\ge2$ at approximately $24.9983\%$ of admissible primes, instead of the naive $1/8$. For $p\equiv1\pmod8$, quadratic reciprocity gives $(2/p)=1$, hence $\chi_{p,8}(2)^4=1$. The possible values already lie in a subgroup of size four. This arithmetic restriction illustrates why an independent-colour prefix model can fail even before examining extreme runs. A rational anchor $w$ can similarly place small rational multiples of consecutive offsets close to fractions of $p$. The resulting signs are controlled by a few small integers. An algebraic anchor instead satisfies $P(w)=0\pmod p$ for a low-degree polynomial $P$; exact identities in $\Q[X]/(P)$ then transfer to the residue window. A polynomial congruence alone is not a compression explanation. It must produce a smaller character-class group or another proved constraint. The distinction is visible in a retained quartic at $p=2\,654\,052\,683$, whose constant-window rank saves no condition. Non-monic templates also enlarge the search class. For $P=aX^2+bX+c$ and $\beta=a\alpha$, \[ \beta^2+b\beta+ac=0,\qquad \alpha+j=(\beta+aj)/a. \] The monic integral equation now has step $a$ rather than step one. Scaling all positions preserves a constant colour but can change whether that colour is neutral. A search restricted to integral monic polynomials with unit-step windows need not include these non-monic families. \subsection{The selected deep tail} The exploratory classification uses order-specific cutoffs \[ (52,36,28,26,26,22,20,20)\quad\text{for }r=2,\ldots,9. \] It selects $11\,557$ profiles and their canonical witnesses. The first classification, before the higher-degree extensions, is printed in Table~\ref{tab:classification}. Categories are assigned in order; they are a reproducible bookkeeping scheme, not mutually exclusive arithmetic possibilities in nature. \begin{table}[htbp]\centering\small\setlength{\tabcolsep}{4pt} \caption{Initial deep-tail classification, before the higher-degree extensions. The final unmodelled list is smaller, as described in the text.}\label{tab:classification} \begin{tabular}{rrrrrrr} \toprule $r$ & Cutoff & Prefix/centre & Rational & Quadratic & Cubic & Remaining \\ \midrule 2 & 52 & 7675 & 430 & 531 & 6 & 44 \\ 3 & 36 & 1538 & 26 & 170 & 1 & 0 \\ 4 & 28 & 573 & 24 & 203 & 1 & 9 \\ 5 & 26 & 118 & 1 & 58 & 0 & 0 \\ 6 & 26 & 43 & 0 & 4 & 0 & 0 \\ 7 & 22 & 36 & 1 & 9 & 0 & 0 \\ 8 & 20 & 12 & 1 & 22 & 0 & 0 \\ 9 & 20 & 7 & 1 & 13 & 0 & 0 \\ \bottomrule\end{tabular}\end{table} The first stage left $53$ profiles: $44$ at order two and nine at order four. Subsequent bounded searches retained models for fifteen and then seven additional binary profiles, and for eight order-four profiles. The remaining list has $23$ entries: $22$ binary profiles and one order-four profile. This count includes the distinction just made between a retained algebraic model and actual compression; it should not be read as a theorem that all other profiles have one unique explanation. The $56$ deepest order-eight and order-nine profiles consist of $19$ prefix/centre cases, two rational cases, and $35$ quadratic cases. The latter use square classes $-47,-31,-23,33,73,97,145,148$ with respective multiplicities $1,4,2,1,10,14,2,1$. After boundary and rational cases are removed, the neutral shares rise sharply in the extreme tails: at order eight they are $612/1767$ for $M\ge16$, $97/117$ for $M\ge18$, and $22/22$ for $M\ge20$; at order nine they are $202/340$, $52/53$, and $13/13$. After the retained quadratic families are also removed at $M\ge16$, the shares become $181/1327$ and $14/152$. These contrasts suggest a mixture of ordinary bulk behaviour and special arithmetic families. Because the families were found from these data, the contrasts are descriptive evidence and not significance tests. \section{Exact character-class certification}\label{sec:certification} \subsection{Neutral and constant windows} Let $F$ contain $\mu_r$, let $\alpha$ be a specified algebraic root, and set $a_j=\alpha+j$ for $0\le j512$, $Q(w)=0\pmod p$, and let $\chi$ be a character with $\chi(-1)=1$. If $\chi(w+j)$ has one constant value for $0\le j\le10$, that value and the characters of $2,3,11$ are $1$. There is $t$ with $t^2=1$ such that the $26$ signs are \[ \chi(w+j)= \begin{cases} 1,&0\le j\le10\text{ or }21\le j\le25,\\ t,&11\le j\le20. \end{cases} \] For every odd character order, all $26$ values are therefore neutral. \end{theorem} \begin{proof} Every identity in Table~\ref{tab:propagation} is an exact polynomial identity after reduction modulo $Q$. Write $u_j=\chi(w+j)$, $c=u_0=\cdots=u_{10}$, $a=\chi(2)$, and $b=\chi(3)$. The first six identities give \[ u_{22}=1,\quad c=a^4,\quad u_{24}=a, \quad u_{21}=a^2/c,\quad u_{23}=a,\quad b=a. \] The additional identity $A_9A_{22}=-18$ gives $c=ab^2=a^3$. Hence $a=b=c=1$. The eighth identity gives $\chi(11)=1$. The first eight also determine $u_{21},\ldots,u_{25}=1$. The ninth gives $u_{12}^2=1$, and identities ten to eighteen give the same value $t=u_{12}$ at all positions $11,\ldots,20$. For an odd order, $t^2=t^r=1$ implies $t=1$. Finally $0<|Q(-j)|\le2161126$ the coefficient and quadratic discriminant exceptions are absent, and $0<|P(-j)|\le411$ for $0\le j<30$ excludes a zero or an ordinary-boundary crossing. The joined neutral group is $C_4^{21}$ and the joined ratio group is $C_4^{20}$, exactly the two-root groups. The first identity puts the class of $-1$ in the original neutral group, so the reflected generators supply the last two positions. For the ratio groups, $B_j/B_0=A_{29-j}/A_{29}$ and $A_{29}/A_0=(A_{26}/A_0)/(B_3/B_0)$, which identifies the ratio groups. Among all rational primes the densities of this particular 30-window event are \[ \delta_{30,n}=2^{-44},\qquad \delta_{30,c}=2^{-42},\qquad \Prob(A_{30,c}\mid A_{28,c})=\frac1{31}. \] Among admissible primes the first two values double. A single 28-window need not continue: the extension is equivalent to the two-root event, not forced by one root alone. For $5X^2+117X+640$ and $2X^2+49X+288$, respectively, the identities are \[ A_0B_{16}=16A_8,\qquad A_{10}B_{11}=A_9. \] Since $16=2^4$, two constant phases $\phi,\psi$ satisfy $\phi\psi=\phi$, and hence $\psi=1$; conjugation gives $\phi=1$. These identities explain why the two joint ratio groups equal their neutral counterparts even though an isolated constant window may be non-neutral. The exact densities among all primes remain \begin{center} \begin{tabular}{lcc} \toprule Polynomial & Neutral density & Constant density\\ \midrule $2X^2+58X+411$ & $31/2^{44}$ & $31/2^{42}$\\ $5X^2+117X+640$ & $524287/2^{54}$ & $2097151/2^{54}$\\ $2X^2+49X+288$ & $131071/2^{56}$ & $262143/2^{56}$\\ \bottomrule \end{tabular} \end{center} The conditional probabilities of two realizing roots given at least one constant realization are respectively $1/31$, $1/2097151$, and $1/262143$. In these three quadratic families a non-neutral constant root cannot coexist with a neutral constant root. Thus subtracting neutral from constant existence densities does count existence of a non-neutral realization, with all-prime values \[ \delta_{\mathrm{nonneutral}}= \frac{93}{2^{44}},\qquad\frac3{2^{35}},\qquad\frac1{2^{39}}, \] in the same order. Such a subtraction is not valid for arbitrary families, where mixed neutral and non-neutral realizations may coexist. These remain limiting probabilities for fixed templates; the originating record does not establish convergence below $2^{32}$. \subsection{Sixteen comparisons suffice for the discriminant-889 model} For $5X^2+117X+640$, the individual ratio group is $C_4^{16}$. It is already generated by the ratios at \[ T=\{1,2,3,4,5,6,7,8,9,10,13,19,21,24,26,27\}. \] Restricting the certified character matrix to these columns gives image order $4^{16}$, equal to the exact full group order. For $p>1126$, $p\equiv1\pmod4$, and a root $w$ of this polynomial, the sixteen comparisons \[ \chi_{p,4}(w+j)=\chi_{p,4}(w),\qquad j\in T, \] therefore force all 28 positions to have the same colour. There are seventeen distinct positions including the reference; they are not consecutive. Sixteen is minimal among generating sets made of individual ratios, since $C_4^{16}$ needs at least sixteen generators. This is not an algorithmic complexity lower bound or a measured scan acceleration. \section{The dihedral quartic and shifted 63-windows}\label{sec:dihedral} Consider the earlier binary quartic \[ P_D(X)=X^4+80X^3+2281X^2+27240X+116640. \] The shift $Y=X+20$ gives $Y^4-119Y^2+4240$. Its roots are $a-20,-a-20,b-20,-b-20$, with $a^2+b^2=119$ and $a^2b^2=4240$. The quartic field has degree $4$, and its normal closure is obtained by adjoining $\sqrt{265}$, giving degree $8$ and group $D_4$. This action has two pair orbits: opposite pairs and adjacent pairs. Put $h_o$ for the opposite-pair group in its quartic field, $h_a$ for an adjacent pair in the normal field, and $h_3,h_4$ for three and four roots in that normal field. Equation~\eqref{eq:orbit} yields \begin{equation}\label{eq:D4} \delta_D=\frac1g-\frac1{2h_o}-\frac1{2h_a} +\frac1{2h_3}-\frac1{8h_4}. \end{equation} The exact orders for its $52$-windows are \[ g=2^{43},\quad h_o=2^{53},\quad h_a=2^{79}, \quad h_3=2^{89},\quad h_4=2^{95}. \] Opposite pairs are certified in the quartic field itself. Extending that field to the normal field can kill the class of $265$, which is why it cannot be silently substituted in the opposite-pair term. The density is \[ \delta_D=2^{-43}-2^{-54}-2^{-80}+2^{-90}-2^{-98}. \] Constant and neutral events coincide. The odd-weight identity \[ 9A_0A_5A_{12}A_{24}A_{27} =[\alpha(\alpha^2+43\alpha+402)]^2 \] forces a constant phase to be neutral; another identity, \[ -9A_{13}A_{16}A_{28}A_{35}A_{40} =(\alpha^3+77\alpha^2+1762\alpha+11280)^2, \] forces $\chi(-1)=1$ in this event. For an opposite root $\alpha'=-40-\alpha$, its positions $0,\ldots,51$ reflect to positions $-11,\ldots,40$ at $\alpha$. With $\chi(-1)=1$, two neutral $52$-windows therefore give one neutral window on $\alpha-11,\ldots,\alpha+51$, of length $63$. Conversely that shifted $63$-window gives the opposite pair. For $p>813402$, the exact nonzero values of $P_D(-j)$ on $-11\le j\le51$ exclude ordinary-boundary crossings; their largest absolute value is $813402$. Let $B$ be the event that at least one opposite pair is neutral. Each pair has density $1/(4\cdot2^{53})$, and their intersection is the four-root event of density $1/(8\cdot2^{95})$. Thus \[ \delta(B)=2^{-54}-2^{-98}. \] Writing $A_{52}$ for existence of a neutral $52$-window at a root, \[ \Prob(B\mid A_{52})=\frac{5864062014805}{12003734944219221} \simeq0.000488519785. \] This differs from the $2^{-10}$ conditional continuation at a marked neutral root. The two quantities have different multiplicities. A realization implies $M_2\ge63$ but does not specify the global maximum or announce a new atlas occurrence. \section{The last order-four profile and bounded negative evidence}\label{sec:residual} The sole order-four profile still without a retained explanatory model is \[ p=2\,940\,141\,229,\quad M_4=28,\quad w=390\,499\,731,\quad\text{phase}=-1. \] Its immediate quadratic projection is a neutral run of exactly $29$ positions, starting at $w-1=390\,499\,730$. Both local ends were checked by exact modular exponentiation. Also $p\equiv5\pmod8$, so $\chi_{p,4}(-1)=-1$; reflection produces an ordinary neutral $28$-window starting at $2\,549\,641\,471$. Thus the canonical non-neutral phase does not preclude a neutral realization of the same maximum at this prime. Reflection alone cannot improve a candidate's constant-window compression. Indeed, for $\beta=-\alpha-(m-1)$, $b_j=\beta+j=-(\alpha+m-1-j)$, and \[ \left\langle[b_j/b_0]:1\le j