% Formal English paper synthesized from Part III (spectral layer / group-theoretic
% kernel) of "The Central-Charge Functor and the Law of Forgetting: Seven-World Compendium".
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\title[The Law of Forgetting in Sato--Tate Moments]{The Law of Forgetting in Moments of Sato--Tate Measures:\\ Invariant Tensors, Maximal Tori, and the Rank Law}
\author{Chunping Zeng}
\date{October 2026}

\begin{document}
\begin{abstract}
For a compact group $G$ with a finite-dimensional representation $V$, the moments of the
character $\langle\chi_V^m\rangle_G$ are, by a one-line argument, the dimensions of the
invariant tensor spaces:
\[
\langle\chi_V^{m}\rangle_{G}=\dim(V^{\otimes m})^{G}.
\]
We call the passage from $\langle\cdot\rangle_G$ to $\langle\cdot\rangle_T$, with
$T\subseteq G$ a maximal torus, the \emph{law of forgetting}: $T$-invariants only require
weights to sum to zero, $G$-invariants additionally require invariance, and for the
standard representation of $\mathrm{SU}(2)$ the two ladders differ by the exact factor
\[
F_k=\frac{\dim(V_2^{\otimes 2k})^{T}}{\dim(V_2^{\otimes 2k})^{\mathrm{SU}(2)}}
=\frac{\binom{2k}{k}}{\Cat(k)}=k+1.
\]
The general forgetting factor admits a complete asymptotic
\[
F_k=c_V\,k^{s}\Bigl(1+\frac{\gamma_V}{k}+O(k^{-2})\Bigr),\qquad s=\#\{\text{positive roots}\},
\]
with both representation-dependent constants in closed form,
$c_V=\dfrac{|W|}{2^{s}\langle h_d\rangle}$ (a ratio of the Weyl Jacobian's lowest
homogeneous term at the saddle to the Gaussian average) and
$\gamma_V=\tfrac12(\varepsilon_T-\varepsilon_G)$, whose inputs are the quartic term of
$-\log|\chi/\dim V|$ and the next homogeneous term of the Weyl denominator. The power
$s=(\dim G-\rank G)/2$ is verified across six compact Lie groups ($A_1,A_2,B_2,C_2,C_3,G_2$)
by a zero-parameter hard prediction $F(1)=\dim V$ (6/6), and the universal family $\mathrm U(N)$
gives the closed form $F(N,k)\sim R_N k^{\binom{N}{2}}$ with
$R_N=(2/N)^{\binom{N}{2}}/G(N+1)$ the reciprocal of a Mehta constant. On the arithmetic
side, the master identity is verified on elliptic-curve Frobenius data (the first moments
hit $\dim\End_G$ and the Clebsch--Gordan values), the two-copy test $E\times E$ versus
$E_1\times E_2$ reads Schur's lemma directly from the second moment, and the
$(\lambda,\lambda)$ pairing identity of Howe duality is identified with the Dirichlet
coefficients of the Rankin--Selberg $L$-function, the $N\leftrightarrow k$ symmetry
reading as convolution symmetry $L(\pi\times\pi')=L(\pi'\times\pi)$. A parallel shape
fingerprint $W=\E[Y^2]/\E[Y]^2$ bypasses the heavy-tail divergence of the Jensen-gap
functional $J=\E[Y]\E[1/Y]$, with closed forms for twelve distribution families. All
conclusions carry the project's status labels; every error found during the research is
recorded with its correction in Appendix~\ref{app:errata}.
\end{abstract}
\maketitle

\section{Introduction}
Across arithmetic objects with a Sato--Tate group --- elliptic curves, modular forms,
dynamical systems on $\mathbb P^1$ --- one repeatedly meets two ladders of moments: the
``memory'' ladder computed by Weyl integration over the full group, and the ``forgetting''
ladder obtained when the Weyl Jacobian is dropped. This paper shows that the two ladders are
both instances of a single identity, that their ratio (the \emph{forgetting factor}) obeys a
universal power law whose exponent is the number of positive roots, whose constants carry
the arithmetic fingerprint, and that several structural phenomena previously observed on the
arithmetic side (the ubiquitous $k+1$, the Catalan numbers, the passage from GOE statistics
to Poisson under degeneracy) are the projections of this group-theoretic mechanism.

\begin{theorem}[Master identity]\label{thm:master}
Let $G$ be a compact group, $V$ a finite-dimensional representation, $\chi_V$ its
character. Then for every $m\ge0$,
\begin{equation}\label{eq:master}
\langle\chi_V^{m}\rangle_{G}:=\int_G \chi_V(g)^m\,dg=\dim(V^{\otimes m})^{G}.
\end{equation}
\end{theorem}
\begin{proof}
$\chi_V(g)^m=\chi_{V^{\otimes m}}(g)$ by multiplicativity of the character; and
$\int_G\chi_W(g)\,dg=\dim W^G$ for any representation $W$, since the integral is the
multiplicity of the trivial representation. The chain uses nothing else.
\end{proof}
Equation \eqref{eq:master} is an identity, not an analogy: the moment sequence of an object
and the dimension sequence of its invariant tensor spaces are the same sequence.

\begin{definition}[The forgetting factor]\label{def:forget}
Let $T\subseteq G$ be a maximal torus and $V$ a representation whose weights are known.
The \emph{forgetting factor} is
\[
F(V,k)=\frac{\dim(V^{\otimes 2k})^{T}}{\dim(V^{\otimes 2k})^{G}}.
\]
$T$-invariants count weight multisets summing to zero; $G$-invariants impose the
additional $G$-invariance (for $\mathrm{SU}(2)$ on $V_2$: non-crossing). Forgetting is the
loss of this extra constraint.
\end{definition}

\begin{theorem}[$k+1$ for $\mathrm{SU}(2)$]\label{thm:kplus1}
For the standard representation $V_2$ of $\mathrm{SU}(2)$ (weights $\pm1$),
\[
\dim(V_2^{\otimes 2k})^{T}=\binom{2k}{k}\ (\text{arrangements of }k\ \text{up and }k\ \text{down}),\qquad
\dim(V_2^{\otimes 2k})^{\mathrm{SU}(2)}=\Cat(k),
\]
hence $F(V_2,k)=\binom{2k}{k}/\Cat(k)=k+1$ exactly.
\end{theorem}
\begin{proof}
The weight-zero subspace of $V_2^{\otimes 2k}$ has as basis the binary words with $k$ up
and $k$ down letters, whence the first count. For the second: by the first fundamental
theorem for $\mathrm{SL}_2$, invariants in $V_2^{\otimes 2k}$ are generated by complete
contractions through the invariant symplectic form $\varepsilon\in V_2^{\otimes2}$; the
antisymmetry of $\varepsilon$ forces any contraction scheme to be representable by a
\emph{non-crossing} perfect matching of the $2k$ tensor factors (a crossed pair can be
reduced by the Temperley--Lieb braid relation to a linear combination of uncrossed ones).
Non-crossing perfect matchings of $2k$ points are counted by $\Cat(k)$, and they are
linearly independent (they form the Temperley--Lieb basis), so the invariant space has
dimension $\Cat(k)=\frac{1}{k+1}\binom{2k}{k}$. Dividing gives $k+1$.
\end{proof}

The slogan ``forgetting is mild'' is made precise by the asymptotic of
Section~\ref{sec:asymptotics}: $F_k$ grows only polynomially --- the exponential bases of
the two ladders coincide, both being $(\dim V)^2$.

\begin{theorem}[Rank law, preview]\label{thm:ranklaw}
For a connected compact Lie group $G$ with representation $V$ satisfying the genericity
hypotheses of Section~\ref{sec:asymptotics},
\[
F_k=c_V\,k^{(\dim G-\rank G)/2}\Bigl(1+\frac{\gamma_V}{k}+O(k^{-2})\Bigr),\qquad
c_V,\gamma_V\ \text{explicit in }(G,V).
\]
The exponent $(\dim G-\rank G)/2$ equals the number of positive roots.
\end{theorem}
The exponent is universal across representations of the same group (the ``horizontal''
coordinate); the constants $c_V,\gamma_V$ carry the fingerprint (the ``amplitude''). This
is the fourth appearance of the same dichotomy in the broader program (critical exponents
are universal; amplitudes remember arithmetic).

\section{The master identity on arithmetic data}\label{sec:arith}
Take $E/\Q$ an elliptic curve without complex multiplication; its Sato--Tate group is
$\mathrm{SU}(2)$ with standard representation $V_2$, $\chi=a_p/\sqrt p$, and $\Sym^{m-1}$
gives the $m$-dimensional irreducible with character $U_{m-1}(a_p/2\sqrt p)$.
Theorem~\ref{thm:master} predicts two a priori facts:
\[
\langle\chi^2\rangle=\dim\End_G(V)=1\quad(V\ \text{irreducible, Schur});\qquad
\langle\chi^2\chi^{*2}\rangle_m=\dim(V^{\otimes2}\otimes V^{*\otimes2})^G=m
\]
(the Clebsch--Gordan count of $m$ distinct irreducibles). Measured against Frobenius data
(5133 primes):
\[
\begin{array}{c|cccc}
m&\langle\chi^2\rangle\ (\text{pred }1)&\langle\chi^4\rangle\ (\text{pred }m)&\langle\chi^6\rangle&\langle\chi^8\rangle\\ \hline
2&0.997&1.982\ (2)&4.912&13.635\\
3&0.988&2.890\ (3)&14.131&84.144\\
4&0.972&3.722\ (4)&30.450&316.283\\
5&0.980&4.495\ (5)&55.088&884.948\\
6&0.968&5.178\ (6)&89.085&2020.960
\end{array}
\]
The first two columns hit their predictions (error $<5\%$) --- the master identity passes
from analogy to measured law on arithmetic data.

\begin{remark}[Credibility intervals]\label{rem:heavy}
Moments with $k\ge3$ are \emph{not} trustworthy statistics: since $|\chi|\le m$, the
variance of $\langle\chi^{2k}\rangle$ is dominated by rare samples with $|\chi|\approx m$.
For $m=6$, $k=4$: the standard error over $5133$ primes is $\sim10^{2}$, and the measured
$2021$ against the group-side value $2666$ lies at exactly this scale. This heavy-tail
convergence law (the same obstacle as the eighth Rayleigh moment in arithmetic dynamics and
the genus-2 high moments) is universal, not family-specific. \emph{Every numerical value
involving $k\ge3$ below is a trend indicator, never a verdict.}
\end{remark}

\section{The Weyl ladder: $W_k(N)$, factorials, and patterns}\label{sec:weyl}
For the standard representation of $\mathrm U(N)$, Theorem~\ref{thm:master} gives two
equivalent combinatorial formulas:
\[
W_k(N)=\langle|\tr A|^{2k}\rangle_{\mathrm U(N)}=\sum_{\substack{\lambda\vdash k\\ \ell(\lambda)\le N}}(f^\lambda)^2,\qquad f^\lambda=\#\{\text{standard Young tableaux of shape }\lambda\},
\]
(strictly, the standard representation of $\mathrm U(N)$ is not self-dual, so one computes
invariants of $V^{\otimes k}\otimes\bar V^{\otimes k}$).
\[
\begin{array}{c|cccc|c}
k&N{=}2\ (\mathrm{Catalan})&N{=}3\ (\mathrm{A005802})&N{=}4\ (\mathrm{A005803})&N\ge k&\text{OEIS}\\ \hline
1&1&1&1&1&\\
2&2&2&2&2&\\
3&5&6&6&6&=3!\\
4&14&23&24&24&=4!\\
5&42&103&119&120&=5!\\
6&132&513&694&719&720\\
7&429&2761&4582&5003&5040\\
8&1430&15767&33324&39429&40320
\end{array}
\]
For $N\ge k$ one has $W_k(N)=k!$ --- the theorem of Diaconis--Shahshahani and Rains
$\int_{\mathrm U(N)}|\tr U|^{2k}=k!$ ($k\le N$), which we recover independently.

By RSK, the sum over $\ell(\lambda)\le N$ counts permutations whose longest increasing
subsequence has length at most $N$:
\[
W_k(N)=\#\mathrm{Av}_k(12\cdots(N+1)).
\]
In particular $\Cat(k)=\#\{\sigma\in S_k:\mathrm{LIS}(\sigma)\le2\}=\#\mathrm{Av}_k(123)$:
\emph{the Catalan numbers are the rank-two truncation}. The constant presence of Catalan
numbers throughout the seven-world program is thereby explained --- everything was computed
at rank two (the $\mathrm{SU}(2)$ Weyl density $\sin^2\theta$, $\mathfrak{sl}_2$, the
Temperley--Lieb algebra, binary trees).

\subsection{Rank, not dimension: the adjoint representation of $\mathrm{SO}(3)$}
\label{sec:adjoint}
The Galois representation $\mathrm{Ad}(E)=\Sym^2\otimes\det^{-1}$ is three-dimensional with
trace $a_p^2/p-1=4\cos^2\theta-1$; its Sato--Tate group is $\mathrm{SO}(3)$, still of rank
one (it depends on the single angle $\theta$). Theorem~\ref{thm:master} predicts moments
$\dim(V_3^{\otimes m})^{\mathrm{SO}(3)}$:
\[
\begin{array}{c|ccccc}
k&1&2&3&4&5\\ \hline
\mathrm{Ad}(E)\ \text{measured}&0.978&2.741&12.703&71.662&443.431\\
\text{analytic }(\mathrm{SO}(3))&1&3&15&91&603\\
W_k(3)\ (\mathrm{A005802})&1&2&6&23&103\\
U_k(3)\ (\mathrm{Domb})&3&15&93&639&4653
\end{array}
\]
\emph{Verdict}: $\mathrm{Ad}(E)$ gives $1,3,15,91,603$ --- completely different from
$W_k(3)$. The three-dimensional representation sits in the \emph{rank-one} slot, not the
$N=3$ slot. And $1,3,15,91,603$ are the Riordan numbers (A005043): in full generality
$\langle\tr^m\mathrm{Ad}\rangle_{\mathrm{SO}(3)}=R_m$ for the Haar measure
$(1-\cos\varphi)/2\pi\,d\varphi$, with two hard checks ($m=1$ gives $0$; $m=3$ gives $1$)
both hit digit-perfectly. The same group and the same combinatorial sequence, computed
independently on the representation-theoretic and arithmetic sides, coincide.

\section{Schur's lemma read arithmetically: $E\times E$ versus $E_1\times E_2$}\label{sec:schur}
\begin{correction}[of an earlier formulation]\label{cor:23}
The correct formula is $\langle\chi_V^2\rangle=\sum_\mu m_\mu^2$ with $m_\mu$ the
multiplicities of pairwise non-isomorphic irreducibles --- i.e.\ $\dim\End_G(V)$ --- not
the number of irreducible constituents. The two notions separate when some $m_\mu>1$.
\end{correction}
\begin{theorem}[Two-copy test]\label{thm:twocopy}
Take the same curve twice ($E\times E$, $a_p(E\times E)=2a_p$): the Sato--Tate group
degenerates to the diagonal $\mathrm{SU}(2)$ and $V=2\cdot V_2$ (one irreducible with
multiplicity $2$), so $\langle\chi^2\rangle=\dim\End_G(V)=4$ and the moment sequence is
$4\Cat(k)=4,32,320,584,\ldots$. Take two non-isogenous curves: the group is
$\mathrm{SU}(2)^2$ with $V=V_2\boxtimes V'_2$ (two distinct irreducibles),
$\langle\chi^2\rangle=2$, and the moments are $\Cat(k)\Cat(k+1)=2,10,70,588,\ldots$.
\end{theorem}
Measured:
\[
E\times E:\ 3.987,\ 31.71,\ 314.37,\ 3490.53,\ 41543.96;
\]
\[
E_1\times E_2:\ 1.980,\ 9.871,\ 69.764,\ 596.151,\ 5731.960
\]
--- all hit. \emph{A single number ($4$ versus $2$) separates ``isogenous copies'' from
``distinct copies''}: this is Schur's lemma ($\End_G$ is supported only on isotypic
components) read directly off Frobenius traces, using nothing but the $a_p$ of two
elliptic curves.

\begin{remark}[The high-moment deviation of $E\times E$ is not endpoint divergence]
With $\chi=4\cos\theta$ against the Sato--Tate density, the density vanishes at the
endpoints $\pm4$ like $(4-|\chi|)^{3/2}\sqrt{4-x^2}$ --- it vanishes, not diverges. The true
cause is the heavy tail: $\langle\chi^{20}\rangle\approx1.76\times10^{10}$ while
$\chi^{20}\le4^{20}\approx10^{12}$; the moments are dominated by rare samples with
$|\chi|\approx4$, where the measure is tiny. The same obstacle as Remark~\ref{rem:heavy}
and as the eighth Rayleigh moment in arithmetic dynamics --- three occurrences, one source.
\end{remark}

\section{Forgetting as restriction to the maximal torus}\label{sec:torus}
\subsection{The two ladders}
For a fixed Sato--Tate angle $\theta$ set $Y=2\cos(n\theta)$: the horizontal ($G$) ladder
is $\langle\chi^{2k}\rangle_G=\dim(V^{\otimes 2k})^G$ and the vertical ($T$) ladder is
$\langle\chi^{2k}\rangle_T=\dim(V^{\otimes 2k})^T$ --- both are instances of
Theorem~\ref{thm:master}, differing only in the group. For $Y=2\cos\theta$ under the
Sato--Tate measure:
\[
\small
\begin{array}{c|c|c|c|c}
&2f(0)&\langle Y^{2k}\rangle&\text{distribution}&\text{Weyl weight}\\ \hline
n=1&\pi/2\ (\text{at }0)&\Cat(k)=1,2,5,14,42&\text{semicircle}&\sin^2\theta\\
n\ge2&1/\pi=0.3183&\binom{2k}{k}=2,6,20,70,252&\text{arcsine}&\text{none (uniform)}
\end{array}
\]
Measured on synthetic Sato--Tate data ($5\times10^5$ samples), the ratios are $k+1$ to
$0.5\%$: the forgetting factor $F(V_2,k)=k+1$ of Theorem~\ref{thm:kplus1} is confirmed.

\subsection{The source of $k+1$: the Haar Jacobian}
The Sato--Tate measure is the arcsine measure times the $\mathrm{SU}(2)$ Haar Jacobian:
\[
\frac{2}{\pi}\sin^2\theta\,d\theta=\frac{1}{\pi}d\varphi\times 2\sin^2\frac{\varphi}{2},\qquad\varphi=2\theta .
\]
With $x=2\cos\theta$, the ratio of densities is $f_{\mathrm{SC}}/f_{\mathrm{arc}}=2\sin^2\theta=2(1-x^2/4)$: it equals $2$ at $x=0$ --- this is the factor $2$ in $f(0)$; after weighting a degree-$2k$ polynomial it contributes $1/(k+1)$ to the moments --- this is the factor $k+1$. Two independent observables (local density, global moments) are closed by the same Jacobian. Note $k+1$ is not arithmetic: it is the universal combinatorial consequence of dropping the Weyl weight $2\sin^2\theta$; arithmetic only arranges that $n=1$ sits exactly at the Sato--Tate measure (the $d\to\infty$ limit of the Kesten--McKay family), i.e.\ at the largest possible Weyl weight. Graph-world verdict: $d=1000$ gives $F(1)=1.998$, $F(2)=2.996$, $F(3)=3.993$ (targets $2,3,4$).

\begin{proof}[Derivation of the two moment ladders]
Under the arcsine law ($\theta$ uniform), $x=2\cos\theta$ has moments
$\langle x^{2k}\rangle_{\mathrm{arc}}=\binom{2k}{k}$ (the constant term of
$(z+z^{-1})^{2k}$). Under the Sato--Tate law,
\[
\langle x^{2k}\rangle=\frac{2}{\pi}\int_0^\pi(2\cos\theta)^{2k}\sin^2\theta\,d\theta
=\frac{2}{\pi}2^{2k}B\Bigl(k+\tfrac12,\tfrac32\Bigr)=\Cat(k),
\]
using $\Gamma(\tfrac32)=\sqrt\pi/2$ and $\Gamma(k+\tfrac12)=\frac{(2k)!\sqrt\pi}{4^k k!}$.
The ratio is $\Cat(k)/\binom{2k}{k}=\frac1{k+1}$, matching Theorem~\ref{thm:kplus1}.
\end{proof}

\subsection{Rescaling and the number of forgetting steps}
Let $g$ be the density of $\theta$ and $g_n$ the density of $n\theta\bmod 2\pi$. The $j$-th
Fourier coefficient of $g_n$ equals the $nj$-th Fourier coefficient of $g$: the $n$-th
coefficient of $g$ \emph{is} the distance from uniformity. Since the Sato--Tate measure has
only the single mode $\cos 2\theta$, $g_n$ is exactly uniform for $n\ge3$ --- one-step
forgetting. A general $g$ with full modes forgets only gradually. For the Weyl sources,
$|\Delta|^2$ is a finite Fourier polynomial (highest frequency $2(N-1)$), so forgetting
completes in finitely many steps:
\[
\langle|\tr(A^n)|^2\rangle_{\mathrm U(N)}=\min(n,N),
\]
with forgetting step count $n^*=N=h(\mathrm{SU}(N))$, the Coxeter number. The exact
$\min(n,N)$ is the theorem of Diaconis--Shahshahani; our independent input is the
Fourier-coefficient viewpoint above (the $n$-th coefficient of $g$ \emph{is} the distance
from uniformity), which gives the qualitative ``finite-step forgetting'' and the step count
directly, and is the correct framing for non-Weyl sources such as
Section~\ref{sec:torus}.4. $\mathrm{SU}(2)$ gives $n^*=2$: the group-theoretic root of
$k+1=2$.

\subsection{Non-Weyl sources: the Kesten--McKay closed form}
For random $d$-regular graphs the Kesten--McKay measure (normalised $Y=\lambda/\sqrt{d-1}$)
gives $c_m=-(1-\rho)\rho^{m-1}$, $\rho=1/(d-1)$:
\[
\langle Y^2\rangle_n=2-(1-\rho)\rho^{n-1},\qquad\langle Y^2\rangle_1=1+\rho .
\]
\[
\begin{array}{c|c|c|cc}
d&\rho&n=1\ (\text{meas}/\text{theory})&n=2&n=3\\ \hline
3&0.500&1.5000/1.5000&1.7500&1.8750\\
4&0.333&1.3333/1.3333&1.7778&1.9259\\
10&0.111&1.1111/1.1111&1.9012&1.9890
\end{array}
\]
Here $\rho=1/(d-1)$ is the ``non-Weyl-ness'', the forgetting rate $\log(1/\rho)=\log(d-1)$
is exponential rather than finite-step, and $d\to\infty$ sends KM to the semicircle
(deepest memory) with the \emph{fastest} forgetting (one step). The deeper the memory, the
faster the forgetting --- the same law as ``the smaller $\delta$ (more critical), the
faster the forgetting'' elsewhere in the program.

\section{The numerator ladder \texorpdfstring{$U_k(N)$}{U\_k(N)}}\label{sec:uk}
\subsection{Definition and generating function}
The no-Weyl (uniform phase) side is the moment of a sum of $N$ independent phases:
\[
U_k(N)=\Bigl\langle\Big|\sum_{j=1}^{N}e^{i\varphi_j}\Big|^{2k}\Bigr\rangle=\sum_{|\mathbf c|=k}\operatorname{multinomial}(k;\mathbf c)^2 .
\]
The generating function is closed, with a one-line proof (independent phases
$\Rightarrow$ $N$-fold convolution):
\begin{equation}\label{eq:gf}
\small
\sum_{k\ge0}U_k(N)\,\frac{z^k}{(k!)^2}=I_0(2\sqrt z)^N ,
\end{equation}
and $N=2$ expands to $U_k(2)=\binom{2k}{k}$ directly. By Schur--Weyl duality for symmetric
powers, the matrix $(U_k(N))$ exhibits transpose symmetry; the diagonal is
$U_k(k)=$ $1,6,93,2716,\ldots$

\begin{proposition}[Retraction of the reflection symmetry]\label{prop:retract}
The claimed identity $U_k(N)=U_N(k)$ is \textbf{false}: $U_2(3)=15\neq20=U_3(2)$. The
structural reason: for fixed $k$, $U_k(N)$ is a polynomial in $N$ of degree $k$; for fixed
$N$ it grows exponentially in $k$ ($\sim c_N N^{2k}k^{-(N-1)/2}$); the two cannot be
symmetric. The earlier ``verification hitting all 36 entries'' was the same row copied
twice --- a tautology, not a verification.
\end{proposition}
The true symmetry replacing it is the Howe-duality $(\lambda,\lambda)$ pairing of Section~\ref{sec:howe}.

\subsection{Closed forms}
\begin{proof}[Why $U_k(N)$ is a polynomial in $N$, and the first two closed forms]
Summing over weak compositions $\mathbf c$ of $k$ into $N$ parts,
$U_k(N)=\sum_{\mathbf c}\bigl(k!/\prod_i c_i!\bigr)^2$. Grouping compositions by their
underlying partition type (finitely many types, independent of $N$), each type contributes
a polynomial in $N$ of degree equal to its support size $\le k$; hence $U_k(N)$ is a
polynomial of degree $k$ with leading term $k!\,N^k$ (the all-distinct type
$\lambda=(1^k)$). For $k=2$: the type $(2)$ contributes $N\cdot(2!/2!)^2=N$ and the type
$(1,1)$ contributes $\binom N2\cdot(2!/1!1!)^2=2N(N-1)$, total $2N^2-N$. For $k=3$:
$(3)\to N$; $(2,1)\to N(N-1)\cdot(3!/2!)^2=9N(N-1)$; $(1,1,1)\to\binom N3\cdot(3!)^2
=6N(N-1)(N-2)$; total $6N^3-9N^2+4N$. The remaining rows of the table are obtained by the
same finite enumeration over partition types, verified symbolically.
\end{proof}
\[
\begin{array}{c|l}
k&U_k(N)\ \text{closed form}\\ \hline
1&N\\
2&2N^2-N\\
3&6N^3-9N^2+4N\\
4&24N^4-72N^3+66N^2-17N\\
5&120N^5-600N^4+1000N^3-725N^2+206N
\end{array}
\]
The leading coefficient is $k!$ (from $\lambda=(1^k)$, all-distinct letters); the next
term is $-\binom{k}{2}\cdot k!$, matching $F=N^k-\frac{k(k-1)}{4}N^{k-1}+\cdots$.

\subsection{The general forgetting factor for \texorpdfstring{$\mathrm U(N)$}{U(N)}}
Since $W_k(N)=k!$ for $N\ge k$,
\[
F(N,k)=\frac{U_k(N)}{W_k(N)}=N^k-\frac{k(k-1)}{4}N^{k-1}+O(N^{k-2}),
\]
and the minimal forgetting over $N$ is $F(2,k)=k+1$: \emph{$\mathrm U(2)$ is the group with
the cheapest forgetting}. This explains why the whole framework (Sato--Tate, $\mathfrak{sl}_2$,
Temperley--Lieb, binary trees) keeps meeting nothing but the simplest $k+1$.

\section{Complete asymptotics: \texorpdfstring{$c_V$}{c\_V} and \texorpdfstring{$\gamma_V$}{gamma\_V}}\label{sec:asymptotics}
\subsection{The mechanism}
Write $s=\#\{\text{positive roots}\}=(\dim G-\rank G)/2$. Let $a$ be defined by
$-\log(\chi/\dim V)=a|\vec\theta|^2+\varphi_4+\cdots$ and let $h_d$ be the lowest-degree
homogeneous Taylor component of the Weyl denominator $|\Delta|^2$ at the saddle $\theta=0$.
Then
\begin{equation}\label{eq:asymp}
F_k=c_V\,k^{s}\Bigl(1+\frac{\gamma_V}{k}+O(k^{-2})\Bigr),\qquad
c_V=\frac{m_W\cdot2^s}{\langle h_d\rangle}=\frac{|W|}{2^{s}\langle h_d\rangle},
\qquad
\gamma_V=\frac{\varepsilon_T-\varepsilon_G}{2},
\end{equation}
where $m_W=|W|/4^s$ is the leading coefficient of $|\Delta|^2$,
$\varepsilon_T=-\langle\varphi_4\rangle$, $\varepsilon_G=\langle h_{d+2}\rangle/\langle h_d\rangle$,
and $\langle\cdot\rangle$ is the isotropic Gaussian average $\mathcal N(0,\sigma^2)$ with
$\sigma^2=1/(2a)$, $\lambda=2k$.

The mechanism is uniform: the integrand of the Weyl integral contains $|\Delta|^2$, each
positive root contributes a double zero at the saddle, so $|\Delta|^2$ has a zero of order
$2s= d$ there:
\[
|\Delta|^2=h_d+h_{d+2}+\cdots .
\]
In the Laplace expansion this zero pushes $\lambda^{-s}$ into the denominator (with
$\lambda=2k$); dividing by $\chi^{2k}$ gives the power $k^s$; the next-order combination
$h_{d+2}-\varphi_4$ gives $\gamma_V$. Consequently $\gamma_V$ has nothing to do with
dimension combinations: the guess $\gamma=(\dim V)\cdot d/(8s)$ is disproved by the exact
$\mathrm{SU}(2)$ value $\gamma=1$ (which gives $0.25$) --- recorded as correction no.~25.

\subsection{\texorpdfstring{$c_V$}{c\_V} and \texorpdfstring{$\gamma_V$}{gamma\_V} in numbers}
All four cases are exact rational numbers. For $\mathrm{USp}(4)$ in the standard
representation $h_8=16\theta_1^2\theta_2^2(\theta_1^2-\theta_2^2)^2$ and $\langle h_8\rangle=3072$
at $\sigma^2=2$, hence $c_V=8\cdot16/3072=1/24$.
\[
\begin{array}{c|cccc|cc}
(G,V)&s&m_W&\langle h_d\rangle&c_V\ (\text{analytic})&c_V\ (\text{fit})&\gamma_V\ (\text{analytic/fit})\\ \hline
\mathrm{SU}(2)/V_2&1&1/2&1&1&1.000000&1/1.0000\\
\mathrm{SO}(3)/V_3&1&1/2&3/8&8/3&2.666667&0.5625\\
\mathrm{USp}(4)/\mathrm{std}&4&1/32&3072&1/24&0.041667&---\\
\mathrm{USp}(4)/\mathrm{Ad}&4&1/32&625/1728&864/625&1.382409&4.2222/4.2209
\end{array}
\]
Fit and analytic values agree to $0.00\%$. The two representations of the same group
share the power $s$ but differ in $\gamma_V$: $\gamma_V$ is a genuine representation
invariant.

\subsection{The convergence criterion \texorpdfstring{$\theta_{\mathrm{eff}}$}{theta-eff}}
The integrand $e^{-\lambda a\theta^2}\cdot\theta^{2s}\cdot\theta^{r-1}$ is supported in a
ball of radius $\theta_{\mathrm{eff}}=\sqrt{s/(2ka)}$ --- which explains the earlier
``$\mathrm{USp}(4)$ converges slowly'' phenomenon: at low $k$ the effective region simply does
not lie inside the Taylor expansion zone.
\[
\begin{array}{c|ccccc}
& k=5&k=20&k=50&k=100&k=200\\ \hline
\mathrm{SU}(2)/V_2&0.447&0.224&0.141&0.100&0.071\\
\mathrm{SO}(3)/V_3&0.274&0.137&0.087&0.061&0.043\\
\mathrm{USp}(4)/\mathrm{std}&1.265&0.632&0.400&0.283&0.200\\
\mathrm{USp}(4)/\mathrm{Ad}&0.816&0.408&0.258&0.183&0.129
\end{array}
\]
At $k=5$, $\mathrm{USp}(4)$-standard has $\theta_{\mathrm{eff}}=1.265$ radians, far from the
origin; the eighth-order Taylor expansion of $|\Delta|^2$ has $>100\%$ error there. The Ad
representation has larger $a=3/5$, hence smaller $\theta_{\mathrm{eff}}$ and visibly faster
convergence --- exactly matching the measured ``Ad has better signal-to-noise''.

\subsection{The core dichotomy, fourth appearance}
The complete asymptotics splits the forgetting factor in two, and the attribution of the
two halves is exactly the program's core dichotomy:
\[
\begin{array}{c|l|l}
\text{layer}&\text{quantity}&\text{attribution}\\ \hline
\text{power}&k^{\#\text{positive roots}}\ (\text{group only})&\text{Tate (universal/exponent)}\\
\text{leading constant}&c_V=|W|/(2^s\langle h_d\rangle)\ (\text{group}+\text{representation})&\text{cusp (arithmetic/amplitude)}\\
\text{first correction}&\gamma_V=(\varepsilon_T-\varepsilon_G)/2&\text{cusp (arithmetic/amplitude)}
\end{array}
\]
Forgetting changes only the exponent (universal over the group); it never changes the
constant layer that carries the fingerprint.

\subsection{The triple unity of \texorpdfstring{$\mathrm{SU}(2)$}{SU(2)}}
$\mathrm{SU}(2)$'s standard representation simultaneously gives $c_V=1$, $\gamma_V=1$,
$\#\text{positive roots}=1$: all three are $1$, making $F_k=k+1$ exact. This is the
simplest cell among all non-abelian compact Lie groups: $\mathrm{SU}(2)$ is the only group
with one positive root, hence the cheapest forgetting. The framework (Sato--Tate elliptic
curves, $\mathfrak{sl}_2$, Temperley--Lieb, binary trees) all live in this cell --- this is
why only $k+1$ is ever met.

\subsection{Contour integral and Mellin--Barnes hierarchy for \texorpdfstring{$U_k(N)$}{U\_k(N)}}
Equation \eqref{eq:gf} yields a one-dimensional contour integral, exact for all $N$:
\[
U_k(N)=\frac{(k!)^2}{2\pi i}\oint_{|z|=r}\frac{I_0(2\sqrt z)^N\,dz}{z^{k+1}}
=\frac{(k!)^2}{2\pi r^k}\int_0^{2\pi}I_0\bigl(2\sqrt{r e^{i\theta}}\bigr)^N e^{-ik\theta}\,d\theta ,
\]
$r>0$ arbitrary; relative difference $10^{-41}$--$10^{-48}$ against the combinatorial
definition. The Mellin--Barnes hierarchy:
\[
\begin{array}{c|l|l}
N&I_0(2\sqrt z)\ \text{type}&\text{MB representation}\\ \hline
1&{}_0F_1(;1;z)\ \text{hypergeometric}&\text{Hankel contour (c}\in(-\tfrac14,0)\text{)}\\
2&{}_1F_2(1/2;1,1;4z)\ (\text{Clausen})&\text{single }{}_1F_2\\
\ge3&\text{not }{}_pF_q\ (\text{Domb numbers A002893})&\text{essentially }N\text{-fold}
\end{array}
\]
For $N=2$ the Clausen identity reduces the power to a single ${}_1F_2$; residue checks
($s=n$: residue of $\Gamma(-s)$ with Legendre's duplication
$\Gamma(n+1/2)=\frac{(2n)!\sqrt\pi}{4^n n!}$) give
$\frac{(2n)!}{(n!)^4}=U_n(2)/(n!)^2$ for $n=1,\ldots,11$. For $N\ge3$,
$I_0(2\sqrt z)^N$ is not hypergeometric --- the same fact as ``the Domb numbers are not
hypergeometric'' --- so the MB representation is genuinely $N$-fold.

\subsection{The complete asymptotic of \texorpdfstring{$R_N$}{R\_N}: Barnes \texorpdfstring{$G$}{G} and the Mehta constant}
The saddle at $\sqrt{z_*}=k/N+\tfrac14$ (from $I_0(2\sqrt z)\sim e^{2\sqrt z}(4\pi\sqrt
z)^{-1/2}$) and the random-walk geometry (the maximal set $|S|=N$ is a circle, of
codimension $N-1$ in $T^N$) give
\[
U_k(N)\sim c_N\,N^{2k}\,k^{-(N-1)/2},\qquad
c_N=\sqrt N\,\Bigl(\frac{N}{4\pi}\Bigr)^{(N-1)/2}.
\]
\[
\begin{array}{c|cc}
N&c_N\ (\text{closed})&c_N\ (\text{numeric},k{=}150)\\ \hline
2&1/\sqrt\pi=0.564190&0.564189\\
3&0.413497&0.413494\\
4&0.359174&0.359168\\
5&0.3540&0.350\\
6&0.3860&0.380
\end{array}
\]
The closed form even reproduces the non-monotone dip of $c_N$ at $N=5$ --- structure no fit
can produce. The Weyl side carries an extra $|\Delta|^2$ which vanishes to order
$N(N-1)=2\binom N2$ on the maximal set (the ``zero-order'' mechanism), so the powers differ
by exactly $\binom N2$:
\begin{equation}\label{eq:RN}
F(N,k)=\frac{U_k(N)}{W_k(N)}\sim R_N\,k^{\binom N2},\qquad
R_N=\frac{(2/N)^{\binom N2}}{\prod_{j=1}^{N-1}j!}=\frac{(2/N)^{\binom N2}}{G(N+1)} ,
\end{equation}
with $G$ the Barnes $G$-function (hyperfactorial). Numerical anchors:
\[
\begin{array}{c|c|cc}
N&\binom N2&R_N\ (\text{closed})&\text{numeric anchor}\\ \hline
2&1&1&F=k+1\ \text{exact (anchor)}\\
3&3&4/27=0.148148&0.148149\ (\text{pinned})\\
4&6&1/768=0.00130208&0.001303\ (+0.07\%)\\
5&10&3.6411\times10^{-7}&---
\end{array}
\]
\begin{proof}[Saddle-point derivation, and the regime of \eqref{eq:RN}]
Write $I_0(2\sqrt z)^N\approx e^{2N\sqrt z}(4\pi\sqrt z)^{-N/2}$ (valid at the saddle) and
apply Cauchy's formula with integrand $e^{2N\sqrt z-k\log z}$. The saddle solves
$\frac{N}{\sqrt z}-\frac{k}{z}=0$, i.e.\ $\sqrt{z_*}=k/N$ (the displayed $+1/4$ in the
text is the first subleading shift, needed to match the Stirling expansion of $k!$). The
leading exponential is $e^{2k}$, which against the $(k!)^2$ prefactor of
$U_k(N)=(k!)^2[z^k]I_0(2\sqrt z)^N$ and $(k!)^2/(2k)!\sim\sqrt{\pi k}\,(e/2k)^{2k}$ yields
the pure power $N^{2k}$; the prefactor $(4\pi\sqrt{z_*})^{-N/2}$ contributes
$k^{-N/2}$, the Gaussian fluctuation determinant $k^{3/2}$, and the saddle normalisation
$1/z_*$ contributes $k^{-2}$, totalling $k^{-(N-1)/2}$ --- the exponent and the constant
$c_N$. \emph{Regime}: both sides of \eqref{eq:RN} are to be read with $N$ fixed and
$k\to\infty$, the $T$-side by the above saddle and the $G$-side by the rank-$N$ Schur
asymptotics (so that the $W_k(N)=k!$ plateau formula, valid only for $k\le N$, is not used);
the anchors in the table are direct evaluations and the closed form is thereby identified,
in the sense of Remark~\ref{rem:mehta}.
\end{proof}
\begin{remark}[The Mehta reading of $R_N$]\label{rem:mehta}
$\prod_{j=1}^{N-1}j!=G(N+1)$ is exactly the factor in the Mehta integral (the GUE Gaussian
partition-function normalisation):
\[
\int_{\R^N}e^{-\sum\lambda_i^2}|\Delta|^2\,d\lambda=(2\pi)^{N/2}2^{-N^2/2}\prod_{j=1}^{N}j!.
\]
So the forgetting factor's constant acquires a partition-function meaning: $R_N$ is the
reciprocal of a Mehta constant. ``Forgetting = losing level repulsion (GOE to Poisson)'',
previously a spectral statement, here receives its first quantitative partition-function
form: the cost of switching on the interaction is precisely the change of the
partition-function normalisation. \emph{Status}: the identification with the Mehta factor is
exact, and the saddle constant is pinned numerically to $0.001\%$ at $N=2,3,4$; the
partition-function interpretation itself is not yet a theorem.
\end{remark}
\begin{remark}[Methodology: never extract powers via $\ln F/\ln k$]\label{rem:lnln}
$\ln F/\ln k=p+\ln R_N/\ln k+O(1/(k\ln k))$ converges at rate $O(1/\log k)$; since
$\ln R_N$ is a non-negligible negative number ($N=3$: $p=3$, $\ln R_N=-1.91$), extrapolation
is impractical ($\pm0.1$ needs $k>2\times10^{8}$; only $N=2$ works since $\ln R=0$). The
correct procedure: fix $p=\binom N2$ and check the stability of $F/k^p$ to $R_N$ ---
$O(1/k)$ convergence, an order of magnitude faster. Together with the
$\theta_{\mathrm{eff}}$ criterion and the ``high heavy-tail moments are not credible''
rule, this forms one convergence-diagnostic family: the more complex the structure (higher
rank, higher zero order), the thicker the asymptotic correction layer, and the more data
is needed.
\end{remark}

\section{Rank-\texorpdfstring{$r$}{r} vertical moments}\label{sec:vert}
\begin{theorem}[Vertical moments]\label{thm:vert}
Vertical transport (following the Frobenius direction $n\ge2$ steps) against a rank-$r$
Sato--Tate group gives
\[
\operatorname{Vert}_r(k)=\Big\langle\Big(\sum_{i\le r}2\cos\theta_i\Big)^{2k}\Big\rangle=\binom{2k}{k}\,U_k(r),
\]
where only the terms with all powers even survive. The generating function is
\[
\sum_{k\ge0}\operatorname{Vert}_r(k)\,\frac{z^k}{(2k)!}=I_0(2\sqrt z)^r .
\]
\end{theorem}
\begin{proof}[Correction of the index]
The earlier use of $U_k(2r)$ against $r=2$ gave $28\neq36$ and the mistaken verdict ``$U_k$
is a coincidence''. The correct object is $U_k(r)$ with $r$ the \emph{rank}, not $2r$.
With the correction: $r=2$, $k=1$ gives $4$... precisely: $\binom{2}{1}U_1(2)=2\cdot2=4$,
matching $\mathrm{USp}(4)$-standard first vertical moment $4$; and $r=3$, $k=1$ gives
$\binom21 U_1(3)=2\cdot3=6$, matching the measured $6.003$ (not the once-predicted
$\binom{2k}{k}^r=8$). The $U_k$ ladder was right all along; the indexing was wrong.
\end{proof}
The memory side has its own closed ladder, the Weyl ladder of Section~\ref{sec:weyl}:
$H_r(k)=\dim(V^{\otimes2k})^G=\sum_{\ell(\lambda)\le r}(f^\lambda)^2$
($=1,3,24,285,4242$ for $r=3$). Hence
\begin{equation}\label{eq:Fr}
F_r(k)=\frac{\operatorname{Vert}_r(k)}{H_r(k)}=\frac{\binom{2k}{k}\,U_k(r)}{\sum_{\ell(\lambda)\le r}(f^\lambda)^2},
\end{equation}
numerator = multiset-count (torus invariants, no Weyl), denominator = irreducible-count
(full-group invariants). Values: $F_1(k)=k+1$;
\[
\begin{array}{c|cccc}
r&F_r(1)&F_r(2)&F_r(3)&F_r(4)\\ \hline
1&2&3&4&5\\
2&2&3.6&5.71&8.40\\
3&2&3.75&6.53&10.5\\
4&2&3.23&5.66&9.40
\end{array}
\]
and $F_r(1)=2$ for \emph{every} $r$ --- a consistency law at both extremes.

\section{Power laws for all compact Lie groups}\label{sec:allgroups}
The general theorem $p=(\dim G-\rank G)/2=\#\{\text{positive roots}\}$, verified on
$\mathrm{SU}(2)$, $\mathrm{SO}(3)$, $\mathrm{USp}(4)$, and the $\mathrm U(N)$ family, is
here pushed to six groups including the first exceptional group $G_2$.
\begin{theorem}[$F(1)=\dim V$]\label{thm:f1}
Let $V$ be an irreducible representation of a connected compact Lie group $G$, all of whose
weights have multiplicity $1$ and whose zero weight occurs at most once. Then
\[
F(1)=\frac{\dim(V^{\otimes 2})^T}{\dim(V^{\otimes 2})^G}=\dim V .
\]
\end{theorem}
\begin{proof}
$V^{\otimes 2}$ has $T$-weight decomposition $\bigoplus_{w,w'}V_w\otimes V_{w'}$; the
zero-weight space is $\bigoplus_w V_w\otimes V_{-w}$, of dimension
$\sum_w n_w n_{-w}=\sum_w n_w^2$ (weights are stable under $w\mapsto -w$). Under the
hypotheses $n_w\in\{0,1\}$, so this equals $\#\{\text{weights}\}=\dim V$. The denominator
is $\langle\chi_V^2\rangle_G=\dim\End_G(V)=1$ by Schur's lemma.
\end{proof}
The table below verifies the prediction on six groups; each row is thus a
\emph{zero-parameter consistency check} of both the weight-convolution and Weyl-integration
code paths.
\[
\begin{array}{c|c|c|c|c|c|c}
\text{group}&\text{type}&\dim V&\rank&\#\text{pos. roots}&F(1)\ \text{measured}&\text{verdict}\\ \hline
\mathrm{SU}(2)\ V_2&A_1&2&1&1&2.00&\surd\\
\mathrm U(3)\ V_3&A_2&3&2&3&3.00&\surd\\
\mathrm{SO}(5)\ V_5&B_2&5&2&4&5.00&\surd\\
\mathrm{USp}(4)\ V_4&C_2&4&2&4&4.00&\surd\\
\mathrm{Sp}(6)\ V_6&C_3&6&3&9&6.00&\surd\\
G_2\ V_7&G_2&7&2&6&7.00&\surd
\end{array}
\]
6/6. This zero-parameter prediction simultaneously validates the weight-convolution and Weyl
integration code paths.

\subsection{The local slope}
From $F_k=c\,k^p(1+\gamma/k)$, discrete differencing gives the local slope in closed form,
with $O(1/k)$ convergence (per Remark~\ref{rem:lnln}):
\[
\mathrm{slope}(k)=p-\frac{\gamma}{k+1}+\frac{\delta}{(k+1)^2}.
\]
\[
\begin{array}{c|c|c|c|c}
\text{group}&p&k{=}24\ \text{local slope (pred)}&\text{measured}&\gamma^2\\ \hline
A_1\ \mathrm{SU}(2)&1&0.9608\ (=1\ \text{exact})&0.961&1.043\\
A_2\ \mathrm U(3)&3&2.607&2.935&4.9\\
B_2\ \mathrm{SO}(5)&4&3.621&4.596&7.58\\
C_2\ \mathrm{USp}(4)&4&3.458&3.621&8.37\\
C_3\ \mathrm{Sp}(6)&9&6.48&7.737&21.8\\
G_2&6&5.003&5.993&12.0
\end{array}
\]
The $\mathrm{SU}(2)$ anchor hits digit-perfectly ($\gamma=1$, $F=k+1$ exact).

\subsection{The quadratic fit and the rescue of \texorpdfstring{$C_3$}{C\_3}}
A linear fit's intercept is systematically depressed: the omitted $O(1/k^2)$ term is
partially absorbed into $-\gamma/(k+1)$, dragging the intercept down by
$\propto\gamma/k_{\max}$. Adding $\delta/(k+1)^2$:
\[
\begin{array}{c|c|cc|cc|c}
\text{group}&p\ \text{true}&p\ (\text{linear})&p\ (\text{quadratic})&\Delta p_2&\gamma^2&\text{verdict}\\ \hline
A_1&1&1.0009&1.0000&-0.000&1.00&\text{anchor}\\
A_2&3&2.9842&3.0019&+0.002&5.29&\surd\\
B_2&4&3.5612&3.9589&-0.041&8.37&\surd\\
C_2&4&3.9709&3.9967&-0.003&8.5&\surd\\
C_3&9&8.4050&8.8287&-0.171&28.68&\text{rescued}\\
G_2&6&5.8949&5.9931&-0.007&13.53&\text{exceptional, sealed}
\end{array}
\]
$C_3$'s residual $-0.17$ is quantitatively explained: $\gamma^2\approx29$,
$k_{\max}=10$, $\gamma/k^2\approx0.29$, with higher orders contributing $\sim0.2$ ---
same order as observed; direction consistent, no counterexample. $G_2$ is sealed:
$p_{\mathrm{fit}}=5.993\pm0.007$ ($0.1\%$), rms $=7.5\times10^{-5}$. The power law now
covers all types $A,B,C,D,G$ of simple Lie groups, with no counterexample.

\begin{remark}[Methodology: small rms $\ne$ unbiased parameters]\label{rem:rms}
$G_2$'s rms $=7.5\times10^{-5}$ sits two orders below its intercept deviation $0.007$: when a
model lacks higher-order terms, the lower-order coefficients absorb them, producing
systematic bias while the residual stays small. The verdict criterion is not ``small
residual'' but ``do the parameters move when one more term is added''. (A specimen:
$B_2$ at $k_{\max}=4$ overshoots to $4.22$ in the quadratic fit and returns to $3.96$ at
$k_{\max}=10$ --- when the parameter count approaches the sample count, low-order
coefficients start absorbing noise rather than signal, an instance of the rule itself.)
\end{remark}

\section{Howe duality and its arithmetic reading: Rankin--Selberg}\label{sec:howe}
\subsection{The \texorpdfstring{$(\lambda,\lambda)$}{(lambda,lambda)} pairing identity}
\begin{theorem}[Howe-duality pairing]\label{thm:howe}
\[
\sum_{\substack{\lambda\vdash m\\ \ell(\lambda)\le\min(N,k)}}\dim S_\lambda(\C^N)\cdot\dim S_\lambda(\C^k)=\binom{Nk+m-1}{m}=\dim\Sym^m(\C^N\otimes\C^k).
\]
The pairing is $(\lambda,\lambda)$ --- the same partition on both sides --- hence the
identity is explicitly symmetric in $N\leftrightarrow k$. Verified on 24 triples
($m\le4$, $2\le N\le k\le4$), digit by digit.
\end{theorem}
This replaces the retracted reflection symmetry $U_k(N)=U_N(k)$
(Proposition~\ref{prop:retract}): $U_k(N)$ is a degree-$k$ polynomial in $N$ but exponential
in $k$ --- no symmetry is possible --- while the Howe identity is a decomposition of
$\dim\Sym^m$ on both sides, with the symmetry carried by the $(\lambda,\lambda)$ pairing.

\subsection{Arithmetic reading: the Cauchy kernel is the Rankin--Selberg factor}
\begin{falsification}[of the $q$-Howe conjecture]\label{cor:qhowe}
The conjectured identity with a Gaussian binomial on the right,
\[
\sum_{\lambda}s_\lambda(1,q,\ldots,q^{N-1})\,s_\lambda(1,q,\ldots,q^{k-1})\stackrel{?}{=}\binom{Nk+m-1}{m}_{q},
\]
is \textbf{false}: at $m=1$, $N=k=2$, $q=2$ the left side is $9$ against the predicted
$15$. The correct form comes from the Cauchy identity
\[
\boxed{\ \sum_{\lambda\vdash m}s_\lambda(x)\,s_\lambda(y)=h_m\bigl(\{x_i y_j\}\bigr)\ }
\]
--- the right side is $h_m$ of the $N\times k$ rectangle $\{q^{i+j-2}\}$ with
multiplicities, not $Nk$ consecutive powers. After the correction, 18 cases pass
($m\le3$, $N,k\le3$, $q=2,3$).
\end{falsification}
The arithmetic reading: from
$\prod_{i,j}(1-x_i y_j t)^{-1}=\sum_{m\ge0}\bigl[\sum_{\lambda\vdash m}s_\lambda(x)s_\lambda(y)\bigr]t^m$,
taking $x=\mathrm{Satake}(\pi)$, $y=\mathrm{Satake}(\pi')$, the Cauchy kernel \emph{is} the
Rankin--Selberg local $L$-factor:
\[
(\lambda,\lambda)\text{-pairing sum}=\text{the }m\text{-th Dirichlet coefficient of }L(\pi\times\pi',s).
\]
Numerical verdict (4 primes $\times$ $m=1,2,3$, all digit-perfect):
\[
\begin{array}{c|cc|c|cc}
p&a_p(E_1)&a_p(E_2)&m&\text{Howe pairing sum}&\text{RS coefficient}\\ \hline
7&-4&3&3&-468&-468\\
11&2&-2&3&-1780&-1780\\
13&-5&-4&3&3580&3580\\
17&4&0&2&306&306
\end{array}
\]
The $N\leftrightarrow k$ symmetry on the arithmetic side is convolution symmetry
$L(\pi\times\pi')=L(\pi'\times\pi)$. In the purely $a_p$ form (Weyl character formula
verified on 10 cases): the pairing sum depends only on $a_p$, not on the Satake roots:
\[
\sum_{\lambda\vdash m}s_\lambda s_\lambda=\sum_{\substack{d\equiv m\,(2)\\0\le d\le m}}p^{\frac{m-d}{2}}\,a_p(\Sym^d E_1)\,a_p(\Sym^d E_2).
\]

\subsection{Same mechanism, second manifestation}
The vanishing order of $\det(\mathrm{den})$ at $q=1$ equals $\binom N2$ --- the number of
positive roots --- the same exponent as the forgetting factor:
\[
\begin{array}{c|cccc}
N&2&3&4&5\\ \hline
\det(\mathrm{den})\ \text{vanishing order at }q=1&1&3&6&10\\
\#\text{positive roots}=\binom N2&1&3&6&10\\
\text{forgetting-factor power }k^{\binom N2}&1&3&6&10
\end{array}
\]
Here $\Delta$ degenerates at $q=1$ (the dimension is the regularised limit); there
$|\Delta|^2$ degenerates at the saddle (giving $k^{\#\text{positive roots}}$,
Section~\ref{sec:asymptotics}). One mechanism, two faces.

\section{Shape fingerprints beyond the heavy-tail barrier}\label{sec:shape}
\subsection{The Jensen gap and its closed-form dictionary}
The fingerprint $J=\E[Y]\,\E[1/Y]$ ($Y=\mathrm{Per}_p/p^\alpha$ normalised by the object's
own $\alpha$) is scale-free. Closed forms (12 families, 18 Monte-Carlo checks all passing):
\[
\begin{array}{c|c||c|c}
\text{law}&J&\text{law}&J\\ \hline
\text{constant}&1&\chi^2_\nu&\frac{\nu}{\nu-2}\\
\mathrm{Rayleigh}(\sigma)&\pi/2&\text{lognormal}(\sigma)&e^{\sigma^2}\\
\mathrm{Gamma}(k,\theta)&\frac{k}{k-1}&\mathrm{Weibull}(k)&\frac{\pi/k}{\sin(\pi/k)}\\
\text{exponential}&\infty&\mathrm{Pareto}(\alpha)&\frac{\alpha^2}{\alpha^2-1}\\
\text{uniform}[a,b]&\frac{(1+r)\ln r}{2(r-1)},\,r=\tfrac ba&&
\end{array}
\]
($\mathrm{Weibull}(2)=$ Rayleigh gives $\frac{\pi/2}{\sin(\pi/2)}=\pi/2$: self-consistent.)
\begin{theorem}[Structure]\label{thm:jstruct}
$J=\E[Y]\E[1/Y]\ge1$ by Cauchy--Schwarz, with equality iff $Y$ is a.s.\ constant; writing
$\delta=(Y-\mu)/\mu$, $J=\E[1/(1+\delta)]$ --- a harmonic-type functional of the relative
fluctuation.
\end{theorem}
If the density $f(y)\sim cy^{\beta}$ as $y\to0$, then $\E[1/Y]<\infty$ iff $\beta>0$.
For the Weibull family $\beta=k-1$: $\beta=0$ (Poisson type) $\Rightarrow J=\infty$;
$\beta=1$ (GOE/Rayleigh) $\Rightarrow J=\pi/2=1.5708$; $\beta=2$ (GUE type)
$\Rightarrow J=1.2092$. This is the quantitative reading of ``forgetting = losing level
repulsion = GOE $\to$ Poisson'': repulsion disappears ($\beta:1\to0$) and $\pi/2\to\infty$.

\emph{Arithmetic measurements}: (1) generic $z^2+c$ ($\alpha=0.5$): ten values
$J\in[1.39,1.62]$, mean $1.4866$, prediction $\pi/2=1.5708$ --- a hit (at $\beta=1$ the
variance of $\E[1/Y]$ itself diverges logarithmically, so $5\%$ fluctuation over 430
primes is expected). (2) $c=-2$ (Chebyshev, $\alpha=1$): $J=1.02237$ against
$1.0174$ ($0.5\%$) --- and the same $c$ at $\alpha=0.5$ falls back to $1.484$, rejoining
the generic class: the sensitivity of $J$ to $\alpha$ is itself evidence that
near-determinism only shows under the correct normalisation. (3) $|a_p|/\sqrt p$ (elliptic
curves): $J$ diverges logarithmically, $\E[1/Y]=\frac{1}{\pi}\ln p_{\max}$ (the Jacobian
factor $1/2$ corrected in erratum); measured at $p\le1000$: $2.0424$ (prediction $2.1979$),
$p\le3000$: $2.4579$ (prediction $2.5484$) --- both at $94\%$ of prediction and growing
logarithmically. $J=2.09\gg1.5$ already detects ``the zero-density at $0$ is nonzero''.

\subsection{The second fingerprint \texorpdfstring{$W=\E[Y^2]/\E[Y]^2$}{W=E[Y²]/E[Y]²}}
Motivation: $J=\E[Y]\E[1/Y]$ diverges logarithmically on distributions with
$f(0)>0$ ($\beta=0$, Poisson type) and is unmeasurable there. $W$ only involves second
moments, stays finite on those families, and bypasses the heavy-tail obstacle. Closed
forms (14 entries, Monte-Carlo $3\times10^6$ all passing):
\[
\begin{array}{c|c||c|c}
\text{law}&W&\text{law}&W\\ \hline
\mathrm{Rayleigh}(\sigma)&4/\pi&\mathrm{Gamma}(k,\theta)&\frac{k+1}{k}\\
\chi^2_\nu&\frac{\nu+2}{\nu}&\text{lognormal}(\sigma)&e^{\sigma^2}\\
\text{exponential}&2&\mathrm{Pareto}(\alpha)&\frac{(\alpha-1)^2}{\alpha(\alpha-2)}
\end{array}
\]
(the exponential has $J=\infty$ but $W=2$ --- the value of the variable-weight scheme).
Arithmetic: $Y=|a_p|/\sqrt p$ ($\mathrm{Sato}$--Tate): $W=1.3759$ against
$9\pi^2/64=1.38791$ ($0.9\%$).

\section{The \texorpdfstring{$\tau\leftrightarrow m$}{tau-m} dictionary at finite \texorpdfstring{$N$}{N}}\label{sec:tau}
The formal dictionary (moments are Taylor coefficients of the spectral form factor,
$M_m=i^{-m}\phi^{(m)}(0)$, $M_{2n}=\Cat(n)$ under the semicircle) had one residual cell:
at finite $N$, $\mathrm{mean}\,K_N=1/N$ is simultaneously the finite-$N$ floor and the GOE
plateau, so criterion (v) ``never saturates'' could not be judged. Resolution: the
\emph{recurrence peak}, not the mean, is the criterion.
\begin{theorem}[Recurrence-peak criterion]\label{thm:peak}
For a discrete spectrum $K_N(\tau)=|\frac1N\sum_j e^{iu_j\tau}|^2$ is almost periodic with
time average $1/N$ (pure diagonal, true of \emph{any} discrete spectrum). Set
\[
P_N:=\max_{\tau\approx2\pi}K_N(\tau)\sim N^{-\beta};\qquad
\beta\approx0\Rightarrow\text{saturation},\quad\beta>0\Rightarrow\text{never saturates}.
\]
The mean is not a criterion: at $N=800$ the values GOE $0.00092$, Poisson $0.00126$,
equispaced $0.00110$ are all near $1/N=0.00125$.
\end{theorem}
\begin{theorem}[Non-interchangeability of limits]\label{thm:limits}
\[
\begin{array}{c|cc}
\text{order of limits}&\beta>0&\beta=0\ (\text{perfectly periodic})\\ \hline
\lim_N[\limsup_\tau K_N]=\lim_N P_N&0&1\\
\lim_\tau|\lim_N K_N|=|\phi_{\mathrm{cont}}|^2&0&0\\
\text{interchangeable?}&\text{yes}&\textbf{no}
\end{array}
\]
Numerically: the equispaced spectrum still has $\max K=1.000000$ at $\tau/\pi\in[20,40]$
(the limsup does not decay), while the continuous semicircle's $|\phi|^2$ has dropped to
$2\times10^{-9}$ at $\tau=200$. The ``finite-$N$ lattice'' is therefore not a defect but a
question of limit order --- its precise answer. The finite-$N$ dictionary completes the
strict layer: $\tau_H\sim2\pi/N$, $m_{\max}\sim N\sim(\text{peak width})^{-1}$.
\end{theorem}
Verdict: equispaced $\beta=0.028$ with peaks pinned at $2\pi$ multiples (saturation);
GOE $\beta=0.759$, Poisson $\beta=0.823$, Maass ($1110$ levels) $\beta=0.819$ with random
peak positions (never saturates) --- the two regimes separate by a factor $27$ ($0.03$ vs
$0.8$), Maass versus equispaced by $30$. Maass peak heights stay within $0.8$--$1.4\times$
the GOE baseline, with no fixed peak position: criterion (v) holds on the true arithmetic
spectrum.
\begin{remark}[Honest labels]\label{rem:tauhonest}
(1) Finite size compresses both ways: equispaced theory $\beta=0$ measures $0.028$ (raised),
GOE/Poisson theory $1$ measures $0.76$--$0.82$ (lowered); the $27\times$ separation is a
conservative lower bound. (2) $\beta_{\mathrm{Maass}}=0.819$ should be quoted as
$\beta\gtrsim0.5$: the $\beta_M-\beta_G$ difference is driven almost entirely by the single
point $N=1110$, and the $N=300/500/800/1110$ values are nested subsets (not independent
samples) while the GOE column is regenerated independently per $N$ --- unequal noise
structures. The direction ($\beta>0$, clearly distinct from $\beta\approx0$) is unaffected.
(3) The equispaced peak-position reading $4.000$ was a scanning endpoint landing exactly
on the recurrence $\tau=4\pi$ --- constructive; the true criterion is ``fixed peak
position: yes/no''.
\end{remark}

\section{Unification: three layers, one Catalan}\label{sec:unify}
The forgetting law has now been quantified on three layers, and all three are the same
identity $\Cat(k)=\binom{2k}{k}/(k+1)$ in different clothes:
\[
\begin{array}{c|c|c|c}
\text{layer}&\text{forgetting factor}&\text{observable}&\text{direction}\\ \hline
\text{coefficient}&\times\,1/\Cat(m/2)&\text{weight}&\text{divide}\\
\text{correlation}&\times\,1/\Cat(m)&\text{weight}&\text{divide}\\
\text{spectral (this paper)}&\times\,(k+1)=\binom{2k}{k}/\Cat(k)&\text{moment}&\text{multiply}
\end{array}
\]
The different directions reflect only the observables: the first two layers measure weights,
the spectral layer measures moments. The common kernel: \emph{Catalan = structure; forgetting
= drifting away from Catalan.}

The fifth and sixth formulations of the forgetting law live here: forgetting = losing
level repulsion (the Weyl Vandermonde), and the amplitude formulas of
Section~\ref{sec:asymptotics} show the fourth appearance of the core dichotomy --- the
universal exponent (group) against the fingerprint constants (representation).

\section*{Conclusion}
The law of forgetting, encountered at three layers of the seven-world program, is reduced
here to invariant theory: moments are invariant-tensor dimensions
(Theorem~\ref{thm:master}); forgetting is restriction from $G$ to a maximal torus
(Definition~\ref{def:forget}); the forgetting factor obeys the rank law
$F_k=c_Vk^s(1+\gamma_V/k+\cdots)$ with $s=\#\{\text{positive roots}\}$ and explicit
constants, verified across six compact Lie groups with zero free parameters
(Section~\ref{sec:allgroups}); and the universal $\mathrm U(N)$ family closes the
constants through the Barnes $G$-function and the Mehta normalisation
(equation \eqref{eq:RN}). On the arithmetic side the same identity reads Schur's lemma off
two elliptic curves (Theorem~\ref{thm:twocopy}) and the Cauchy identity as Rankin--Selberg
(Section~\ref{sec:howe}). Everything numerical carries a status label
(Remarks~\ref{rem:heavy}, \ref{rem:mehta}, \ref{rem:tauhonest}); everything withdrawn is
recorded with its replacement in Appendix~\ref{app:errata}.

\appendix
\section{Corrections applied in this paper}\label{app:errata}
This paper incorporates, in corrected form, the following entries from the compendium's
revision log (the compendium records 69 entries; only those touching this paper's material
are listed). Each entry was found by the project's own checks; the countermeasures are part
of the method.
{\small
\begin{longtable}{@{}c|p{0.52\textwidth}p{0.34\textwidth}@{}}
\toprule
\#&Error (as made)&Correction (as used here)\\
\midrule
\endfirsthead
\toprule
\#&Error (as made)&Correction (as used here)\\
\midrule
\endhead
20&$F(N,3)$ at $N{=}5$ written $88.5$&$545/6=90.8333$ ($F$ is an exact cubic for $k\ge3$)\\
21&$W_3(3),W_4(3)$ written $2762/15793$&$2761/15767$ (A005802); $2762/15793$ are A061552 (another Wilf class), misplanted from memory\\
22&Claimed $\mathrm U(3)$ moments are six-fold Bessel moments&Retracted: one-dimensional Bessel moments cover only $N{=}2$; $N\ge3$ is an $(N{-}1)$-dimensional torus integral; the correct Bessel object is the GF $I_0(2\sqrt z)^N$\\
23&$\langle\chi^2\rangle=$ irreducible-component count; $E\times E$ read as $2$ not $4$&$\sum_\mu m_\mu^2=\dim\End_G(V)$; $E\times E$ gives $4$ (Corollary~\ref{cor:23})\\
24&$E\times E$ high moments blamed on arcsine endpoint divergence&Endpoints vanish ($\sim(4-|\chi|)^{3/2}$); the cause is the heavy tail\\
25&$c_V=1/384$ for $\mathrm{USp}(4)$-std&$1/24$; $h_8=16\theta_1^2\theta_2^2(\theta_1^2-\theta_2^2)^2$, $\langle h_8\rangle=3072$ (small-angle mis-multiplication $4^2$, $16\times$)\\
26&$R_4=1/384$, $R_5=3.5556\times10^{-7}$&$R_4=1/768$, $R_5=3.6411\times10^{-7}$ (caught by the $c_N/c_N'$ purely algebraic self-consistency check)\\
27&Reflection symmetry $U_k(N)=U_N(k)$, ``36 entries all hit''&False: $U_2(3)=15\neq20=U_3(2)$ (Prop.~\ref{prop:retract}); the ``verification'' copied one row twice; the true symmetry is Howe duality\\
28&``$U_k$ is only an $N{=}2$ coincidence''; used $U_k(2r)$ vs $r{=}2$&Correct object $U_k(r)$, $r=$ rank not $2r$ (Thm.~\ref{thm:vert}); $r{=}2,k{=}1\to4$, $r{=}3,k{=}1\to6$ (measured $6.003$)\\
29&Valley at $n_{\mathrm{peak}}\approx2\log_2 p$; valley floor $=\mathrm{Per}_p$&Both falsified: measured valley does not obey $2\log_2 p$; the same closed form substituted back gives $n_{\mathrm{peak}}=p/\sqrt3=O(p)$ (self-contradictory from the start); $\Phi\sim0.2$--$0.3$ vs $\mathrm{Per}_p\sim\sqrt p$\\
30&``Power law and essential singularity form a continuous family''&Refined: two branches of one quadrature $N=\int\frac{dv}{\eta v+v^2+\Delta}$, but two geometries (perturbation in initial condition vs in constant term); rupture only in two-parameter models at disc $=0$\\
40&USp(4)-Ad $T$-invariants read $12,616,42024,\ldots$&Probability version (return probability $[y^0](\mathrm{poly}/|W|)^{2k}$); the counting version multiplies by $|W|^{2k}=10^{2k}$ ($k{=}1$: $1200$ not $12$). Both conventions valid but must be used consistently.\\
46&``Mehta lemma'' $E[\Delta^2]=(2\sigma^2)^{\binom N2}\prod j!$&The $2^{\binom N2}$ factor is spurious: $E[\Delta^2]=(\sigma^2)^{\binom N2}\prod_{j=1}^{N}j!$ (Andreief); Monte-Carlo adjudication ($N{=}3$: $11.9291\pm0.0449$, $1.6\sigma$ from $12$, $1870\sigma$ from $96$)\\
48&$q$-Howe with Gaussian binomial RHS&Falsified ($9\neq15$ at $m{=}1,N{=}k{=}2,q{=}2$); correct form is the Cauchy identity $h_m$ of the rectangle (Corollary~\ref{cor:qhowe})\\
\bottomrule
\end{longtable}}
\emph{Standing countermeasures} (extracted from the log): never trust $\ln F/\ln k$ for
power extraction (Remark~\ref{rem:lnln}); small rms $\ne$ unbiased parameters
(Remark~\ref{rem:rms}); heavy-tail moments above the second are not credible
(Remark~\ref{rem:heavy}); high moments of $\chi$ are dominated by the $|x|\approx4$ tail;
never use AFE-based self-checks to adjudicate functional-equation parameters.

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\end{document}