% paper_B_maass_spectrum.tex % Adapted from 总册 第七章 (算术量子混沌:简并度作为第三个轴) and 第九章 9.2 (tau<->m 形式化、上包络判据)。 % 内部编号独立成篇;修正记录按总册主编号引用(master log no. NN)。 % Compile: pdflatex twice (hyperref). No TikZ; booktabs only. \documentclass[11pt]{article} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage[margin=1in]{geometry} \usepackage{booktabs} \usepackage[hidelinks]{hyperref} \numberwithin{equation}{section} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{verdict}[theorem]{Verdict} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{correction}[theorem]{Correction} \DeclareMathOperator{\li}{li} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\vol}{Vol} \DeclareMathOperator{\Sha}{Sha} \title{Degeneracy and the level-one Maass spectrum:\\a measured verdict on arithmetic quantum chaos} \author{Chunping Zeng} \date{October 2026} \begin{document} \maketitle \begin{abstract} We isolate the spectral axis of the seven-world program: the length spectrum of the modular surface, its degeneracy, and the statistics of the level-one Maass spectrum. Three results. (1)~\emph{The true degeneracy is the narrow class number}: the number $g(D)$ of closed geodesics of discriminant $D$ equals the number of $\rho$-orbits of Zagier-reduced indefinite binary quadratic forms, which equals $h^{+}(D)$ (master log nos.~41,~43); on all $36$ fundamental discriminants $5\le D\le 120$ the equality $\#\rho\text{-orbits}=\operatorname{round}(h^{+}_{\mathrm{Dirichlet}})$ holds with zero counterexample. (2)~\emph{The prime geodesic theorem has error exponent $1/2$ under the corrected counting} --- all non-square discriminants $D\equiv 0,1\pmod 4$, primitive forms only, $\rho$-orbits rather than reduced forms: $N(X)/\operatorname{li}(e^{X})=0.9963$ at $X=12$ and $|\mathrm{err}|/e^{X/2}$ remains bounded and decreases overall from $1.547$ to $0.138$ (master log no.~44). (3)~\emph{The level-one Maass spectrum is long-range GOE and short-range non-Wigner}: on $1110$ even levels, number variance $\Sigma_{2}(20)=1.203\,[1.02,1.26]$ and $\Sigma_{2}(30)=1.194\,[1.06,1.32]$ against Poisson values $20$ and $30$ and compatible with GOE values $1.049$ and $1.131$, while the spacing distribution fails the Wigner surmise ($P(s<0.05)=0.0334$ against GOE $0.0020$, a $17\times$ small-gap excess; spacing coefficient of variation $0.802$). This is the standard signature of arithmetic quantum chaos: spectral rigidity without local Wigner repulsion. We also record the archimedean parity law $\Gamma^{\mathrm{a}}_{2k}=2\Gamma^{\mathrm{a}}_{2k+1}\sim r^{-2k}$ for Maass $L$-functions, the synthetic-spectrum self-certification of the pipeline, and the corrected saturation criterion (master log no.~49). Every measured number carries a confidence interval; every correction applied is recorded in Appendix~A. \end{abstract} \section{Introduction} The seven-world framework attaches to each arithmetic system three axes: the analytic axis (poles versus natural boundaries), the amplitude axis (fingerprint-carrying critical values), and the spectral axis treated here --- length spectra, their degeneracies, and the eigenvalue statistics of Laplacians. The spectral axis is the most delicate: it is the only one on which the program's predictions can be checked against an externally produced dataset of a genuine arithmetic spectrum. This paper reports the completed status of that axis, with two independent parts. \medskip \noindent\textbf{Part I: length spectra and degeneracy (Sections 2--4).} Two of the program's own corrections (master log nos.~41,~42,~43) changed the counting twice: the unit in Pell's equation is $p-q\bar\omega^{n}$, not $p+q\omega^{n}$, and the degeneracy is the number of $\rho$-orbits, not the number of reduced forms. Under the corrected definitions the structure theorem $g(D)=h^{+}(D)$ holds with a closed formula (Proposition~\ref{prop:hplus}), the degeneracy grows robustly (over $D\le 20000$: $348914$ reduced forms in $21721^{+}$ geodesics, mean $57.4$ forms per discriminant), and the prime geodesic theorem's error exponent $1/2$ is confirmed (Section~\ref{sec:pgt}). The degeneracy then drives the diagonal weight of the trace formula up by a factor $18.86$, giving the Bogomolny chain degeneracy $\Rightarrow$ GOE$\to$Poisson. \medskip \noindent\textbf{Part II: the Maass level-one verdict (Sections 5--9).} Using the LMFDB/Zenodo level-one Maass dataset, with the even spectrum ($1110$ levels) as primary evidence and the odd spectrum discarded because of a large hole (Section~\ref{sec:data}), we find: number variance decisively GOE at long range ($\Sigma_{2}$ against Poisson by a factor $15$--$35$ with non-overlapping bootstrap intervals), the $\Delta_{3}$ statistic pointing the same way with its residual excess reported, not absorbed (Section~\ref{sec:honest}), spacing distribution Poisson-leaning at short range ($P(s)$ Kolmogorov--Smirnov statistics $0.175$ against the Wigner surmise versus $0.073$ against the exponential law), and a $17\times$ excess of small gaps, quantitatively attributed to Hecke near-degeneracies. The same section contains the corrected saturation criterion for the spectral form factor (master log no.~49): the mean value $K_{N}=1/N$ is pure diagonal and carries no information; the criterion is the decay of the upper envelope, on which the discrete and semicircular spectra differ by five orders of magnitude at $T=40$. \medskip Every table below is reproduced from the master compendium, where each entry carries its own provenance; the correction register in Appendix~A isolates the entries touching this paper's material. \subsection*{Status labels} Measured numbers are verdicts, not theorems, and are quoted with their bootstrap confidence intervals. The pipeline is self-certified against synthetic spectra of both candidate statistics before being applied to the arithmetic data (Section~\ref{sec:stats}). Residual discrepancies are reported rather than absorbed (Section~\ref{sec:honest}). \section{The length spectrum: degeneracy as the third axis} \label{sec:degeneracy} \subsection{Closed geodesics and Zagier reduction} Let $D>0$ be a non-square discriminant. Indefinite binary quadratic forms $[a,b,c]$ of discriminant $b^{2}-4ac=D$, $a>0$, admit Zagier's reduction operator $\rho$; each $\rho$-orbit is a closed geodesic on $X=\mathrm{SL}_{2}(\mathbb{Z})\backslash\mathcal{H}$ of length \begin{equation} \ell(D)=2\operatorname{arccosh}(t/2), \end{equation} where $t$ comes from the fundamental solution of Pell's equation $t^{2}-Du^{2}=4$. All $\rho$-orbits of the same $D$ are individually verified to be isometric ($\sum_{\mathrm{cycle}}\log\frac{b+\sqrt{D}}{2c}=\ell/2$ on $D=12,13,60,105,120$). \begin{correction}[Master log no.~41, 43: the definition of degeneracy] The naive count $Z(D)$, the total number of Zagier-reduced forms, overcounts: it weights each orbit by its length. Over $D\le 20000$ ($6081$ fundamental discriminants) one has $\sum Z = 348914$ but $\sum_{D} h^{+}(D)=21721^{+}$ --- an average orbit-length factor of $\approx 16\times$. The \emph{true degeneracy} of the length spectrum is \begin{equation}\label{eq:degeneracy} g(D)=\#\{\rho\text{-orbits}\}=h^{+}(D), \end{equation} the narrow class number, \emph{not} the wide class number $h(D)$ and \emph{not} $Z(D)$. Example: $Z(8)=2$ ([1,4,2] and [2,4,1]) but a single $\rho$-orbit, so $g(8)=h^{+}(8)=1$. The earlier degeneracy scripts also carried two bugs (perfect-square duplication; \texttt{isqrt} in place of $\sqrt{D}$); after correction $Z(D)$ grows steadily (mean $57.4$ forms per discriminant over $D\le 20000$, maximum $83$). The structural fact behind the count: each narrow ideal class contains exactly one $\rho$-cycle of Zagier-reduced forms (Zagier's reduction theory), so $g(D)=h^{+}(D)$ is a theorem, not merely a measured match. \end{correction} \begin{correction}[Master log no.~42: the Pell unit] The fundamental unit is $p-q\bar\omega^{n}$, not $p+q\omega^{n}$; consequently for $D\equiv 1\pmod 4$ the trace is $t=2p-q$ (not $2p+q$). With the corrected unit, all $8/8$ benchmarks pass: $t(5)=3$, $t(8)=6$, $t(12)=4$, $t(13)=11$, $t(17)=66$, $t(29)=27$. \end{correction} \subsection{The closed formula for the degeneracy} \begin{proposition}\label{prop:hplus} For a fundamental discriminant $D$, \begin{equation}\label{eq:hplus} h^{+}(D)=\frac{2\sqrt{D}\,L(1,\chi_{D})}{\ell(D)}=\frac{\sqrt{D}\,L(1,\chi_{D})}{\log\eta^{+}(D)},\qquad L(1,\chi_{D})=-\frac{1}{D}\sum_{a1$ is the fundamental unit of norm $+1$, and $\ell(D)=2\log\eta^{+}(D)$ is the geodesic length. \end{proposition} \begin{proof} Dirichlet's class number formula reads $h\log\varepsilon_{0}=\tfrac{\sqrt{D}}{2}L(1,\chi)$ with $\varepsilon_{0}>1$ the fundamental unit. If $N(\varepsilon_{0})=-1$ then $\eta^{+}=\varepsilon_{0}^{2}$, $\ell=4\log\varepsilon_{0}$ and $h^{+}=h$; if $N(\varepsilon_{0})=+1$ then $\eta^{+}=\varepsilon_{0}$, $\ell=2\log\varepsilon_{0}$ and $h^{+}=2h$. In both cases $h^{+}\log\eta^{+}=\sqrt{D}\,L(1,\chi)$, which is \eqref{eq:hplus}. The digamma form of $L(1,\chi_{D})$ is standard. \end{proof} \begin{correction}[Closed formula for $h^{+}$] An earlier form of \eqref{eq:hplus} read $2\sqrt{D}\,L(1,\chi_{D})/\log\varepsilon_{0}$; this equals $4h$ for $D=5$ instead of $h^{+}(D)=1$; the denominator must be the norm-$+1$ regulator ($\log\eta^{+}=\ell(D)/2$), not $\log\varepsilon_{0}$. The displayed formula is the corrected one, verified against $\rho$-cycle counting on all $36$ fundamental discriminants $5\le D\le 120$. \end{correction} \begin{center} \begin{tabular}{rrrr} \toprule $D$ & $g(D)=h^{+}(D)$ & $t$ & $\ell(D)$\\ \midrule $5$ & $1$ & $3$ & $1.924847$ (shortest)\\ $12$ & $2$ & $4$ & $2.633916$\\ $17$ & $1$ & $66$ & $8.378850$\\ $44$ & $2$ & $20$ & $5.986446$\\ $52$ & $1$ & $1298$ & $14.337159$\\ \bottomrule \end{tabular} \end{center} Here $t$ is the minimal solution of $t^{2}-Du^{2}=4$ (the proper automorph) and $\ell(D)=2\operatorname{arccosh}(t/2)$; all entries independently recomputed. \begin{verdict}\label{verdict:degeneracy} On all $36$ fundamental discriminants $5\le D\le 120$: $\#\rho\text{-orbits}=\operatorname{round}\bigl(h^{+}_{\mathrm{Dirichlet}}\bigr)$, zero counterexample; every equality holds as an exact integer (independently recomputed by $\rho$-cycle counting and by \eqref{eq:hplus}). \end{verdict} Arithmetic chaos is thus the unbounded degeneracy $g(D)\to\infty$ of the length spectrum --- in contrast to finite-type shifts ($g=1$) and to prime numbers ($g=1$). Degeneracy is the third axis, after poles/natural boundaries and amplitudes. \section{The prime geodesic theorem under the corrected counting} \label{sec:pgt} \begin{correction}[Master log no.~44: the counting] The correct prime geodesic count is \begin{equation}\label{eq:counting} N(X)=\sum_{\substack{D\ \mathrm{nonsquare}\\ D\equiv 0,1\,(4)\\ \ell(D)\le X}}\#\{\rho\text{-orbits}\}, \end{equation} with three ingredients that the naive count omits: (a)~\emph{all} discriminants, including nonmaximal orders (these contribute $\approx 60\%$ of the count); (b)~only primitive forms ($\gcd(a,b,c)=1$); (c)~$\rho$-orbits in place of reduced forms. Both naive countings fail decisively: fundamental discriminants only give $N/\operatorname{li}\approx 0.42$--$0.47$; reduced-form totals give $N/\operatorname{li}=12.2$ at $X=9$. \end{correction} \begin{verdict}\label{verdict:pgt} Under the counting \eqref{eq:counting}, $N(X)/\operatorname{li}(e^{X})\to 1$ and $|\mathrm{err}|/e^{X/2}$ is bounded, decreasing overall (with fluctuations): \begin{center}\small \setlength{\tabcolsep}{4pt} \begin{tabular}{rrrrr} \toprule $X$ & $N(X)$ & $\operatorname{li}(e^{X})$ & $N/\operatorname{li}$ & $|\mathrm{err}|/e^{X/2}$\\ \midrule $3$ & $3$ & $9.9$ & $0.302$ & $1.547$\\ $4$ & $10$ & $19.6$ & $0.509$ & $1.303$\\ $5$ & $29$ & $40.2$ & $0.722$ & $0.918$\\ $6$ & $74$ & $86.0$ & $0.861$ & $0.597$\\ $7$ & $167$ & $191.5$ & $0.872$ & $0.740$\\ $8$ & $408$ & $440.4$ & $0.927$ & $0.593$\\ $9$ & $1004$ & $1037.9$ & $0.967$ & $0.376$\\ $10$ & $2451$ & $2492.2$ & $0.984$ & $0.278$\\ $11$ & $5961$ & $6071.4$ & $0.982$ & $0.451$\\ $12$ & $14904$ & $14959.5$ & $0.996$ & $0.138$\\ \bottomrule \end{tabular} \end{center} The error exponent $1/2$ of the prime geodesic theorem is thereby confirmed; the earlier plateau of $N(X)$ beyond $X\approx 20$ under the $D\le 300$ truncation was an artefact of the wrong counting, not of the theorem. \end{verdict} \begin{remark}[Critical line versus natural boundary; master log no.~2] For Maass/Selberg one has $\kappa=0$ (trivial bound $|\alpha|=1=p^{0}$), so the critical line is $\kappa+\tfrac12=\tfrac12$ with zeros at $s=\tfrac12\pm ir$. The equality with the holomorphic case is the equality of \emph{critical lines}, not of $\kappa$: the holomorphic cusp forms of weight $2$ have $\kappa=\tfrac12$ and critical line $\operatorname{Re}s=1$. The rate exponent of QUE and of the prime geodesic theorem is the critical line $\kappa+\tfrac12$, equal to $1/2$ on both sides of the comparison. \end{remark} \section{From degeneracy to statistics: the Bogomolny chain} The trace-formula amplitude of a geodesic is $A_{\gamma}=g_{\gamma}\ell/(2\sinh(\ell/2))$, and the diagonal weight is $\Psi=\sum A_{\gamma}^{2}\propto g_{\gamma}^{2}$. Comparing the length spectrum with degeneracy against the same spectrum with $g_{\gamma}\equiv 1$: \begin{center} \begin{tabular}{lrrrrrr} \toprule $T=\varepsilon/L_{\max}$ & $0.1$ & $0.2$ & $0.3$ & $0.4$ & $0.5$ & $0.8$\\ \midrule with degeneracy $K_{\mathrm{deg}}/\Psi_{\mathrm{tot}}$ & $0.024$ & $0.572$ & $0.936$ & $0.995$ & $0.9996$ & $1.000$\\ without degeneracy $K_{\mathrm{nodeg}}/\Psi_{\mathrm{tot}}$ & $0.195$ & $0.792$ & $0.979$ & $0.9988$ & $0.9999$ & $1.000$\\ \bottomrule \end{tabular} \end{center} The total weight ratio is $18.86$: degeneracy amplifies the diagonal weight by a $g^{2}$ effect and, because large $g$ concentrates on long orbits, shifts weight toward large $\tau$. The Bogomolny chain, quantitatively: degeneracy growth $\Rightarrow$ diagonal weight $\propto g^{2}$ amplified $\Rightarrow$ the (Poisson) diagonal term overtakes the off-diagonal RMT correction $\Rightarrow$ GOE$\to$Poisson. This is formulation (five) of the law of forgetting --- loss of level repulsion --- measured from the length spectrum side; the eigenvalue side is Section~\ref{sec:verdict}. \section{The archimedean parity law for Maass \texorpdfstring{$L$}{L}-functions} \label{sec:parity} \begin{proposition}[Parity law]\label{prop:parity} For a Maass form with spectral parameter $r$, the archimedean terms of the logarithmic derivative of $\Lambda(s)=\pi^{-s}\Gamma\!\left(\frac{s+ir}{2}\right)\Gamma\!\left(\frac{s-ir}{2}\right)L(s)$ satisfy \begin{equation}\label{eq:gammader} \Gamma^{\mathrm{a}}_{1}=-\log\pi+\tfrac12\Bigl[\psi\Bigl(\tfrac{s+ir}{2}\Bigr)+\psi\Bigl(\tfrac{s-ir}{2}\Bigr)\Bigr],\qquad \Gamma^{\mathrm{a}}_{n}=2^{-n}\Bigl[\psi^{(n-1)}\Bigl(\tfrac{s+ir}{2}\Bigr)+\psi^{(n-1)}\Bigl(\tfrac{s-ir}{2}\Bigr)\Bigr]\quad(n\ge 2). \end{equation} As $r\to\infty$, the conjugate pair $(s\pm ir)^{-(n-1)}$ gives $i^{-(n-1)}+(-i)^{-(n-1)}=i^{-(n-1)}\bigl[1+(-1)^{n-1}\bigr]$: odd $n$ survive at leading order, even $n$ cancel exactly one order faster. The unified law is \begin{equation}\label{eq:parity} \Gamma^{\mathrm{a}}_{2k+1}\sim(-1)^{k+1}\frac{(2k-1)!}{r^{2k}};\qquad \Gamma^{\mathrm{a}}_{2k}\sim 2(-1)^{k+1}\frac{(2k-1)!}{r^{2k}}. \end{equation} \end{proposition} \begin{proof} Asymptotic expansion $\psi^{(n-1)}(z)\sim(-1)^{n}(n-2)!\,z^{-(n-1)}$ and the conjugate-pair algebra above. For even orders the constant displayed in the law is the $s=2$ value, cf.\ the table note below; the exponent $2\lfloor n/2\rfloor$ is $s$-independent. \end{proof} \begin{center} \begin{tabular}{lrrrrrrrr} \toprule $n$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$\\ \midrule fitted exponent $p$ & $2.000$ & $2.000$ & $4.000$ & $4.000$ & $6.000$ & $6.000$ & $8.000$\\ fitted constant $C$ & $2$ & $1$ & $12$ & $6$ & $240$ & $120$ & $10078$\\ \bottomrule \end{tabular} \end{center} ($n=8$: prediction $2\cdot 7!=10080$ against measured $10078$.) The even-order constants are fitted at $s=2$ and are proportional to $s$ (at $s=\tfrac12$ they read $(2k-1)!/2$); the exponents and the odd-order constants are $s$-independent. For contrast, a \emph{single} $\psi^{(n-1)}$ has $p=n-1$ and $C=(n-2)!/2$ always: even orders are suppressed by $1/r$ and odd orders enhanced by exactly $2\times$. Observability: for the lowest Maass eigenvalue $r_{1}=9.5336952613$ the $n=1$ archimedean term dominates (ratio $1.89\times$), while $n=2,3$ contribute $\approx 10\%$; $n=1$ is again special in that $\Gamma^{\mathrm{a}}_{1}$ grows like $\log r$ ($0.48\to 6.01$ for $r:10\to 2560$), marking $n=1$ as the Dixmier-trace order, in the family of Weil's $W_{3}$ exclusion and the Bost--Connes $\delta=2$. \section{Data and acceptance} \label{sec:data} The dataset is Zenodo \texttt{10.5281/zenodo.15490636} (LMFDB Collaboration and Lowry-Duda, 2025), six columns \texttt{label:N:R:s:w:coeffs} with $R$ the spectral parameter ($\lambda=\tfrac14+R^{2}$), $s=0$ even, $s=1$ odd. Level one contributes $1110$ even and $1092$ odd levels (matching the LMFDB statistics page, $2202=1110+1092$). \medskip \noindent\emph{Acceptance of the even spectrum.} No anomalous gap, and the Weyl-law fit of the cumulative count returns $0.04166$, exactly $\tfrac{1}{24}=(\tfrac12)(\tfrac{1}{12})$: the even levels occupy precisely half of the Weyl total $N(r)\approx r^{2}/12$. This is complete internal evidence. \medskip \noindent\emph{Rejection of the odd spectrum.} The odd spectrum has a hole at $[99.579,110.170]$ of gap/local-mean-spacing $=46$ (about $45$ missing levels at $\approx 31\%$ of the spectrum). Mixing it in would artificially depress the number variance and fake GOE-like rigidity. All verdicts below use the even spectrum only; re-including the odd segment requires re-extracting that stretch. \section{Unfolding and statistics} \label{sec:stats} The unfolding does not presuppose the Weyl law: a log-log spline is fitted to the cumulative count and $r^{2}/12$ is printed as a control. The statistics are $\Sigma_{2}(L)$ (number variance), $\Delta_{3}(L)$ (Dyson--Mehta), and $P(s)$ (spacing distribution), all with bootstrap error bars --- at $L=30$ the standard deviation of $\Sigma_{2}$ over $1110$ levels is already $O(1)$. \medskip \noindent\emph{Pipeline self-certification on synthetic spectra} (mandatory, and done first). GOE synthetic spectrum ($1200$ levels): $\Sigma_{2}(10)=0.896$ (theory $0.909$), $\Sigma_{2}(20)=1.080$ ($1.049$). Poisson synthetic spectrum: $\Sigma_{2}(20)=19.95$ (theory $20$). Spacing KS correctly identifies GOE against Poisson; the pipeline separates the two statistics by a factor $6$ already at $L=5$ ($0.77$ versus $5$). \section{Verdict: long-range GOE, short-range non-Wigner} \label{sec:verdict} \begin{verdict}\label{verdict:maass} The level-one even Maass spectrum is long-range GOE and short-range non-Wigner. \end{verdict} \noindent\emph{Long range.} With $\Sigma_{2}$ the primary statistic (validated unbiased at the same sample size on synthetic spectra): \begin{center} \begin{tabular}{lrrr} \toprule $L$ & $\Sigma_{2}$ measured (bootstrap CI) & $\Sigma_{2}$ GOE & $\Sigma_{2}$ Poisson\\ \midrule $20$ & $1.203\ [1.02,1.26]$ & $1.049$ & $20$\\ $30$ & $1.194\ [1.06,1.32]$ & $1.131$ & $30$\\ \bottomrule \end{tabular} \end{center} $\Sigma_{2}$ differs from Poisson by a factor $15$--$35$ with non-overlapping intervals: decisive GOE. The $\Delta_{3}$ statistic points the same way ($0.5345$ and $0.5574$ at $L=20,30$, against GOE synthetic values $0.4216$ and $0.4627$ and Poisson $1.333$ and $2.000$ --- the residual $30$--$80\%$ excess over GOE is reported in Section~\ref{sec:honest}). \medskip \noindent\emph{Short range.} The spacing distribution fails the Wigner surmise: $\mathrm{KS}(\text{Wigner})=0.175$ against $\mathrm{KS}(\text{exponential})=0.073$: Poisson-leaning spacings. The hard evidence is the small-gap excess: $P(s<0.05)=0.0334$ against GOE $0.0020$ ($17\times$) and $P(s<0.10)=0.0730$ against GOE $0.0078$ ($9.3\times$). Spacing coefficient of variation: $0.802$, between GOE ($0.523$) and Poisson ($1.0$). \medskip \noindent\emph{Consistency.} Spectral rigidity ($\Sigma_{2}$ GOE) records chaotic classical dynamics; non-Wigner spacings record arithmetic near-degeneracies forced by Hecke symmetry: small spacings are over-produced, short-range repulsion is suppressed, long-range rigidity untouched. Rigid but non-Wigner spacing is the standard definition of arithmetic quantum chaos. \subsection{The saturation criterion, corrected} \begin{correction}[Master log no.~49: upper envelope in place of max/mean] The previous saturation criterion used the max/mean ratio, which fails: the semicircular form factor $\phi=J_{1}(2t)/t$ oscillates (ratio $\sim 5$ from Bessel oscillation), and the true spectrum's ratio $\approx 9.0\approx\log 6000$ is the $\chi^{2}$ extreme-value/mean ratio of a complex Gaussian --- neither is structure. The corrected criterion is the \emph{absolute decay of the upper envelope}, with the $\chi^{2}$ baseline removed: \begin{center} \begin{tabular}{lrrrr} \toprule $T$ & semicircle $\max K$ & $1/(\pi T^{3})$ theory & discrete $\max K$ & ratio\\ \midrule $20$ & $3.97\times 10^{-5}$ & $3.98\times 10^{-5}$ & $1.000$ & $2.5\times 10^{4}\times$\\ $40$ & $4.81\times 10^{-6}$ & $4.97\times 10^{-6}$ & $1.000$ & $2.1\times 10^{5}\times$\\ \bottomrule \end{tabular} \end{center} The semicircle envelope decays like $1/(\pi T^{3})$ (never saturates: the forgetting side); the discrete spectrum keeps $\max K=1$ at recurrences (saturating: memory side). By $T=40$ the two differ by five orders of magnitude: the criterion is decisive. For a finite spectrum, $\mathrm{mean}\ K_{N}=1/N$ is a pure diagonal floor ($9.0\times 10^{-4}$ at $N\approx 1110$) and is shared by GOE, Poisson, and equispaced alike --- it is \emph{not} a criterion. The recurrence-peak scaling $P_{N}\sim N^{-\beta}$ separates the two regimes by a factor $27$ ($\beta=0.028$ equispaced against $0.76$--$0.82$ random/chaotic); the full dictionary is in the companion paper on the law of forgetting \cite{forgetting}, whose Section~12 closes this question for the genuine arithmetic spectrum. \end{correction} \section{Arithmetic reading: structure survives} Maass forms share with holomorphic cusp forms the $\mathrm{SU}(2)$ Sato--Tate semicircle: $\langle\lambda^{m}\rangle=1,2,5,14$ for $m=2,4,6,8$, matching the Catalan moments digit by digit; their central charges are identical to the holomorphic case, and the parity splitting lives only in the archimedean terms $\Gamma^{\mathrm{a}}_{n}$ of Proposition~\ref{prop:parity}. The three-regime table: \begin{center}\small \begin{tabular}{lll} \toprule regime & $\Sigma_{2}$ behavior & position\\ \midrule Poisson & $\sim L$ & forgetting\\ GOE & $\sim\log L$ & chaos, no arithmetic structure\\ Maass & GOE long range, non-Wigner short range & chaos $+$ arithmetic structure\\ \bottomrule \end{tabular} \end{center} Maass is not an instance of forgetting but an instance of structure survival at the rigidity end; the direction of forgetting --- pushing $\Sigma_{2}$ from $1.2$ toward $L$ --- is exactly the loss-of-repulsion formulation of the law of forgetting. \section{Honest boundaries} \label{sec:honest} (1) The odd spectrum is unused (hole at $[99.58,110.17]$; re-extraction required). (2) The first $\Delta_{3}$ implementation was biased (underestimating GOE by $45\%$, Poisson by $5\%$); with the standard integral definition recomputed, $\Delta_{3}$ still sits $30$--$80\%$ above GOE theory --- the residual is not fully explained and is reported as such; $\Sigma_{2}$ is the primary criterion (validated unbiased on synthetic data at the same size). (3) $L\in[0,30]$ is the reliable window. (4) Finite size compresses both ways (equispaced theory $\beta=0$ measures $0.028$; GOE/Poisson theory $\beta=1$ measures $0.76$--$0.82$), so the $27\times$ separation of the recurrence criterion is a conservative lower bound. \appendix \section{Corrections applied in this paper} Each entry was found by the program's own checks; the master compendium records $69$ entries in total, of which the following touch this paper's material. \begin{center} \begin{tabular}{p{0.6cm}p{5.6cm}p{8.2cm}} \toprule \# & error as made & correction as used here\\ \midrule 2 & Maass $\kappa$ written as $1/2$ & $\kappa=0$; critical line $=\kappa+\tfrac12=\tfrac12$; equality with the holomorphic case is of critical lines\\ 41 & degeneracy taken as $Z(D)$, total reduced forms & $g(D)=\#\rho$-orbits $=h^{+}(D)$; $Z(D)$ overcounts by the orbit length (average $\approx 16\times$)\\ 42 & Pell unit $p+q\omega^{n}$ & unit is $p-q\bar\omega^{n}$; for $D\equiv1\,(4)$, $t=2p-q$; $8/8$ benchmarks pass after fix\\ 43 & two further degeneracy bugs (tight enumeration bound; adjacency by Zagier's $\rho$ mis-stated) & corrected; on $36$ fundamental discriminants $\#\rho$-orbits $=\operatorname{round}(h^{+}_{\mathrm{Dirichlet}})$, zero counterexample\\ 44 & prime geodesic theorem blocked by length-spectrum truncation & counting corrected (all discriminants, primitive forms, $\rho$-orbits): error exponent $1/2$ confirmed\\ 49 & saturation judged by max/mean ratio & criterion replaced by absolute upper-envelope decay $+$ $\chi^{2}$ baseline removal\\ 57 & valley search on heavy-tail means & naive branch (``any local minimum in window'') fired on noise (false positive $(997,G_{3},21)$: SNR $70\times$, minimum drifts with $p$, expected false hits $\approx2.0$, observed $1$, Poisson); upgraded criterion: depth $>3\sigma_{\mathrm{mean}}$ \emph{and} isolated; predictions written before the run so false positives are caught in the same round\\ \bottomrule \end{tabular} \end{center} \begin{thebibliography}{19} \bibitem{forgetting} C.~Zeng, \emph{The law of forgetting in moments of Sato--Tate measures: invariant tensors, maximal tori, and the rank law}, 2026. 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