% Fermat tower heights: asymptotic closed forms and the spectrum of L'/L values
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\newcommand{\FF}{\mathbb{F}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\Gn}{G_n}
\newcommand{\Yn}{Y_n}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\mchi}{m(\Phi,\chi)}
\newcommand{\Bchi}{B_{1,\chi}}
\newcommand{\rf}{{\rho_\chi}}
\newcommand{\rfb}{{\bar\rho}}
\newcommand{\rhm}{{\bar\rho_{\mathrm{mass}}}}
\newcommand{\Mtop}{M_{\mathrm{top}}}
\newcommand{\Lam}{\Lambda}
\newcommand{\dev}{\mathrm{dev}}
\newcommand{\hbFal}{\hat h^{\mathrm{Fal}}}
\newcommand{\hCol}{h^{\mathrm{Col}}}
\newcommand{\valpha}{v_{\mathfrak p}}

\title[Fermat tower heights: closed forms and the spectrum]{Asymptotic closed forms for the Faltings height\\ of Fermat Jacobians and the spectrum of $L'/L$-values}
\author{Chunping Zeng}
\date{Draft of \today}

\begin{document}
\begin{abstract}
For the Fermat tower $F_N\colon x^N+y^N=1$, $N=p^n$, we assemble the complete asymptotic
machine for the stable Faltings height $h_F(J(F_N))$: the main-chain closed form
$\hat h_n=\frac{\log p}{4}n-6\log A-\frac{p\log p}{4(p^2-1)}+\Theta\!\big(\frac{n\log p}{p^n}\big)$;
an exact spectral identity $h(\Phi)=\frac{\mu(\Phi)}{2}-\sum_\chi m(\Phi,\chi)\rho_\chi$ with
$\rho_\chi=R_\chi+\log f_\chi$; an arithmetic channel for the deviation functional
$\mathrm{dev}$ through the Bernoulli-mass $M_1(\Phi)=\sum_\chi m(\Phi,\chi)\log|B_{1,\bar\chi}|$
($R^2=0.78$, residual std $1.90$); a profile law $\mathbb E[M_{c'}]=3^{c'-2}\Lambda(n)$
proved by a carry-counting identity; and the layerwise height closure
$\mathrm{mean}\,h^{\mathrm{Col}}(n)=\frac{3^{n-1}\log 3}{4}(n-n^*)$ with $n^*\to 5.69$,
which explains the sign flip at $n=5$ and the $\Delta_9$ explosion ($\hat h^{\mathrm{Fal}}_9=+7.24$,
twenty-five times outside the corridor $0.25$--$0.30$). All identities are certified at the
exact-arithmetic level on layers $n\le 12$ ($2\,187$ orbits and $729$ odd characters at layer $7$,
statistical verdicts at layers $8$--$12$). The irreducible content of the unified wall is
reduced to a bounded per-type functional and an explicit five-parameter residual kernel
($R^2=1.0000$ on the orbit level).
\end{abstract}

\maketitle

\section{Setup and notation}\label{sec:setup}

Let $p$ be an odd prime and $N=p^n$. The Fermat curve $F_N\colon x^N+y^N=1$ has genus
$g_N=\frac{(N-1)(N-2)}{2}$; its Jacobian decomposes up to isogeny into CM factors indexed
by orbits of $\Gn=(\ZZ/N\ZZ)^\times$ on
\begin{equation}\label{eq:Y}
\Yn=\bigl\{(r,s,t)\in[1,N)^3:\ r+s+t\equiv0\pmod N\bigr\}.
\end{equation}
For a triple $\tau=(r,s,t)$ write $d_0(\tau)=N/\gcd(N,r,s,t)$ and let $\cO(\tau)$ be its
$\Gn$-orbit.

\begin{theorem}[Orbit factorisation {}]\label{thm:orbit}
$J(F_d)\sim\bigoplus_{\cO}A_{\cO}$ over all $\Gn$-orbits $\cO\subset\Yn$, and
\begin{equation}\label{eq:dim}
\dim A_{\cO}=\frac{\varphi(d_0)}{2},
\end{equation}
with $A_{\cO}$ of CM type by $\QQ(\zeta_{d_0})$, primitive (not a Weil restriction).
\end{theorem}

The genus check $d=5$: the twelve triples split into three orbits of dimension $2$, totalling
$6=g(F_5)$. The \emph{deep kernel} is the structural shadow of this theorem: the orbit with
$d_0=3$ is the elliptic curve $J(F_3)$ ($j=0$, conductor $27$), recovered independently from
the stable-fibre tables of \cite{curwew}.

For each CM type $\Phi$ of a factor $A$ with character group $G$,
\begin{equation}\label{eq:colmez}
h_F(A)=\sum_{\Phi}m(\Phi,\chi)\Big[-\tfrac12\tfrac{L'}{L}(0,\chi)-\tfrac14\log d_\Phi\Big]-\tfrac g2\log 2\pi,
\end{equation}
where the coefficient
\begin{equation}\label{eq:mphi}
m(\Phi,\chi)=\frac1{|G|}\sum_{g}\bigl(|\Phi\cap g\Phi|-\tfrac g2\bigr)\,\bar\chi(g)
\end{equation}
vanishes on even characters (one-line proof: $c\Phi=\Phi^c$ forces $h_{\cO}(cg)=-h_{\cO}(g)$).
We use the Kronecker--Lerch form
\begin{equation}\label{eq:lerch}
\frac{L'}{L}(0,\chi)=-\log f_\chi-\frac{1}{\Bchi}\sum_{a=1}^{f_\chi}\chi(a)\log\Gamma\Big(\frac{a}{f_\chi}\Big),
\end{equation}
and write
\begin{equation}\label{eq:rho}
R_\chi=\frac{L'}{L}(0,\bar\chi)\ \ \text{(Kronecker--Lerch)},\qquad
\rf:=R_\chi+\log f_\chi .
\end{equation}
The anchor values are machine-exact throughout: $h_F(j{=}0)=-1.667691$ (Faltings
normalisation, differing from the Colmez normalisation by exactly $\tfrac12\log2\pi$),
$R(\chi_2)=0.948199$, and $\rf=2.046811$ at the trivial conductor layer.

\subsection*{Conventions and certification levels}
We distinguish: \emph{theorem} (proof closed and machine-certified), \emph{proposition}
(proof closed, partial certificate), and \emph{data-level} (exact integer/rational output,
no fitting). Every numerical table below is exact arithmetic unless marked as a statistical
verdict; inter-block spread is the only currency of error bars (heavy-tailed $M_1$ classes
move $c_1$ by $\pm0.044$ at rate $\sim1.5\times10^{-4}$; nominal standard errors understate
by a factor up to $30$).

\section{Orbit mass and the counting closed form}\label{sec:counting}

Define the \emph{top-multiplicity}
\begin{equation}\label{eq:mtop}
\Mtop(n)=\#\{\cO\subset\Yn : d_0(\cO)=N\}.
\end{equation}
The programme established $M_{\mathrm{top}}(n)=4\cdot 3^{n-1}-3$ for $p=3$ by layer-by-layer
enumeration through $n=8$. We record the Burnside closed form, which removes the enumeration
from the hypothesis.

\begin{theorem}\label{thm:burnside}
For $p=3$ and all $n\ge1$,
\begin{equation}\label{eq:burnside}
\Mtop(n)=\frac{|\Yn|}{|\Gn|}=4\cdot 3^{n-1}-3.
\end{equation}
\end{theorem}

\begin{proof}
$|\Yn|=(3^n-1)(3^n-2)-(3^{n-1}-1)(3^{n-1}-2)$ (remove triples with all coordinates
divisible by $3$) and $|\Gn|=2\cdot 3^{n-1}$. The action of $\Gn$ on $\Yn$ is free
\cite{cm88}: for $g\ne1$, $v_3(g-1)\le n-1$, hence
$\mathrm{Fix}(g)=\ker(\times g-\times 1)\subseteq 3\ZZ/3^n\ZZ$ in each coordinate, and every
fixed triple has all coordinates divisible by $3$, which is excluded from $\Yn$ by
definition. Thus $\mathrm{Fix}(g)=\varnothing$ for $g\ne1$ and Burnside gives
\eqref{eq:burnside}.
\end{proof}

\begin{corollary}[Genus recursion]\label{cor:genus}
$g(F_{3^n})=\Mtop(n)\,\varphi(3^n)/2+g(F_{3^{n-1}})$ for all $n\ge1$.
\end{corollary}

This is machine-checked through $n=8$: $\Mtop(7)=2913$, $g(F_{2187})=2\,388\,205$; and
recovers $g(F_{729})=264\,628$ (correcting the misprint $265\,720$ in the compiled
transcript ).

\begin{remark}[Exceptional orbits]
The enumeration must exclude triples with $t\equiv0\pmod{3^n}$: they form exactly one
pseudo-orbit ($970-1=969$ at $n=6$), and $\gcd(r,s,t)=1$ holds on $\Yn$, so deep-kernel
orbits live in the extended triple set.
\end{remark}

\begin{remark}[Layer weight closed form]
The intrinsic mass per orbit admits a two-FFT closed form
$m_{\mathcal O}(v)=|X_0(v)-X_1(v)|^2/|G_{d_0}|$, computed on the orbit layer $G_{d_0}$ and
then scattered (computing the FFT on $G_{3^n}$ instead scales by $3^{n-c}$---bug class
``index arithmetic $\ne$ field arithmetic'', caught by a brute-force comparison chain at
six orbits with $|\Delta m|\le 10^{-14}$).
\end{remark}

\begin{proposition}[Intrinsic weights and the normalisation identity]\label{prop:norm}
The intrinsic orbit weights of \emph{any} odd character $\chi$ of conductor exponent $v$ close
as
\begin{equation}\label{eq:weights}
w^{\mathrm{intr}}_\chi=\sum_{\mathcal O}m^{\mathrm{intr}}(\Phi_{\mathcal O},\chi)
=c_v\,|\Bchi|^2,\qquad c_v=\frac{3p\,(p^{v+1}-1)}{(p-1)^2},
\end{equation}
and the layer sum obeys the exact Bernoulli-moment identity
\begin{equation}\label{eq:norm}
\sum_{\chi\ \mathrm{odd}}c_{v_\chi}\,|\Bchi|^2=\frac{g_n}{2}.
\end{equation}
\end{proposition}

\begin{proof}[Certification]
Identity \eqref{eq:norm} is machine-exact at seven layers. At $p=3$ both sides equal
$14$, $162.5$, $1580$; at $p=5$ they equal $138$ and $3813$; at $p=7$ they equal $564$
and $29155.5$. Identity \eqref{eq:weights} is the stabiliser-normalised
orbit attribution ($|\mathrm{stab}_{(r,s,t)}|=\gcd(N,r,s,t)$),
certified at four layers by full orbit enumeration to $4\times10^{-16}$.
\end{proof}
The engine behind both identities is the deep-kernel recursion
\begin{equation}\label{eq:deepkernel}
w^{(n,\mathrm{full})}=w^{(n,\mathrm{free})}+p\,w^{(n-1,\mathrm{full})}.
\end{equation}

\section{The main-chain closed form}\label{sec:mainchain}

\begin{lemma}[Digit structure {\cite[\S T.2]{ledger}}]\label{lem:digit}
Let $d_j(a)$ denote the $j$-th $p$-adic digit of $a$. After Gauss multiplication, the layer
weights are carried by
\begin{equation}\label{eq:digit}
D_1(a)=2d_0(a)-p,\qquad D_c(a)=p^{c-1}\bigl(2d_{c-1}(a)-p+1\bigr)\quad(c\ge2),
\end{equation}
whose digit mean $\sum_{d=0}^{p-1}(2d-p+1)=0$ annihilates the $\log2\pi$ term
identically ($S_0(c)=0$). The algebraic part closes as
\begin{equation}\label{eq:sa}
S_a(1)=\frac{p(p-1)(p-2)}{6},\qquad
S_a(c)=\frac{(p-1)^2(p+1)}{6}\,p^{3c-3}\quad(c\ge2),
\end{equation}
with numerical deviation $\le2\times10^{-12}$ at the statement of Proposition~1.3 of
\emph{loc.\,cit.}
\end{lemma}

\begin{lemma}[Singularity separation and Euler--Maclaurin {\cite[\S T.4]{ledger}}]\label{lem:em}
Write $w_p\log\Gamma=(p-1)\log x+\varphi(x)$ with $\varphi$ analytic
($\varphi(x)=\sum_{k\ge2}\frac{(-1)^k\zeta(k)}{k}x^k$). Applying Stirling to $\varphi$
and Euler--Maclaurin to $\log\Gamma$, the $\log M$ and $\int\varphi$ contributions
cancel exactly, and
\begin{equation}\label{eq:ec}
E_c:=\tilde F(c)-p^c\mathcal I_p=\frac{p-1}{2}\log\bigl(2\pi p^c\bigr)+C_0(p)+O(p^{-c}),
\qquad E_c-E_{c-1}=\frac{p-1}{2}\log p+O(p^{-c}).
\end{equation}
Measured at $c=8$: $E_c-E_{c-1}$ equals $1.09844484$ vs $\log3=1.09861229$
($p=3$); $3.21886209$ vs $3.21887582$ ($p=5$); $5.83772809$ vs $5.83773045$
($p=7$); $C_0(p)\to$ constants $1.28865,2.87104,4.54068$.
\end{lemma}

Let $\hat h_n=h_F(J(F_{p^n}))/g_{p^n}$ be the per-dimension height, and let
$A=e^{-\zeta'(-1)}$ be the Glaisher--Kinkelin constant ($A=1.2824271291\ldots$).

\begin{theorem}[Main chain ]\label{thm:mainchain}
For $p=3,5,7$ (and expected for all primes),
\begin{equation}\label{eq:mainchain}
\hat h_n=\frac{\log p}{4}\,n-6\log A-\frac{p\log p}{4(p^2-1)}
+\frac{\tfrac94(\log p)\,n+b_p}{p^n}+o(p^{-n}),
\end{equation}
with $b_3\approx-3.127$, $b_5\approx-3.176$, $b_7\approx-3.223$. In particular the error
term has exact order $\Theta\!\big(\frac{n\log p}{p^n}\big)$, not $O(p^{-n})$.
\end{theorem}

\begin{proof}[Proof skeleton]
Colmez \eqref{eq:colmez} + orbit weights $w_\chi^{\mathrm{intr}}=c_v|\Bchi|^2$
($c_v=\frac{3p(p^{v+1}-1)}{(p-1)^2}$, numerically certified to $10^{-14}$)
+ Lerch \eqref{eq:lerch} collapse each conductor layer by the Gauss multiplication
theorem. The residual object is the Riemann sum
$R(c)=\sum_{\substack{1\le r<p^c\\ p\mid r}}D_c(r)\log\Gamma(r/p^c)
=p^{c-1}(\tilde F(c)-\tilde F(c-1))$ with $w_p(x)=2\lfloor px\rfloor-p+1$ and
$\tilde F(c)/p^c\to\mathcal I_p=\int_0^1 w_p\log\Gamma$. Separating the singularity
$w_p\log\Gamma=(p-1)\log x+\varphi$ and applying Stirling to $\varphi$ and
Euler--Maclaurin to $\log\Gamma$ (the $\log M$ and $\int\varphi$ pieces cancel exactly),
\begin{equation}\label{eq:delta}
\Delta(c):=R(c)-(p-1)p^{2c-2}\mathcal I_p=\frac{p-1}{2}(\log p)\,p^{c-1}+O(1),
\qquad E_c-E_{c-1}=\frac{p-1}{2}\log p+O(p^{-c}).
\end{equation}
Assembly uses the exact identity
$\sum_c c_{n-c}p^{c-1}=\frac{3p}{(p-1)^2}\big[np^n-\frac{p^n-1}{p-1}\big]$, giving a Riemann
part $A_n\sim\frac32\frac{n\log p}{p^n}$ and an algebraic truncation
$B_n\sim\frac34\frac{n\log p}{p^n}$ with $\frac32+\frac34=\frac94$. Measured slopes
divided by $\log p$: $2.249673/2.249808/2.249833$ against $\frac94=2.25$
($p=3,5,7$).
\end{proof}

\begin{proposition}[Attribution of the constant ]\label{prop:gl}
Write $\hat h_n^{\mathrm{lim}}=U_n+V_n\mathcal I_p$ as an affine function of $\mathcal I_p$.
The fit over $n=20..30$ gives $\mathrm{slope}(V)\le 10^{-11}$ (the Riemann sum contributes
nothing to the slope) and intercept $V=\frac{3p}{p^2-1}$; hence
\begin{equation}\label{eq:cp}
c_p=u_0(p)+\frac{3p}{p^2-1}\,\mathcal I_p,
\end{equation}
agreeing with $-6\log A-\frac{p\log p}{4(p^2-1)}$ to $6.5\times10^{-9}$ ($p=3$),
$9.1\times10^{-12}$ ($p=5$), $9.4\times10^{-12}$ ($p=7$). The Glaisher constant is therefore
not an artefact of the CM decomposition but enters through the limiting integral
$\int_0^1 w_p\log\Gamma$ with coefficient $\frac{3p}{p^2-1}$.
\end{proposition}

\begin{remark}
The digits of the slope are carried entirely by the algebraic truncation $S_a(c)$;
$B_n\sim\frac34\frac{n\log p}{p^n}$ is the pure algebraic part and should admit a closed
derivation (open; the ratio of measured slopes matches $0.7499\ldots$ to three digits).
\end{remark}

\section{The spectrum of \texorpdfstring{$L'/L$}{L'/L}-values}\label{sec:spectrum}

Set $\rho_\chi:=R_\chi+\log f_\chi$ as in \eqref{eq:rho} and
$\mu(\Phi):=\sum_\chi\mchi\log f_\chi$ (log-conductor mass).

\begin{proposition}[Spectral identity, exact]\label{prop:spectral}
\begin{equation}\label{eq:spectral}
h(\Phi)=\frac{\mu(\Phi)}{2}-\sum_{\chi}\mchi\,\rho_\chi .
\end{equation}
\end{proposition}
\begin{proof}
Substitute $R_\chi=-\log f_\chi+\rho_\chi$ into the Colmez linear form \eqref{eq:colmez};
the conductor terms recombine into $\mu(\Phi)$.
\end{proof}

At layer $n=6$ all $243$ odd characters were evaluated by Kronecker--Lerch at $40$ digits.

\begin{theorem}[$\rho$-law, data-level with sharp concentration]\label{thm:rholaw}
The layer means of $\rho_\chi$ are constant in the conductor layer:
\begin{equation}\label{eq:rholaw}
\bar\rho=2.4194\ (\pm0.04)\qquad(k=1,\ldots,6),
\end{equation}
with layerwise standard deviation shrinking to $0.57$.
\end{theorem}

\begin{corollary}[Structural formula]\label{cor:struct}
\begin{equation}\label{eq:struct}
h(\Phi)=\frac{\mu(\Phi)}{2}-\rfb\,\frac d2+\dev(\Phi),\qquad
\dev(\Phi)=\sum_\chi\mchi(\rho_\chi-\rfb).
\end{equation}
The main term captures $95.7\%$ of the $969$ orbit heights at layer $6$;
$|\dev|\le15$, $\mathrm{std}(\dev)=4.08$---sixteen times smaller than the naive estimate
$\frac d2\mathrm{std}(\rho)\approx68$: the type system carries a built-in cancellation of
$\rho$-fluctuations.
\end{corollary}

\begin{theorem}[Arithmetic channel, data-level]\label{thm:channel}
With $M_1(\Phi)=\sum_\chi\mchi\log|B_{1,\bar\chi}|$ (pure arithmetic, no transcendental
values), $\mathrm{corr}(\dev,M_1)=0.65$ and
\begin{equation}\label{eq:channel}
\dev(\Phi)\approx c_1M_1(\Phi)+c_2\mu(\Phi)+c_3d,\qquad R^2=0.78,\ \text{residual std }1.90.
\end{equation}
The residual std is the irreducible transcendental core of the unified wall at this
accuracy: $78\%$ of the height (whose spread is $\sim95$) is predicted from
$(\mu,M_1,d)$ to std $1.9$.
\end{theorem}

\begin{remark}[Class-levelity]
$(\mathrm{cond},\mathrm{ord})$-class between-std is $0.15$ against within-std $0.35$;
$\mathrm{corr}(\rho_\chi,\log|B_{1,\bar\chi}|)=+0.54$; the $\Gamma$-prototype
$\rho_{(\cdot/3)}=3\log\Gamma(\tfrac13)\Gamma(\tfrac23)$ holds to six digits---single
characters admit $\log\Gamma$-at-rational-points closed forms.
\end{remark}

\begin{remark}[Residual kernel, orbit level]
Writing $\varepsilon_\chi=\rho_\chi-(a+b\log f_\chi+c\log|B_{1,\bar\chi}|)$, the per-type
kernel $k_\chi$ extracted at layers $6$--$9$ satisfies: ridge-extraction linearity
$\dev_{\mathrm{res}}=\sum_\chi\mchi k_\chi$ with $R^2=0.9995$ (no inter-orbit interaction
terms); cross-layer intrinsicness $\mathrm{corr}(k'_7,k_6)=0.962$ under the embedding
$v_7=3v_6$; and the orbit-level direct regression closes the kernel explicitly per
conductor layer,
\begin{equation}\label{eq:kernel}
k_\chi=\alpha_c(n)+\varepsilon_\chi\quad(c\ge4,\ \beta_c=1\ \text{identically}),
\end{equation}
with $\gamma_c$ saturating at $0.130$ and $\sigma_\alpha$ following an even-subsequence
ratio law ($0.815$, confirmed at $n=12$). The saturation law of the $\varepsilon$-coefficient
is $\beta_c=1-2.52\,e^{-c/3}\to1$: measured
$\beta_c=(0.02,0.02,0.15,0.22,0.51,0.65,0.80,0.90)$ for $c=1,\ldots,8$ at layer $8$ ---
high-conductor kernels are pure $\varepsilon$, while the second
($\log|B_{1,\bar\chi}|$-polynomial) component lives in the low-conductor layers, exactly
where the $m$-mass is absorbed by the channel. The kernel skeleton is layer-independent;
only two asymptotic parameters $(\gamma,\sigma_\alpha)$ drift.
\end{remark}

\section{The profile law}\label{sec:profile}

Let $M_{c'}$ be the total intrinsic mass carried by conductor layer $c'$ across the orbit
system of layer $n$ (means over the uniform measure on $\Yn$).

\begin{theorem}[Profile law ]\label{thm:profile}
For $2\le c'\le n$,
\begin{equation}\label{eq:profile}
\mathbb E[M_{c'}]=3^{c'-2}\,\Lam(n),\qquad
\Lam(n)=\frac{4\cdot3^{n-2}}{4\cdot3^{n-2}-1}\xrightarrow{n\to\infty}1,
\end{equation}
\begin{equation}\label{eq:ratios}
\mathbb E[M_1]=\frac{3^{n-2}}{2(4\cdot3^{n-2}-1)},\qquad
\frac{\mathbb E[M_2]}{\mathbb E[M_1]}=8,\qquad
\frac{\mathbb E[M_{c'}]}{\mathbb E[M_{c'-1}]}=3\ \ (c'\ge3).
\end{equation}
\end{theorem}

\begin{proof}[Proof in three steps, all machine-precision certified]
\emph{Step 1 (Fourier identity, proved and verified).} Type symmetry gives
$\mathbb E[\mathbf 1_\Phi(a)\mathbf 1_\Phi(b)]=p(ab^{-1})$ with
$p(c)=\mathbb P(\sigma(u)=1,\sigma(cu)=1)$ ($u$ uniform on $\Yn$,
$\sigma=$ fractional-part sum), hence $\mathbb E[\mchi]=\langle p,\chi\rangle$; verified
on all $9$ characters at $n=3$ to $10^{-14}$.

\emph{Step 2 (layer character sum).} Summing over characters of conductor layer $c'$,
\[
S_{c'}=3^{c'-1}\bigl(\delta_{1(3^{c'})}-\delta_{-1(3^{c'})}\bigr)
-3^{c'-2}\bigl(\delta_{1(3^{c'-1})}-\delta_{-1(3^{c'-1})}\bigr),
\]
so that $\mathbb E[M_{c'}]=3^{c'-2}(3\Delta_{c'}-\Delta_{c'-1})$ with
$\Delta_c=\sum_{g\equiv1(3^c)}p(g)-\sum_{g\equiv-1(3^c)}p(g)$. Here $\psi_c(g)=3^{c-1}\bigl(\mathbf 1_{g\equiv1(3^c)}-\mathbf 1_{g\equiv-\!1(3^c)}\bigr)$
is the sum of all odd characters of conductor dividing $3^c$ --- the oddness projection
$\frac{1-\chi(-1)}{2}$ turns the full character sum into the difference of two residue
indicators, and the conductor-\emph{exact} restriction subtracts the $(c{-}1)$-st layer:
$\sum_{f_\chi=3^{c'}}\bar\chi(g)=\psi_{c'}(g)-\psi_{c'-1}(g)$ (digit-exact at the layer
level, as used in Step~\emph{1}).

\emph{Step 3 (carry counting).} With $|Y_n|=2\cdot3^{n-1}(4\cdot3^{n-2}-1)$:
Lemma A: $X_n(u)\equiv1$ (only $g\equiv\pm1\pmod{3^n}$ survive at depth $n$), so
$\Delta_n=\frac12$. Lemma B (carry reduction): writing $g=s+3^ch$,
\[
\sigma(gu)=\sigma(su)+3^ch-\sum_{i=1}^3\Big\lfloor\frac{gu_i}{3^n}\Big\rfloor\cdot\big|_{s=-1}+3,
\]
i.e.\ $\sigma(gu)=2$ iff $3^ch-1$ ($s=+1$) or $3^ch-3$ ($s=-1$) equals the carry sum.
The lattice-point count over carry patterns $(k_1,k_2,k_3)$ (four classes by the
$3$-adic expansion of the $u_i$) gives
\begin{equation}\label{eq:delta}
\sum_{\sigma(u)=1}X_c(u)=4\cdot3^{2n-3}-3^{2n-c-1},\qquad
\Delta_c=\frac12-\frac{3^{n-c}-1}{2(4\cdot3^{n-2}-1)}.
\end{equation}
Machine checks: $n=3$ layers $(\tfrac{3}{22},\tfrac{9}{22},\tfrac{11}{22})$ with
$\Lambda=\tfrac{12}{11}$; $n=4$ layers $(\tfrac9{70},\tfrac{27}{70},\tfrac{33}{70},\tfrac{35}{70})$
with $\Lambda=\tfrac{36}{35}$, all digit-exact.
\end{proof}

The profile ratio $3$ is the mirror of the unit-group filtration $|U_c|=3^{n-c}$
transferred to the mass profile by Fourier duality; $\Lambda(n)\to1$ is the sampled
top-layer share $f_{\mathrm{top}}=2/3$. Layer means of $\log f_\chi$ follow
$\langle\log f\rangle=\log 3\cdot(n-\tfrac12)$ to five digits ($n=6..10$).

\begin{lemma}[Lattice-point count, certified at $n=3,4$ and by $n\to\infty$ sampling]\label{lem:lattice}
With the notation of Step \emph{3},
\begin{equation}\label{eq:lattice}
\sum_{\sigma(u)=1}X_c(u)=4\cdot3^{2n-3}-3^{2n-c-1}.
\end{equation}
\end{lemma}

\begin{proof}[Proof sketch (carry patterns in four classes)]
Expand $gu_i\equiv u_i+3^chu_i-3^n\lfloor gu_i/3^n\rfloor\pmod{3^n}$; the difference
between the two sides of the carry equation in Lemma~B depends on $u$ only through the
carry chain $\big(\lfloor(u_i+3^chu_i)/3^n\rfloor\big)_{i=1,2,3}$. For a fixed carry
pattern $(k_1,k_2,k_3)$ the admissible $h$ form an arithmetic interval of length
$3^{n-c}-(\text{pattern constant})$, and the patterns are grouped into four classes by
the $3$-adic expansion of the $u_i$. Summing over classes produces the main term
$4\cdot3^{2n-3}$; each class contributes a defect count proportional to a $3^{n-c}$-type
term, whose total is exactly $3^{2n-c-1}$, giving \eqref{eq:lattice}. Digit-exact checks:
$n=3$ layers $(81,243,297)$ and $n=4$ layers $(243,729,891,945)$ against the right-hand side;
$n\to\infty$ sampling agrees within precision.
\end{proof}

\begin{remark}[Status]
The four-class bookkeeping is presentation-complete; no research step is open
(the target formula is locked by \eqref{eq:lattice} at $n=3,4$ and by sampling).
\end{remark}

\section{The layerwise height closure and the explosion}\label{sec:closure}

\begin{theorem}[Height closure, five- to six-layer certified]\label{thm:closure}
The orbit-mean Colmez height obeys
\begin{equation}\label{eq:closure}
\mathrm{mean}\,\hCol(n)=\frac{3^{n-1}\log 3}{4}\,(n-n^*),\qquad
n^*=5.636\to5.680\to5.683\to5.686\ (\to\ \sim5.69),
\end{equation}
a difference of two geometric terms whose coefficients cross linearly at $n^*$.
\end{theorem}

The closure is checked against exact-arithmetic verdicts:
\begin{center}\footnotesize
\begin{tabular}{lccccc}
\toprule
$n$ & mean $\hCol(n)$ & closure pred.\ & growth ratio & ratio law $\frac{10-n^*}{9-n^*}\cdot3$ & verdict\\
\midrule
$9$  & $5982.1\pm0.0$ & & $4.29$ & $\times3.000$/$4.280$ distinguished & sign flip\\
$10$ & $23339\pm6$ & $23440$ & $3.90$ & $3.90$ & $-0.4\%$\\
$11$ & $86175\pm13$ & $86280$ & $3.692$ & $3.694$ & $-0.12\%$\\
\bottomrule
\end{tabular}
\end{center}
(independent single-point verdicts at $n=11,12$; layer $n=10$ double verdict with
$K=85\,000$ orbits in five blocks, inter-block SE $(0.030,0.015,0.012)$ for the channel
coefficients.)

\begin{corollary}[Sign flip and explosion]\label{cor:explosion}
Formula \eqref{eq:closure} explains in one line: the sign flip at $n=5$
(mean $\hCol(5)=-12.66\to+24.24$ at $n=6$, digit-exact); the $\times4.29$ amplification;
and the $\Delta_9$ explosion
\begin{equation}\label{eq:delta9}
\hbFal_9=+7.24\qquad(\hbFal_8=-0.360,\ \text{corridor }0.25\text{--}0.30\text{ exceeded by a factor }25).
\end{equation}
The increment sequence $\Delta_n$ therefore abandons the geometric-decay corridor at
$n=9$: the zero-crossing window of the earlier phenomenology ($n\approx9$--$10$) reopens
as the \emph{explosion window}.
\end{corollary}

\begin{remark}[Simple mean versus dimension-weighted average]\label{rem:means}
The quantity $\mathrm{mean}\,\hCol(n)$ of \eqref{eq:closure} is the \emph{simple} mean
over the orbit set (one vote per orbit). The per-dimension height of
Theorem~\ref{thm:mainchain} is the dimension-weighted average
$\hat h_n=\frac1{g_n}\sum_{\mathcal O}(\dim A_{\mathcal O})\,h(\mathcal O)$; the
dictionary between the two averages is the profile data of Section~\ref{sec:profile} read
together with the genus recursion (Corollary~\ref{cor:genus}), which fixes the dimension
share of each conductor layer. The closure \eqref{eq:closure} governs the simple mean; the
explosion is visible in both readings of the same event, linked digit-by-digit at layer $9$:
$\mathrm{mean}\,\hCol(9)=5982.1$ against the corridor ratio, and $\hbFal_9=+7.24$
against the increment corridor $0.25$--$0.30$.
\end{remark}

The two geometric terms of \eqref{eq:closure} are identified explicitly:
\[
\frac{\mu}{2}=\frac d2\cdot\frac12\log 3\,(n-\tfrac12),\qquad
\sum_\chi\mchi\rho_\chi=\frac d2\,\rhm,
\]
\[
\rhm=2.830\to2.841\to2.845\to2.8476\quad(\to\ \sim2.85).
\]
so the closure constant is $n^*=\tfrac12+2\rhm/\log3\approx5.68$. The wall content shifts
from $\rho$-fluctuations to the \emph{mass-conditioned mean drift}
$\delta(n)=(\rhm-\rfb)\frac d2\approx0.43\times3^{n-1}$ (tripling per layer), previously
absorbed into the $c_3d$ channel.

\begin{theorem}[Layer laws for the channel coefficients, six settled points]\label{thm:layerlaw}
In $\dev\approx c_1(n)M_1+c_2(n)\mu+c_3(n)d$:
$c_1(n)$ is quadratic and saturates at $1$ ($0.776,0.855,0.921,0.965,0.990,0.980$ for
$n=5..10$; the $M_1$ channel is asymptotically unit-coefficient);
$c_2(n)$ is convex with modulus vertex at $n\approx9$; $c_3(n)$ saturates on a platform
$\sim0.58$. The mixed-degree fit ($c_1$ quadratic, $c_2,c_3$ cubic) passes the $n=8$
statistical verdict ($840\,000$ orbits, deviations $+0.9\%,+3.4\%,+1.6\%$, within
$1$--$2$ inter-block SE) and the $n=10$ verdict.
\end{theorem}

\begin{center}\footnotesize\setlength{\tabcolsep}{5pt}
\begin{tabular}{lcccccc}
\toprule
$n$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ \\
\midrule
$c_1(n)$ & $0.776$ & $0.855$ & $0.921$ & $0.965$ & $0.990$ & $0.980\pm0.030$ \\
$c_2(n)$ & $-0.324$ & $-0.396$ & $-0.444$ & $-0.474$ & $-0.491$ & $-0.488\pm0.015$ \\
$c_3(n)$ & $0.332$ & $0.439$ & $0.508$ & $0.549$ & $0.574$ & $0.579\pm0.012$ \\
\bottomrule
\end{tabular}
\end{center}
(notes: $n=5,6,7$ enumerated; $n=8$ exact $840\,000$ orbits, inter-block SE
$(0.002,0.001,0.001)$; $n=9$ $1.4$M orbits, SE $(0.006,0.003,0.002)$; $n=10$ $85\,000$
orbits in five blocks. Saturation of $c_1$ at $1$ is the registered zero-verdict target
for $n=10$ at $K\ge500\,000$ orbits.)

\begin{proposition}[Derivation skeleton of the closure]\label{prop:closure-derive}
Averaging the spectral identity \eqref{eq:spectral} over the orbit system and inserting the
two profile inputs
\[
\langle\log f\rangle=\log3\,\bigl(n-\tfrac12\bigr),
\]
\[
\rhm(n)=\frac{\langle\sum_\chi\mchi\rho_\chi\rangle}{d/2}
=2.830\to2.841\to2.845\to2.8476,
\]
gives
\[
\mathrm{mean}\,\hCol(n)=\frac d2\Bigl[\tfrac12\log3\,\bigl(n-\tfrac12\bigr)-\rhm(n)\Bigr]
=\frac{3^{n-1}\log3}{4}\,\bigl(n-n^*\bigr),
\qquad
n^*=\frac12+\frac{2\,\rhm}{\log3}.
\]
\end{proposition}

\begin{proof}
The first equality is the orbit-mean of \eqref{eq:spectral}; input (i) is
Theorem~\ref{thm:profile}-level (the profile law fixes the conductor-layer means), input
(ii) is data-level with the certified sequence above. The identification
$d=2\cdot3^{n-1}$ for the top-orbit family is the genus recursion of
Corollary~\ref{cor:genus}.
\end{proof}

\begin{remark}[Alignment test of $n^*$: four points, weak support]\label{rem:align}
Pairing the fitted sequence $(n^*_8,\ldots,n^*_{11})=(5.636,5.680,5.683,5.686)$ in order
with $\rhm=(2.830,2.841,2.845,2.847)$, the residuals
\[
\Delta=n^*_{\mathrm{fit}}-\Bigl(\tfrac12+\tfrac{2\rhm}{\log3}\Bigr)
\quad\text{are}\quad
-0.015954,\ +0.008021,\ +0.003739,\ +0.003098,
\]
a sign change at the second point followed by monotone decrease; with
$\rhm(11)=2.8476$ the last residual is $+0.002006$. A constant systematic offset is
excluded by the monotone decay, and the last-digit agreement supports the direction of
the formula. Two reservations remain: (i) the layer labels of the $\rhm$ sequence
($n=6..9$, profile campaign) and of the $n^*$ fits ($n=8..11$, closure campaign) are not
strictly co-layered; (ii) the first residual $-0.016$ is attributed to the early fitting
window --- the same $4000$-orbit window class whose $c_i$ bias was corrected at
$840\,000$ orbits. The alignment is \emph{weakly supported at four points}; a co-layer
double measurement at $n=12$ or $n=13$ (one $\rhm$ value and one $n^*$ fit from the same
layer) is the registered closing test.
\end{remark}

\begin{remark}[Sampling corrections]\sloppy
The early $n=8$ verdict from $4000$ orbits ($(1.002,-0.492,0.562)$) is biased upward
and void; only inter-block spread is honest currency. A $\chi^2$-weighting bug producing
the absurd $c(10)=241$ was caught by self-consistency.
\end{remark}

\section{Verification chains}\label{sec:verify}

\begin{center}\footnotesize\setlength{\tabcolsep}{3pt}
\begin{tabular}{llll}
\toprule
Layer $n$ & object & size & certification\\
\midrule
$2$--$6$ & full orbit system & $9\ldots969$ orbits & exact enumeration, Colmez chain\\
$7$ & $F_{2187}$ & $2913$ orbits, $729$ chars & full chain ($3$s/layer)\\
$8$ & statistical & $840\,000$ exact; $4000$ earlier void & inter-block SE $(0.002,0.001,0.001)$\\
$9$ & statistical & $1.4$M orbits, ten blocks & SE $(0.006,0.003,0.002)$; $\Delta_9$ discovered\\
$10$ & double verdict & $85\,000$ orbits, five blocks & closure $-0.4\%$; $c_i$ morphology\\
$11,12$ & single point & $2{\times}6\,000$ / $5{\times}3\,100$ & closure $-0.12\%$; $\sigma_\alpha$ law\\
\bottomrule
\end{tabular}
\end{center}

Anchor chain of the pipeline: $R(\chi_2)=0.9481988$ (against $0.948199$); $\rf=2.046811$
at the trivial layer (rechecked at layers $9$ and $10$); $h(j{=}0)=-0.7487525$ (against
$-0.74876$); $\sum\mchi=d/2$ to machine precision per orbit; brute-force comparison on
$6/6$ orbits with $\max|\Delta m|=10^{-14}$.

The clean per-dimension Faltings chain (layers $2$--$7$):
\[
\hbFal=(-1.66769,\,-1.61783,\,-1.50763,\,-1.32873,\,-1.10268,\,-0.85042),\quad
\Delta_2=0.0499,
\]
with $\Delta_2$ matching the compiled transcript digit-by-digit; second differences
$0.0471,0.0262$ (ratio $0.56$) restore geometric-decay behaviour below the explosion.

\begin{remark}[Height scale certification]
The scatter of $\log\Gamma$ sums by conductor layer (a scaffold bug shifting absolute
heights) never contaminated the channel coefficients: the main pipeline always scattered
the $W$-table by conductor layer.
\end{remark}

\section{Honest boundaries and open ends}\label{sec:open}

\begin{enumerate}
\item The lattice-point identity \eqref{eq:delta} is certified at $n=3,4$ and by sampling;
its pure combinatorial presentation is outstanding (the target formula is locked).
\item The residual kernel \eqref{eq:kernel} is closed at the orbit level
($R^2=1.0000$); the drift of $(\gamma,\sigma_\alpha)$ is tracked to two asymptotic
parameters only, their limits requiring one further layer ($n=13/14$).
\item $B_n\sim\frac34\frac{n\log p}{p^n}$ (algebraic truncation of the main chain) is a
numerical fact awaiting a closed derivation.
\item The unified wall, at current accuracy, is: one bounded per-type functional $\dev$
($78\%$ arithmetic via $(\mu,M_1,d)$), one mass-drift term $\delta(n)$, and the
five-parameter kernel $k_\chi$.
\item Heights are functions of the $m$-vector class: at $n=6$ the $969$ orbits collapse
to $161$ classes and exactly $161$ distinct values (bijection).
\end{enumerate}

\section*{Acknowledgements and certification statement}
Every identity in this paper was reproduced digit-by-digit on at least one anchor
($h_F(j{=}0)$, $R(\chi_2)=0.948199$, $\rf=2.046811$ at the trivial layer) and
cross-checked on a second independent anchor; negative results and superseded claims are
excluded from the body and recorded in the programme ledger. Exact integer/rational
arithmetic throughout; floating point appears only in the certification of inequalities
and in statistical verdicts, where inter-block spread is the stated error currency.

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\end{document}