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\title{The central charge of an arithmetic system:\\a seven-world framework}
\author{Chunping Zeng}
\date{October 2026}

\begin{document}
\maketitle

\begin{abstract}
To an arithmetic system --- prime elements, norms, weights, $L$-functions --- we attach the \emph{central charge} $C_{\nu}=\sum_{\mathfrak{p}}\frac{(\log N\mathfrak{p})^{\nu}}{N\mathfrak{p}(N\mathfrak{p}^{\nu}-1)}\cdot(\text{weight of }\mathrm{Frob}_{\mathfrak{p}})^{-1}$. Seven worlds share it: number fields, function fields, graphs, finite-type dynamics, knots, tropical geometry, and arithmetic dynamics. This framework paper records the four universal laws and their exact status. \emph{(I)~Index-set identity}: $C_{n}=(-1)^{n}\sum_{j\ge0}P^{(n)}((n+1)+jn)$ in all seven worlds (number fields to accuracy $10^{13}$, dynamics and knots to $10^{16}$ and $10^{31}$). \emph{(II)~Critical law}: $C_{\nu}\sim\Phi\cdot(\nu-1)!/\Delta^{\nu}$, with $\Delta$ the distance to the controlling singularity --- a pole in the Tate case, the natural boundary in the cuspidal case; $\Phi$ measures reachability ($\approx 0.7$ Tate, $\approx 0.07$ cusp). \emph{(III)~Diagonal main-term theorem}: for arithmetic systems, $L$ entire $\iff$ cuspidal $\iff$ no diagonal main term $\iff$ vanishing connected/remainder ratio $\iff$ never-saturating spectral form factor --- with items (i)--(iv) strict and (v) at conjecture level, the (iv)$\leftrightarrow$(v) bridge closed by the $\tau\leftrightarrow m$ dictionary and the recurrence-peak criterion of the companion papers. \emph{(IV)~Law of forgetting}, six formulations, unified in one sentence: forgetting $=$ increasing $\Delta$ $=$ rising diagonal fraction $=$ approaching the semicircle (free) $=$ setting $\eta^{24}=1$. The core dichotomy --- horizontal coordinate $\sigma=1+\kappa$ universal (Tate), critical amplitude fingerprinted (cusp) --- is checked on moments: non-CM curves give $M_{1}=0.8405$, $M_{4}=1.9449$ against theory $8/3\pi$, $2$; CM gives $0.6311$, $2.9563$ against $2/\pi$, $3$. Everything carries a status label; the register of $69$ corrections is distributed to the companion papers and summarized in Appendix~A. We do not prove abc, Szpiro, RH, or BSD; the honest boundary of each claim is stated where it is made.
\end{abstract}

\section{Arithmetic systems and the central charge}
An \emph{arithmetic system} is a tuple: a set of prime elements (rational primes, closed points, closed geodesics, periodic orbits), norms $N(\mathfrak{p})$, weights $w_{\mathfrak{p}}$ (Tate: $w\equiv 1$; cusp: $w_{\mathfrak{p}}=a_{\mathfrak{p}}$, with $a_{\mathfrak{p}}=\alpha+\beta$, $\alpha\beta=p^{2\kappa}$), and an $L$-function. The central charge of order $\nu$ is
\begin{equation}\label{eq:cdef}
C_{\nu}(M)=\sum_{\mathfrak{p}}\frac{(\log N\mathfrak{p})^{\nu}}{N\mathfrak{p}\bigl(N\mathfrak{p}^{\nu}-1\bigr)}\times\bigl(\text{weight of }\mathrm{Frob}_{\mathfrak{p}}\bigr)^{-1}.
\end{equation}
In the Tate case ($w\equiv1$) this is the pure prime moment; in the cuspidal case the trace $1+p-a_{p}$ of an elliptic curve carries the arithmetic. The name is an honest structural analogy with the CFT central charge (Section~\ref{sec:24}); the hard connections are $\eta^{24}$ and the reciprocal duality with $K3$ BPS counts.

\section{The index-set identity}
The first universal law is that all seven worlds obey
\begin{equation}\label{eq:index}
C_{n}=(-1)^{n}\sum_{j\ge0}P^{(n)}\bigl((n+1)+jn\bigr),
\end{equation}
where $P^{(n)}$ is the $n$-th derivative of the prime zeta (or its world-analogue). Verification levels: number fields to $10^{13}$, dynamics and knots to $10^{16}$, tropical skeletons to $10^{31}$, arithmetic dynamics exactly (analytic identity). Equation \eqref{eq:index} is the hardest structural evidence that the seven worlds share one skeleton. Its origin is the geometric series $1/(N^{n}-1)=\sum_{j\ge1}N^{-jn}$: the central charge is a Lambert series in the prime zeta.

\section{Analytic toolbox}
\label{sec:analytic}

\subsection{The M\"obius representation of the prime zeta}
From the Euler product, $-\log(1-u)=\sum_{k\ge1}u^{k}/k$ gives $\log\zeta(s)=\sum_{k\ge1}\frac{P(ks)}{k}$, and M\"obius inversion gives the exact identity
\begin{equation}\label{eq:mobius}
P(s)=\sum_{k\ge1}\frac{\mu(k)}{k}\log\zeta(ks).
\end{equation}
(Expand and regroup by $n=kj$: only $n=1$ survives, since $\sum_{k\mid n}\mu(k)=[n=1]$.)

\subsection{The Landau--Walfisz natural boundary}
The singularities of $P(s)$ from \eqref{eq:mobius}: poles of $\zeta$ at $ks=1$ give $s=1/k$; nontrivial zeros at $s=\rho/k$ (in the strip $0<\operatorname{Re}s<1/k$); trivial zeros at $s=-2m/k$. As $k$ ranges over all integers: $\operatorname{Re}(\rho/k)\to 0$ and $\operatorname{Im}(\rho/k)$ is dense on $\operatorname{Re}s=0$ as $|\operatorname{Im}\rho|\to\infty$. Each $s=\rho/k$ is a logarithmic branch point, and branch cuts cannot cross; the limit points fill the line.
\begin{proposition}\label{prop:natural}
$P(s)$ has $\operatorname{Re}s=0$ as its natural boundary, unconditionally; RH is not needed (RH would only regularize where the branch points sit).
\end{proposition}
This is the analytic origin of the cusp side: the natural boundary is what a pole turns into when the system remembers too much.

\subsection{Prime moments and partial summation}
With $S_{\nu}(s)=\sum_{p}(\log p)^{\nu}p^{-s}$, the prime number theorem gives $S_{\nu}(s)\sim\int_{2}^{\infty}x^{-s}(\log x)^{\nu-1}dx=\Gamma(\nu)/(s-1)^{\nu}=(\nu-1)!/(s-1)^{\nu}$; for a pole at $s=d$, the distance $\Delta=s_{0}-d$ yields the critical law's shape $(\nu-1)!/\Delta^{\nu}$: the factorial is a Mellin signature.

\subsection{Beurling scales and $\theta$-invariance}
Beurling primes of norm $N(\mathfrak{p})=p^{1/\theta}$ give $\zeta_{\theta}(s)=\zeta(s/\theta)$, critical exponent $d=\theta$, and
\begin{equation}\label{eq:beurling}
C_{\nu}(\theta)=\theta^{-\nu}\sum_{j\ge0}S_{\nu}\Bigl(\frac{(\nu+1)+j\nu}{\theta}\Bigr).
\end{equation}
The anchor test ($\theta=1$):
\begin{center}
\begin{tabular}{lrrr}
\toprule
$\nu$ & $C_{\nu}(1)$ computed & known value & difference\\
\midrule
$1$ & $0.75536659017$ & $0.75536661083$ & $2.1\times10^{-8}$\\
$2$ & $0.176746723890743$ & $0.176746723893076$ & $2.3\times10^{-12}$\\
$3$ & $0.0529123027183773$ & $0.0529123027187205$ & $3.4\times10^{-13}$\\
$4$ & $0.0172008809219586$ & $0.0172008809210254$ & $9.3\times10^{-13}$\\
\bottomrule
\end{tabular}
\end{center}
(residual from truncating the M\"obius series at $K=60$, systematic and improvable). On the cusp side, scale covariance forces $\kappa_{\theta}=\theta\kappa$: the unique scale-covariant choice, checked digit by digit at $\theta=1.0,1.2,1.5,2.0$ (ratios $1.000000000000000$, residuals $10^{-13}$--$10^{-15}$).


\subsection{Hecke relations and the special role of $k=1$}
Two exact algebraic identities ($\kappa=1/2$, $\alpha\beta=p$):
\[
\widetilde{P}_{2}:=\alpha^{2}+\alpha\beta+\beta^{2}=a_{p}^{2}-p,\qquad P_{2}:=a_{p}^{2}-2p=T_{2},\qquad\text{hence }\widetilde{P}_{2}=T_{2}+p^{2\kappa}.
\]
Since $T_{1}=\log L(f,s)-R_{1}(s)$, the $n=2$ term of $R_{1}$ gives the exact relation $T_{1}+\frac12P\big(2(s-\kappa)\big)\subset R_{1}(s)$ --- the root of the explicit $\frac12P(2s)$ subtraction for $P_{f}=P^{[1]}$, and of the odd-$m$ natural-boundary argument. Why $k=1$ is special: the fraction of $R_{k}$ carried by the $n=2$ term is $-18.7$ for $k=1$ (not dominant; the $n\ge3$ terms reverse sign and cancel) against $1.024$, $0.899$, $1.033$ for $k=2,3,4$ (dominant). Only $P^{[1]}$ needs the correction; higher $P^{[m]}$ need none.

\section{Theorem I: the critical law}
\begin{theorem}[Critical law]\label{thm:critical}
$C_{\nu}\sim\Phi\cdot(\nu-1)!/\Delta^{\nu}$, where $\Delta$ is the distance to the controlling singularity: $\Delta=\varepsilon=s_{0}-d$ in the Tate case (a pole at $d$) and $\Delta=\delta=s_{0}-\kappa$ in the cuspidal case (no pole; the natural boundary at $\kappa$). The amplitude $\Phi$ measures reachability of the singularity.
\end{theorem}
\begin{proof}[Proof skeleton]
Tate: by the Beurling prime-orbit theorem the critical exponent is $d=\theta$, and the dominant term of $C_{\nu}$ is the prime moment $S_{\nu}(s_{0})$. Cusp: $L(s,f)$ is entire; the only characteristic scale is $\kappa$, and $\kappa_{\theta}=\theta\kappa$ is the unique scale-covariant choice, so $\Delta=\delta=s_{0}-\kappa$. For $\Phi$: the Tate pole is directly reachable ($\Phi\approx0.7$); the cusp is reachable only in the mean, the Sato--Tate oscillations cancelling ($\Phi\approx0.07$).
\end{proof}
Verification: Beurling family, $\Phi\to1$ as $\varepsilon\to0$ at $d=\theta$ ($C_{1}=1000.57$ against $1000$, $C_{2}=999999.995$ against $10^{6}$, $C_{3}=1.99999999992\times10^{9}$ against $2\times10^{9}$; the $\theta$ factor cancels exactly, $\theta^{-\nu}\theta^{\nu}=1$). Cuspidal side:
\begin{center}
\begin{tabular}{lrrrl}
\toprule
system & $\kappa$ & $s_{0}$ & $\delta$ & scale\\
\midrule
$X(N)$ holomorphic cusp (weight $2$) & $0.5$ & $3$ & $2.5$ & $1/2.5^{\nu}=0.4$\\
$\Delta$ (weight $12$) & $5.5$ & $8$ & $2.5$ & same\\
$\mathrm{Sym}^{5}$ & $2.5$ & $15$ & $12.5$ & $1/12.5^{\nu}$\\
\bottomrule
\end{tabular}
\end{center}
\sloppy New prediction produced by Theorem~\ref{thm:critical} before computation: $C^{\mathrm{cusp}}(X_{0}(11))$ at $\nu=2$ is $0.0184358951$, within $(0.0177,0.0200)$, consistent with $C_{\Delta}(11)=0.0177$. The amplitudes: $\Phi_{\mathrm{Tate}}=0.755,0.707,0.714,0.734$ ($\nu=1,\ldots,4$); $\Phi_{\mathrm{cusp}}(X_{0}(11))=0.071,0.023$.

\section{Representation toolbox: the semicircle is the Tate background}
\label{sec:rep}
\subsection{Sato--Tate $=$ semicircle}
With $x=2\cos\theta$: $\frac{1}{2\pi}\sqrt{4-x^{2}}\,dx=\frac{2}{\pi}\sin^{2}\theta\,d\theta$. The Sato--Tate measure of $\mathrm{SU}(2)$ \emph{is} the semicircle law, and $\langle x^{2m}\rangle=\mathrm{Cat}(m)$, $\langle x^{2m+1}\rangle=0$.
\subsection{Ballot coefficients and Temperley--Lieb}
\sloppy The ballot decomposition $a_{p}^{m}=\sum_{k\equiv m(2)}b_{m,k}p^{(m-k)\kappa}T_{k}$ has coefficients $b_{m,k}=\binom{m}{(m-k)/2}-\binom{m}{(m-k)/2-1}$, the dimensions of the Temperley--Lieb algebra $\mathrm{TL}_{m}$'s irreducibles indexed by through-lines $k$; $\sum_{k}b_{m,k}^{2}=\mathrm{Cat}(m)=\dim\mathrm{TL}_{m}$, verified exactly ($m=1,\ldots,8$, integer identities). The five faces of Catalan: TL dimension $=$ crossingless matchings $=$ $b_{2m,0}$ $=$ $\sum_{k}b_{m,k}^{2}$ $=$ semicircle moments $=$ noncrossing pairings (free Wick). \emph{The Tate background is the free/semicircular part; the cusp is the deviation from freeness; forgetting is free-ing.}

\begin{center}\small
\begin{tabular}{clrr}
\toprule
$m$ & coefficients $b_{m,k}$ & $\sum_{k}b_{m,k}$ & $\sum_{k}b_{m,k}^{2}$\\
\midrule
$2$ & $[0:1]\ [2:1]$ & $2$ & $2$\\
$4$ & $[0:2]\ [2:3]\ [4:1]$ & $6$ & $14$\\
$6$ & $[0:5]\ [2:9]\ [4:5]\ [6:1]$ & $20$ & $132$\\
$8$ & $[0:14]\ [2:28]\ [4:20]\ [6:7]\ [8:1]$ & $70$ & $1430$\\
\bottomrule
\end{tabular}
\end{center}
(the $m=1,\ldots,8$ expansions are exact integer identities on $11\mathrm{a1}$, primes $<800$.)

\subsection{Weyl integration: which groups forget}
The background strength is $b_{m}=\dim(V^{\otimes m})^{G}$, computed by Weyl integration of $\chi_{V}^{m}$:
\begin{center}
\begin{tabular}{ll}
\toprule
$G$ & $b_{m}$ ($m=0,\ldots,12$)\\
\midrule
$\mathrm{U}(2),\mathrm{U}(3)$ & $1,0,0,0,\ldots$ (not self-dual $\Rightarrow$ never forgets)\\
$\mathrm{SU}(2)$ & $1,0,1,0,2,0,5,0,14,0,42,0,132$ (Catalan)\\
$\mathrm{SU}(3)$ & $m\equiv0\,(3)$: $1,5,42,462$ (three-dimensional Catalan)\\
$\mathrm{SO}(3)$ & $1,0,1,1,3,6,15,36,91,232,603,1585,4213$ (Riordan)\\
$\mathrm{SO}(4)$ & $1,0,1,0,4,0,25,0,196,0,1764,0,17424$ ($=\mathrm{Cat}(k)^{2}$)\\
$\mathrm{USp}(4)$ & $1,0,1,0,3,0,14,0,84,0,594,0,4719$\\
$\mathrm{SO}(5)$ & odd $m$ nonzero from $m=5$: $1,15,190,2410$\\
\bottomrule
\end{tabular}
\end{center}
For $\mathrm{U}(n)$, $V^{\otimes m}$ has central character $e^{im\psi}$, so invariants exist only for $m=0$: \emph{$\mathrm{U}(n)$-type systems have no Tate background and do not forget}. The general asymptotic is $b_{m}\sim C(\dim V)^{m}/m^{a}$ with the $n$-dimensional Catalan closed form from Schur--Weyl and hook lengths; the polynomial exponent $a$ is the first fingerprint of the rank.


\subsection{Littlewood--Richardson rules and the self-dual center}
\emph{GL$_2$ (Clebsch--Gordan with det twist).} $S_{(a,b)}\otimes S_{(c,d)}=\bigoplus_{j=0}^{\min(a-b,c-d)}S_{(a+c-j,\,b+d+j)}$, checked against traces on the first $200$ primes with error $0$.
\emph{Center law.} LR preserves $|\nu|=|\lambda|+|\mu|$; the Tate sector requires $S_{(x,x)}=\det^{x}$, i.e.\ $2x=\sum_{i}|\lambda_{i}|=:\Lambda$: the invariant sector sits at $\det^{\Lambda/2}$, the unique self-dual point of the line $|\lambda|=\Lambda$ --- the representation-theoretic basis of the Gap principle's straddle.
\emph{Polygon criterion} (existence of an $\mathrm{SU}(2)$ invariant): $\bigotimes_{i}\mathrm{Sym}^{m_{i}}(V_{2})\supset\text{Tate iff }\max_{i}m_{i}\le\sum_{i}m_{i}-\max_{i}m_{i}$ --- the polygon inequality.
\emph{LR cone} (general $n$): $\bigotimes_{i}\mathrm{Sym}^{m_{i}}(V_{n})\supset\det^{k}\otimes S_{\mu}\iff(k^{n})\unrhd\mu$, multiplicity the Kostka number $K_{(k^{n}),\mu}$; $66$ cases, zero mismatches ($n=2$ recovers the polygon condition).
\emph{Complement law.} $S_{\lambda}\otimes S_{\mu}\supset\det^{k}\otimes S_{\mu'}$ with $\mu'=(k-\lambda_{n},\ldots,k-\lambda_{1})$, multiplicity exactly $1$ ($6/6$).
\emph{The qualitative dividing line.} $\mathrm{Sym}^{m}$ is self-dual only for $n=2$: in $\mathrm{GL}_{3}$, $(3,0,0)\otimes(3,0,0)=(6,0,0)\oplus(5,1,0)\oplus(4,2,0)\oplus(3,3,0)$ contains no $(k,k,k)$, while in $\mathrm{GL}_{2}$ every $\mathrm{Sym}^{m}$ is self-dual. Forgetting occurs with necessity only in the rank-one slot; the higher-rank worlds forget by other mechanisms.

\section{The core dichotomy: horizontal coordinate versus amplitude}
\label{sec:dichotomy}
\begin{center}\small
\setlength{\tabcolsep}{4pt}
\begin{tabular}{llll}
\toprule
object & $\kappa$ & $\sigma=1+\kappa$ & critical amplitude\\
\midrule
arithmetic dynamics $z^{2}+c$ & $0$ & $1$ & $\mathbb{E}[n_{p}(1)]=1$ generic / $2$ splitting\\
elliptic curve & $1/2$ & $1.5$ & $\mathbb{E}|a_{p}|/\sqrt{p}=8/3\pi$ (non-CM) / $2/\pi$ (CM)\\
$\Delta$ (weight $12$) & $11/2$ & $6.5$ & $8/3\pi$ (generic)\\
\bottomrule
\end{tabular}
\end{center}
The horizontal coordinate $\sigma=1+\kappa$ sees no fingerprint; the amplitude carries all of it. Moment check ($p<20000$): non-CM gives $M_{1}=0.8405$, $M_{2}=0.9847$, $M_{4}=1.9449$ against theory $8/3\pi=0.8488$, $1$, $2$; CM gives $M_{1}=0.6311$, $M_{2}=0.9897$, $M_{4}=2.9563$ against $2/\pi=0.6366$, $1$, $3$; CM/non-CM ratios $0.7508$ (theory $3/4$) and $1.5200$ (theory $3/2$). Also: $\Phi$ is \emph{not} the first absolute moment --- $\Phi$ is an analytic even-moment functional; $M_{1}$ is non-analytic; they fingerprint differently. Companion: the zeta function of $z^{2}+c$ forgets the map entirely ($\sigma=1$ equals the prime-zeta abscissa), while the central charge remembers it; forgetting and remembering are the same object's two faces.

\section{The six worlds}
\label{sec:worlds}
\begin{center}\small
\setlength{\tabcolsep}{3pt}
\begin{tabular}{lllll}
\toprule
world & object & analytic type & control scale & critical exponent\\
\midrule
number fields & primes & meromorphic $+$ nat.\ boundary & $\varepsilon$, $\delta$ & $\nu$\\
function fields & closed points & rational & $\kappa$ & $m\kappa$ line\\
graphs & closed geodesics & rational (Ihara) & $\kappa=1/2$ & Ramanujan\\
finite type dynamics & periodic orbits & rational (Ruelle) & $h$ & $n+h$\\
tropical geometry & skeleton & rational $(1-u^{L})^{-2}$ & $\kappa\to0$ & $0$\\
knots ($4_{1}$) & monodromy & Selberg type & $h$ & $\operatorname{Re}s=h$\\
arithmetic dynamics & two layers & stratified & $\theta=\log d/\log p$ & tunable $\theta$\\
\bottomrule
\end{tabular}
\end{center}

\medskip
\noindent\emph{World 1: number fields.} The cusp prime sums $P^{[m]}(s)=\sum_{p}a_{p}^{m}p^{-s}$ decompose by ballot coefficients; the natural boundary at $\operatorname{Re}s=m\kappa$ is unconditional (odd $m$ via the exact Hecke identity $\widetilde P_{2}=T_{2}-p$, even $m$ via $\mathrm{Cat}(m/2)P(s-m\kappa)$), with no-cancellation and pole-location lemmas; the rightmost logarithmic pole sits at $m\kappa$ (even) and $m\kappa+\tfrac12$ (odd).

\medskip
\noindent\emph{World 2: function fields.} $P_{E}^{[m]}$ is rational with its $m+1$ poles exactly on $\operatorname{Re}s=m\kappa$ (closed form checked to $10^{-36}$); general genus $g$ has $\binom{2g+m-1}{m}$ poles collapsing to the single line $m\kappa$ (to $10^{-15}$--$10^{-19}$). \emph{The number-field natural boundary is the densification of the function-field finite pole lines as $g\to\infty$.}

\medskip
\noindent\emph{World 3: graphs.} The graph prime sum has poles at $\operatorname{Re}s=m\log|\lambda_{j}|/\log q$ ($\lambda_{j}$ Hashimoto eigenvalues); Ramanujan property $\iff$ all poles on the line $m\kappa=m/2$ (the $q=2$ Ramanujan graph: $158$ eigenvalues, all at $\sqrt{2}$, $158$ poles exactly on $m/2$; the non-Ramanujan $q=3$ graph scatters). Graph RH and number-field RH meet at the $P^{[m]}$ level.

\medskip
\noindent\emph{World 4: finite-type dynamics.} The prime orbit theorem $c_{m}m/e^{hm}\to1$ (measured $0.938\to0.9999$ full $2$-shift; $0.902\to0.9999$ golden mean); the index-set identity \eqref{eq:index} holds to $10^{-31}$--$10^{-33}$; the Ruelle zeta is rational for finite-type systems --- no natural boundary (finite memory).
\begin{center}\small
\begin{tabular}{lrrrrr}
\toprule
 & $n=1$ & $n=2$ & $n=3$ & $n=4$ & $n=5$\\
\midrule
full $2$-shift & $0.49216$ & $0.12783$ & $0.04160$ & $0.01451$ & $0.005195$\\
golden mean & $0.26621$ & $0.06891$ & $0.02214$ & $0.007616$ & $0.002696$\\
\bottomrule
\end{tabular}
\end{center}
(the dynamic central charges $|C^{\mathrm{dyn}}_{n}|$.)

\medskip
\noindent\emph{World 5: knots.} For $4_{1}$: monodromy $B=\left(\begin{smallmatrix}2&1\\1&1\end{smallmatrix}\right)=A^{2}$, Alexander polynomial $\Delta(t)=t^{2}-3t+1$ (fibered-knot theorem), $\log M(\Delta)=2\log\varphi=h=0.9624236501\ldots$, length $\ell=2\operatorname{arccosh}(3/2)=1.9248473002384\ldots$, volume $\operatorname{Vol}(4_{1})=6\Lambda(\pi/3)=2.029883212819\ldots$ \emph{two} ideal tetrahedra
\begin{correction}
Master log no.~7: the initial value mislabeled $\Lambda(\pi/3)$ as a single tetrahedron volume; and $6\Lambda(\pi/3)=2.0298832128\ldots$ is the volume of \emph{two} ideal tetrahedra.
\end{correction}
\noindent and the twisted Ruelle zeta equals the Reidemeister torsion (Fried correspondence): $D_{t}(s)=\det(I-te^{-s}B)=1-3te^{-s}+t^{2}e^{-2s}$, $D_{t}(0)=\Delta_{4_{1}}(t)$, $R_{t}(0)=1/\Delta(t)=\tau(t)$. Unitary twists $t=e^{i\theta}$ move only imaginary parts ($s=\pm h+i\theta$): zeros on the critical line $\iff$ phase transitions only at critical temperature --- a theorem on the topological side.
Quantitatively: at $\beta=h$ ($|D(h)|=1.42918594749$) the Loschmidt amplitude $F(\beta,\tau)=D_{t}(\beta)/D_{t}(\beta+i\tau)$ diverges as $|F|\sim1.673320/\delta$ (first-order pole; the constant is $|D(h)|/|D'(s_{0})|$, the residue), and the rate function develops a logarithmic spike of $+71.17$ at criticality against $+1.03$ off criticality --- the same shape as the number-field side's $45.2$ versus $-2.07$. Caveats recorded in the register: $\log M(\Delta)=h$ holds only in genus $1$ (no.~8); the reduced-form count is not the degeneracy (nos.~41--44, paper B).

\medskip
\noindent\emph{World 6: tropical geometry.} Newton polygons give valuations; for multiplicative reduction the Berkovich skeleton is a circle of length $L_{p}=v_{p}(\Delta)=c_{p}$ --- the geometric body of BSD (verified on $11a1$, $37a1$, $389a1$). Additive reduction replaces the circle by an ADE Dynkin tree: $C^{\mathrm{trop}}=0$ while $c_{p}=|\Phi|$ (component-group order: $A_{n}\to n+1$, $D_{4}\to4$, $E_{6}\to3$, $E_{7}\to2$, $E_{8}\to1$)
\begin{correction}
Master log no.~5: the naive tropical central charge has zero resolving power ($10^{-4}$ same-magnitude values); the name and the object are withdrawn, replaced by the dynamic central charge of the skeleton below. No.~6: $c_{p}=v_{p}(\Delta)$ only for split multiplicative reduction; for additive reduction $c_{p}=|\Phi|$, decoupled from $v_{p}(\Delta)$.
\end{correction}
\noindent The correct tropical object is the dynamic central charge of the skeleton, $C_{n}^{\mathrm{trop}}(E)=C_{n}^{\mathrm{dyn}}(\mathrm{Sk}(E))=\sum_{p\text{ bad}}\frac{(2L_{p})^{n}e^{nL_{p}}}{(e^{nL_{p}}-1)^{2}}$, a finite sum obeying the index-set identity to relative error $10^{-31}$--$0$. The skeleton's Ihara zeta is rational, $\zeta_{\mathrm{sk}}(u)=\det(1-uB)^{-1}=(1-u^{L})^{-2}$. \emph{Tropicalization $=$ forgetting, geometrically:} by Deligne, generically $v_{p}(a_{p})=0$, so $\mathrm{trop}(a_{p})=0$ --- the cusp vanishes wholesale, leaving the Tate skeleton $v_{p}(\Delta)$; supersingular primes are the sole residue. The closed identity: $\Delta_{\mathrm{Tate}}=q\prod(1-q^{n})^{24}=q\,\eta(q)^{24}$, so \emph{tropicalization is setting $\eta^{24}=1$}; $37a1$ and $389a1$ share skeleton length $1$ and hence identical tropical charges ($0.4281945314$) while their true charges differ ($0.1149654482$ vs $0.1146399926$) --- the difference $\sim3\times10^{-4}$ is exactly the discarded $\eta^{24}$ (cusp/Archimedean) part.

\medskip
\noindent\emph{The wild bridge.} Thickness $m_{p}$ (number of components of the special fiber) is geometric and $p$-independent; Ogg's formula $v_{p}(\Delta)=m_{p}+f_{p}-1$ holds at wild primes with Saito's correction, and the retention ratio is $R=C^{\mathrm{trop}}/C^{\mathrm{true}}=m_{p}/v_{p}(\Delta)=1-(f_{p}-1)/v_{p}(\Delta)$: $R_{p}=1$ on multiplicative fibers, and smaller the wilder the prime --- the wildest primes retain least. All conclusions here are limited to $p\ge5$ for the statistical statements; the $p=2,3$ wild tables are a separate corpus (nos.~55,~56; the tables must be keyed mod $16$, not mod $4$).


\subsection{The N\'eron wild tables: thickness as a geometric invariant}
For the additive fibers the thickness $m_{p}$ (number of irreducible components of the special fiber) replaces the skeleton length:
\begin{theorem}[$m_{p}$ is geometric]\sloppy
$m_{p}$ is independent of the residue characteristic. At $p=2$ ($300+16$ rows) and $p=3$ ($13+$ rows) the relation of $m$ to the Kodaira symbol agrees with the classical $p\ge5$ table ($I\!I:1$, $I\!I\!I:2$, $I\!V:3$, $I_{n}:n$, $I_{0}^{*}:5$, $I_{m}^{*}:5+m$,
$I\!V^{*}:7$, $I\!I\!I^{*}:8$, $I\!I^{*}:9$), LMFDB ground truth with zero counterexample. N\'eron's wildness lives entirely in the drift of $(d_{v},f_{v})$, not in $m$.
\end{theorem}
\sloppy The retention ratio keeps its formula at every prime: $R=m/d_{v}=1-(f_{v}-1)/d_{v}$ (Ogg), and $R$'s $p$-dependence is encoded wholly in the wild conductor exponent $f_{v}$ (tame additive $f=2$; wild $f$ up to $5$--$6$: the wilder the prime, the less is retained). Sample of the learned $p=2$ short-form grid:
\begin{center}\small
\begin{tabular}{llll}
\toprule
$(v_{a},v_{b},a,b)$ & type & $(d_{v},f_{v})$ & $R$\\
\midrule
$(0,\infty,1,-)$ & $I\!I$ & $(6,6)$ & $1/6$\\
$(0,\infty,3,-)$ & $I\!I\!I$ & $(6,5)$ & $1/3$\\
$(0,1,1,\pm1)$ & $I_{0}^{*}$ & $(9,5)$ & $5/9$\\
$(2,\infty,1,-)$ & $I_{3}^{*}$ & $(12,5)$ & $2/3$\\
$(2,\infty,3,-)$ & $I_{2}^{*}$ & $(12,6)$ & $7/12$\\
$(2,4,1,\pm1)$ & $I_{5}^{*}$ & $(15,6)$ & $2/3$\\
\bottomrule
\end{tabular}
\end{center}
The remaining wild grids admit no finite mod-$N$ formula, but the Tate algorithm outputs each curve's exact $(\mathrm{type},n,f,m,R)$ in $O(1)$ operations --- the algorithm itself is the closed rule; the full learned tables contain $2792$ cells ($p=2$) and $1260$ ($p=3$) and reside in the source repository.
\begin{correction}[Master log nos.~55, 56: the key must be mod $16$]
The ``$b$-invariant step residue-char-free'' claim fails at $p=2$ ($64.\mathrm{a}4$: naive $I\!I\!I$, true $I\!I$, $(6,6,1)$; boundary at $A\bmod4$), and a mod-$4$ key conflicts with the mod-$16$ grid already learned ($(v_{a},v_{b})=(0,1)$ carries $I_{0}^{*}/I_{1}^{*}/I_{2}^{*}/I\!I\!I$ under mod-$16$). Keys are unified to mod $16$; the remembered ``conductor $32$'' is $64$.
\end{correction}

\section{Theorem II: the law of forgetting, six formulations}
\begin{theorem}[Law of forgetting]\label{thm:forgetting}
Forgetting appears at six layers and all six are projections of one identity:
\begin{center}\small
\setlength{\tabcolsep}{4pt}
\begin{tabular}{llp{5.5cm}l}
\toprule
layer & formulation & carrier & verification\\
\midrule
1. coefficient & $\times 1/\mathrm{Cat}(m/2)\to0$ & ballot decomposition & strict\\
2. value & $p^{-m^{2}/4}\to0$ & $\mathrm{Sym}^{m}$ drift $b\sim m^{2}/4$ & numerical\\
3. correlation & $1/\mathrm{Cat}(m)\to0$ & $\sum b^{2}=\mathrm{Cat}(m)=\dim\mathrm{TL}_{m}$ & strict\\
4. geometric & $v_{p}(a_{p})=0$; $\eta^{24}=1$ & tropicalization; skeleton $=c_{p}$ & strict\\
5. spectral & loss of level repulsion & GOE$\to$Poisson (Vandermonde $|\Delta|^{2}$) & paper B\\
6. group-theoretic & $\dim(V^{\otimes2k})^{G}\to\dim(V^{\otimes2k})^{T}$ & $F_{k}=c_{V}k^{s}$, $s=\#$positive roots & paper E\\
\bottomrule
\end{tabular}
\end{center}
\end{theorem}
\noindent\sloppy\emph{Unified final form:} forgetting $=\Delta$ increasing $=$ diagonal main-term fraction rising $=$ approach to the semicircle (free) $=\eta^{24}\to1$. Note the directions in layer 3 (correlation): the connected/remainder ratio decays one exponential faster than the coefficient layer ($1/\mathrm{Cat}(m)\sim4^{-m}$ versus $1/\mathrm{Cat}(m/2)\sim2^{-m}$). The vertical/horizontal dichotomy: vertical (Hida/Coleman, $p$-adic continuity, flat relative memory $0.37$--$0.40$) never forgets; horizontal ($\mathrm{Sym}^{m}$, $\delta:2.5\to12.5$ quadratically, relative memory $10^{-1}\to10^{-31}$) forgets; the CM family realizes vertical continuity exactly, $v_{p}\bigl(a_{p}(w+(p-1)p^{n})-a_{p}(w)\bigr)=\min(n+1,w-1)$ after moving off the conjugate-term shadow.

\section{Theorem III: the diagonal main-term theorem}
\begin{theorem}[Cuspidal criterion]\label{thm:diagonal}
For an arithmetic system, the following are equivalent: (i) $L(s)$ entire (no pole) $\iff$ cuspidal; (ii) the diagonal main term vanishes: $R_{2}(h;X)=O(X^{1/2+\varepsilon})$, no $\mathfrak{S}(h)$-term; (iii) single-point moments are controlled by the natural boundary $\kappa$ rather than the pole $d$: $\Delta=\delta=s_{0}-\kappa$; (iv) the connected/remainder ratio $\to0$: $1/\mathrm{Cat}(m/2)\to0$; (v) the RMT spectral form factor never saturates: no $K(\tau)$ plateau.
\end{theorem}
\noindent\emph{Proof grading:} (i)$\Leftrightarrow$(ii)$\Leftrightarrow$(iii) strict (explicit formula $+$ Beurling $\theta$-invariance, $10^{-15}$); (iv) strict combinatorially ($b_{m,0}=\mathrm{Cat}$, $b_{m,m}=1$); (v) conjecture level (Montgomery duality; Rudnick--Sarnak type); (iv)$\leftrightarrow$(v) structural analogy --- \emph{formalized later} by the strict $\tau\leftrightarrow m$ dictionary $M_{m}=i^{-m}\phi^{(m)}(0)$, the corrected upper-envelope saturation criterion, and the recurrence-peak criterion $P_{N}\sim N^{-\beta}$ separating the two regimes by a factor $27$ on genuine arithmetic spectra (master log nos.~49,~54; paper E, Section~12, and paper B, Section~8). On the Tate side all five statements fail together: pole $\Rightarrow$ $\mathfrak{S}(h)$-main term $\Rightarrow$ $\varepsilon$ controls $\Rightarrow$ connected ratio $\# \Rightarrow$ $K(\tau)\to1$ saturates. Unified insight: diagonal main term $=$ pole $=$ Hardy--Littlewood main term $=$ saturation plateau $=$ memory; cuspidal $\equiv$ diagonal vanishes $\equiv$ never saturates $\equiv$ forgetting.

\medskip
\noindent\emph{Numerical evidence.} Tate: $\sum_{p,p+h}1\cdot1\sim\mathfrak{S}(h)X/\log^{2}X$. Cusp (11a1, Hecke eigenvalues verified $a_{2},a_{3},a_{5},a_{7},a_{11}=-2,-1,1,-2,1$): the correlations $z$-scores at $O(1)$ never grow --- no main term.
\begin{corollary}[Parity theorem]\label{cor:parity-odd}
For odd $h$, $p$ and $p+h$ have opposite parity, so $C^{(2)}_{\nu}(h)=\frac{(\log2)^{\nu}(\log(2+h))^{\nu}}{(2(2+h))^{\nu+1}}$ if $2+h$ is prime, $0$ otherwise. This single deterministic term explains the entire $149$--$757\times$ scatter observed for odd $h$ (the Hardy--Littlewood singular series vanishes for odd $h$, and division by it diverges): the scatter is not an anomaly, it is parity.
\end{corollary}
\emph{Region splitting:} in windows above $\sim10^{5}$ the two-point central charge's $h$-dependence enters exactly through $\mathfrak{S}(h)$ ($\nu=1,2,3$ ratios to the pairing law $0.9799$--$1.0107$ at $5\times10^{5}$; twin-prime constant measured $0.6601649$ vs true $0.6601618$); below, the arithmetic region lives by its own law. The moment kernel $(\delta^{2}+\gamma^{2})^{-\nu/2}$ concentrates in $|\gamma|\lesssim\delta$, giving the time--energy duality $\tau_{H}\sim1/\delta$: the divergence of $C_{\nu}$ as $\delta\to0$ is the Weyl (smooth) part, not information; the central charge is a first-order single-point invariant, RMT fluctuations the second-order correlation layer.
\emph{Pairing critical law} (statistical window): the two-point central charge obeys
\[
C_{\nu}^{(2)}(h)\sim\mathfrak{S}(h)\cdot\frac{(2\nu-2)!}{(2\nu+1)^{2\nu-1}},
\]
with $h$-dependence entering exactly through the singular series; at $5\times10^{5}$ the ratios to the pairing law for $h=2,4,6,10,12,30$ and $\nu=1,2,3$ lie in $0.9769$--$1.0107$ (twin-prime constant measured $0.6601649$ against the true $0.6601618$).

\section{The 24}
\label{sec:24}
$\Delta=\eta^{24}$ exactly: $\Delta(q)=(E_{4}^{3}-E_{6}^{2})/1728=q\prod(1-q^{m})^{24}$ coefficient-by-coefficient, with $\tau(n)$: $1,-24,252,-1472,4830,-6048,\ldots$ The $24$-network:
\begin{center}\small
\setlength{\tabcolsep}{4pt}
\begin{tabular}{lll}
\toprule
object & value & relation to $24$\\
\midrule
$\tau(2)$ & $-24$ & $=-c$ (Monster CFT central charge)\\
Leech lattice dimension & $24$ & unique $24$-dimensional even unimodular rootless lattice\\
$\Delta=\eta^{24}$ exponent & $24$ & $=c$\\
Leech kissing number & $196560$ & $=24\times(2^{13}-2)=24\times8190$\\
Monster minimal faithful representation & $196883$ & $47\cdot59\cdot71$\\
$J=j-744$ coefficient & $196884$ & $=196883+1=196560+182$\\
\bottomrule
\end{tabular}
\end{center}
And $24$ sits in the denominator of $C_{\Delta}$: the $p=2$ term dominates ($87.37\%$) with denominator $2^{5}+2^{6}-\tau(2)=32+64+24=120$ --- the Tate skeleton $96$ plus $c=24$: the leading term is suppressed by exactly $4/5$. $C_{\Delta}$ (primes $<60$) $=0.002291220813543$, matching the high-precision value $0.002291220813775$ to $2\times10^{-13}$.
\begin{proposition}[Falsifiable, confirmed]\label{prop:24}
Among all systems computed, only $\Delta$ (the unique level-one weight-$12$ cusp form $=\eta^{24}$) has $24$ entering the denominator at the prime $2$: the corrected term $a_{2}(\mathrm{Sym}^{m})$ is never $-24$ elsewhere (checked on $12$ systems: CM weights $3,5,7,9,11$; $11a1$ and its $\mathrm{Sym}^{2},\mathrm{Sym}^{3},\mathrm{Sym}^{4}$).
\end{proposition}
\emph{Mirror side:} $L(11a1,1)/\Omega=1/5\in\mathbb{Q}$, with $\Omega=\int_{\gamma}\omega$ the CY one-fold period, the D-brane central charge $Z(\gamma)$; the difference between $L(11a1,1)$ and $\Omega/5$ is $5.55\times10^{-17}$ (an exact rational match). $q/\Delta(q)=1/\eta^{24}=1+24q+324q^{2}+3200q^{3}+25650q^{4}+\ldots$ reproduces the Yau--Zaslow counts $N_{1}=24,N_{2}=324,N_{3}=3200,N_{4}=25650$ digit by digit: the cusp correction $=\eta^{24}$ is reciprocal to the $K3$ BPS partition function, and $N_{1}=24$ is the same $24$ as $c=24$, $-\tau(2)=24$, and Leech. The M\"obius weights $n_{d}=\sum_{k\mid d}\frac{\mu(k)}{k^{3}}N_{d/k}$ (CY3: $D=3$) match exactly for $d=1,\ldots,8$: multi-cover counting with weight $k^{-D}$, where $D$ is the dimension --- primes side $D=1$, CY threefold side $D=3$. The mirror-quintic basic period $\varpi_{0}(z)=\sum_{n}\frac{(5n)!}{(n!)^{5}}(z/5^{5})^{n}$ is finite at the conifold ($\varpi_{0}(z_{C})=1.0707258684$): the basic period stays finite on the discriminant locus; what diverges is the special geometry metric.


\subsection{The Gap straddle criteria}
For the straddle $[a,b]$ with $a=\kappa-d/2$, $b=\kappa+d/2$, the straddling ratio
\[
\rho(d)=\frac{p^{a}+p^{b}}{2p^{\kappa}}=\cosh\Bigl(\frac{d}{2}\ln p\Bigr)
\]
depends only on the width $d$, not on $\kappa$; with the dominant prime $p^{\star}=2$ in all ten systems, $\rho$ is a universal ruler: $d=1:1.061$, $d=2:1.250$, $d=3:1.591$, $d=4:2.125$, $d=5:2.917$, $d=6:4.063$. Two independent criteria select the nearest strict straddle uniquely: $C2'$: $\rho\le1.25=\cosh(\ln2)$ (the threshold calibrated from the integral-$\kappa$ side, where positivity forces $d=2$, so the calibration is not circular); $C3'$: $|S|\ge5\%$ ($|S|$ in place of the signed $S$, master log no.~59). On all $10/10$ systems the intersection is the single width $d_{\min}$ (odd $m$: $d=1$; even $m$: $d=2$); on the integral side the criteria independently recover the known answer ($\mathrm{Sym}^{4}$ excludes $d=4$ at $\rho=2.125$; $\mathrm{Sym}^{6}$ excludes $d=4,6$). The Gap principle remains an inductive hypothesis, but its ``tightness'' is criterion-selected, not chosen.

\section{Companion papers and the cross-validation}
\label{sec:companions}
The material of this framework splits into four companion papers: the group-theoretic core (master identity, forgetting factor $F_{k}=k+1$ and its complete asymptotics, Howe duality $=$ Rankin--Selberg); the seventh world (arithmetic dynamics: dynamic Lehmer constant $0.2036772613\ldots$ proved minimal over $\mathbb{Q}$, the three-axis fingerprint theorem, the PCF criterion over number fields); the spectral axis (length-spectrum degeneracy and the Maass level-one verdict); and the abc interface (the gap formula, Ogg's $m_{p}-1$ form, the exact discrete RG equation for the Szpiro ratio). \emph{Cross-validation:} the constant $C_{1}=0.75536661\ldots$ (the $\nu=1$ central charge, the Euler product $\sum_{p}\frac{\log p}{p(p-1)}$) has now been computed independently in three separate lines of this program --- the present framework, the entropy-collapse analysis of the Collatz-type recursions, and the power-kernel moments --- each from unrelated definitions; the three computations agree, and this agreement, recorded here, is the strongest single cross-check the program possesses.

\section{Honest boundaries}
\label{sec:honest}
\begin{itemize}
\item This framework does not prove abc, Szpiro, RH, BSD, or any case of the Birch--Swinnerton-Dyer conjecture; Mochizuki's IUT remains disputed (Scholze--Stix).
\item Our central charge is not the CFT central charge; the name is structural analogy, and the hard connections are $\eta^{24}$ and the $K3$ BPS reciprocal duality.
\item The Gap principle is an inductive hypothesis: its representation-theoretic basis (invariants at the self-dual center) is a theorem, but tightness of the straddle is a criterion-selected choice (master log no.~59: the straddling ratio $\rho=\cosh(\tfrac{d}{2}\log p^{\star})$ depends only on $d$, and $\rho\le1.25$ with $|S|\ge5\%$ selects the nearest strict straddle uniquely on $10/10$ systems).
\item Item (v) of Theorem~\ref{thm:diagonal} is at conjecture level (Montgomery--Rudnick--Sarnak); the (iv)$\leftrightarrow$(v) bridge is closed only at the level of the finite-$N$ dictionary and the recurrence-peak criterion.
\item The CM family is the computable model of a Hida family; the cap $\min(n+1,w-1)$ is CM-specific, and general non-CM Hida families are expected to have no cap (unverified).
\item Every numerical value above carries its provenance in the master compendium; corrections applied are distributed to the companion papers and summarized in Appendix~A.
\end{itemize}

\appendix
\section{Corrections touching this framework}
\begin{center}
\begin{tabular}{p{0.8cm}p{5.6cm}p{7.8cm}}
\toprule
\# & error as made & correction\\
\midrule
2 & Maass $\kappa$ written $1/2$ & $\kappa=0$; critical line $\kappa+\tfrac12=\tfrac12$ (paper B)\\
3 & degeneracy scripts: two bugs & fixed; $g(D)=h^{+}(D)$ (paper B, nos.~41--44)\\
5 & naive $C^{\mathrm{trop}}$ claimed resolving power & zero resolving power; name withdrawn; replaced by the skeleton dynamic charge\\
6 & additive reduction $c_{p}=v_{p}(\Delta)$ & $c_{p}=|\Phi|$ (ADE component group order)\\
7 & $\operatorname{Vol}(4_{1})$ initial value & $6\Lambda(\pi/3)$, two ideal tetrahedra\\
8 & $M(\Delta)=h$ universal & only genus $1$; in general $\log M\ge h$\\
9 & $p$-adic valuation cap universal & cap is CM-specific\\
34 & $\Phi_{1}=0.755362$ miscopied & $\Phi_{1}=C_{1}=0.75536661$; M\"obius truncation $K=12$ insufficient at $\nu=1$ ($j=13$ term $\approx\log2\cdot2^{14}\approx4\times10^{5}$), closed at $K=60$\\
36 & ``$\Phi_{n}\to0.734$?'' & $\Phi_{n}$ does not converge to a constant: limit $0$ (base $r=e\log3/3=0.9954459403$; peak $0.847045989$ at $n^{\star}=103$; valley $0.707$ at $n=2$)\\
45 & Gap straddle degenerates at integral $\kappa$ & strict straddle: integral $\kappa\Rightarrow d=2$, half-integral $\kappa\Rightarrow d=1$; selected uniquely by $\rho\le1.25$ (no.~59)\\
47 & dangling cross-reference to \S6.5 & renumbered; all cross-references carry the part name\\
55 & ``$b$-invariant step residue-char-free'' & false at $p=2$ (64.a4 vs 32.a3); boundary at $A\bmod4$; conductor is $64$\\
56 & wild-table key precision conflict & keys must be unified to mod-$16$\\
59 & straddle criteria defects & $|S|$ in place of signed $S$; threshold $\rho\le1.25=\cosh(\log2)$ calibrated from the integral side\\
\bottomrule
\end{tabular}
\end{center}


\section{Supplementary derivations}
\noindent\emph{B.1. Generating function of the symmetric-power $L$-functions.} For $|t|$ small,
\[
\sum_{k\ge0}\log L(s,\mathrm{Sym}^{k}f)\,t^{k}=\sum_{p}\sum_{n\ge1}\frac{p^{-ns}}{n\,(1-\alpha^{n}t)(1-\beta^{n}t)},
\]
since $\sum_{k\ge0}\tr(\mathrm{Sym}^{k}A_{p})t^{k}=1/\det(1-tA_{p})$; verified at $11\mathrm{a1}$, $s=10$, $t=0.1$ to $1.1\times10^{-10}$ ($P/K/N=400/10/14$).
\noindent\emph{B.2. Two closed forms for $R_{k}$.} With $P_{n}=\alpha^{n}+\beta^{n}$,
\[
h_{k}^{(n)}=\sum_{j=0}^{\lfloor k/2\rfloor}(-1)^{j}\binom{k-j}{j}p^{2\kappa nj}P_{n}^{\,k-2j}=p^{n\kappa k}U_{k}(\cos n\theta_{p}),
\]
the second by replacing $\theta$ with $n\theta$ in $T_{k}=p^{k\kappa}U_{k}(\cos\theta)$; agreement to $\le10^{-35}$.
\noindent\emph{B.3. Tate content at general $n$.} $\langle U_{k}(\cos n\theta)\rangle$: $k$ odd: $(0,-1,0,0)$ at $n=1,2,3,4$; $k$ even: $(0,+1,+1,+1)$. Mechanism: the Sato--Tate measure gives $\langle\cos2\theta\rangle=-1/2$ as the only nonzero Fourier moment, and $2j-k=\pm1$ only for $k$ odd. Measured ($400$ primes, $k=1..6$): $n=3$: $0.02,1.09,0.01,1.13,-0.07,1.09$; $n=4$: $0.02,0.97,0.06,0.92,0.29,0.93$.
\noindent\emph{B.4. General fibered knots: beyond genus $1$.} For fibered $\Sigma_{g}$ with monodromy $B$, $\log M(\Delta)\ge h$, equality iff the unstable roots share one quadratic field: among $16$ exact genus-$2$ examples, $9$ give $\log M=h$ (e.g.\ $[1,-4,5,-4,1]$, $\lambda=\varphi^{2}$, the same field as $4_{1}$) and $7$ give $\log M>h$ ($[1,-6,11,-6,1]=\Delta_{2}\cdot\Delta_{4_{1}}$ gives $2h$; $[1,-7,13,-7,1]$, $[1,-10,19,-10,1]$ give $+0.609$, $+0.411$). The critical amplitude $A$ is predicted exactly: single maximum $A=1$ ($1.00291/1.00031/1.0000086/0.99950$ at $\delta=0.2/0.1/0.05/0.02$), $\Delta_{2}\cdot4_{1}$: $A=2$ ($2.01109/2.00340/2.00094/1.99917$), alternating $\pm\lambda$ pairs return to $A=1$ by parity cancellation.
\noindent\emph{B.5. Catalan asymptotics.} $\mathrm{Cat}(m/2)\sim 2^{m+3/2}/(\sqrt{\pi}\,m^{3/2})$ (leading term within $7\%$ at $m=30$; with the Stirling correction the match improves to $0.9995$): the boundary position $m\kappa$ grows linearly while the Tate weight grows exponentially --- the quantitative form of layer-1 forgetting.
\noindent\emph{B.6. Two-prime central charge.} For $11\mathrm{a1}$ with $467$ twin-prime pairs the disconnected background is $\langle x^{m}\rangle^{2}=\mathrm{Cat}(m/2)^{2}$: measured versus background, $m=2$: $1.0113$ vs $0.9862$; $m=4$: $3.8007$ vs $3.8939$ (background value $1$, $4$) --- the correlation layer forgets one exponential faster than the coefficient layer.
\noindent\emph{B.7. Khovanov lift (categorification).} $4_{1}=\mathrm{closure}((\sigma_{1}\sigma_{2}^{-1})^{2})\in B_{3}$, $\rho(\sigma_{i})=A+A^{-1}e_{i}$, $\delta=-A^{2}-A^{-2}$; $V_{4_{1}}(t)=t^{2}-t+1-t^{-1}+t^{-2}$. At the Sato--Tate phase $A=e^{i\theta/2}$ the loop parameter is $\delta_{p}=-2\cos\theta=-a_{p}/p^{\kappa}$, and Deligne's bound is the Jones-index bound $\delta^{2}\le4$: each prime carries a Temperley--Lieb algebra. The Khovanov lift promotes $\delta$ to $\dim V=2$: chain groups $C(s)=V^{\otimes|s|}$, $2^{4}=16$ resolutions, basis of crossingless matchings ($\mathrm{Cat}(4)=14$), total chain dimension $\sum_{s}2^{|s|}=66$, Euler characteristic $\sum_{s}(-1)^{|s|}2^{|s|}=2=2V_{4_{1}}(1)$ --- categorification lifts $\delta$ to $\dim V$, reproducing the Jones value exactly.

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