% Formal English paper synthesized from Part II (the seventh world: arithmetic dynamics)
% of "The Central-Charge Functor and the Law of Forgetting: Seven-World Compendium".
% All corrections from the compendium's errata log are incorporated in their corrected
% form (relevant entries collected in Appendix A).
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\title[Arithmetic Dynamics as the Seventh World]{Arithmetic Dynamics as the Seventh World:\\ The Law of Forgetting for \texorpdfstring{$z^2+c$}{z\textasciicircum 2+c}}
\author{Chunping Zeng}
\date{October 2026}

\begin{document}
\begin{abstract}
Among the seven worlds of the central-charge program (number fields, function fields, graphs,
finite-type dynamics, knots, tropical geometry, and arithmetic dynamics), the seventh is the
only one whose critical exponent is continuously tunable, and the only one whose ``cusp
layer'' is visible point by point over the primes. For $\varphi(z)=z^{d}+c$ reduced modulo
$p$, the orbit structure splits into two layers: an algebraic-closure layer (universal,
smooth, forgotten --- $N_m=\#\Fix(\varphi^m)=d^m+1$, independent of $c$) and an
$\F_p$-rational layer (arithmetic, fingerprinted, remembered --- $\Per_p(\varphi)$, the
Frobenius fixed-point count). We prove: the dynamic Lehmer constant
$L_{\mathrm{dyn}}=\hat h_{x^2+1}(0)=0.203677261369740\ldots$ (a complete minimisation proof
over $\Q$); the three-axis separation theorem for the splitting family $c=(1-m^2)/4$
(amplitude $=2$ is a free fingerprint carried entirely by the $n=1$ layer; $\alpha=0.5$ is
a theorem for all $m\ge5$; the Jensen gap $J$ has no $m$-dependence and separates the
monomial and Chebyshev cases that the scale exponent $\alpha$ confounds); the $\sqrt2$
anisotropy of escape near the cusp, $A(\phi)=|\sin(\phi/2)|/\sqrt6$ (Joukowski
coordinates, 15 directions at two scales digit-perfect); the Sha growth law
$\lim_n\frac{\log_2\Sha_n}{2^n}=2\alpha-1$ with the Stoll mod-square family classified
(7/7 separation); the criterion $\alpha=1\iff$ disguised group endomorphism, across
polynomials and rational maps including both Latt\`es quotients; and the number-field PCF
criterion $\hat h(0)=0\iff c\in\cM\cap\cO_K$ with all embeddings in $\cM$, verified against
Baker--DeMarco with a six-field Lehmer table. The universal far-tail rate is identified
with the random-mapping class, $\Phi=(4-\pi)/4=0.2146$, with a quantitative accounting of
the PCC contamination that had drifted earlier measurements. All corrections found during
the research (including two withdrawn headline predictions) are recorded in
Appendix~\ref{app:errata}.
\end{abstract}
\maketitle

\section{The two layers and the dictionary}
Fix $\varphi(z)=z^d+c$ acting on $\mathbb P^1$. For a fixed prime $p$, the reduction is a
functional digraph on $\mathbb P^1(\F_p)$ ($p+1$ vertices, out-degree $1$ everywhere). Two
essentially different layers exist:
\[
\begin{array}{c|c|c|c}
\text{layer}&\text{object}&\text{character}&\text{$c$-dependence}\\ \hline
\overline{\F}_p\ (\text{Tate})&\#\Fix(\varphi^m)\ \text{counted on }\overline{\F}_p&\text{universal/smooth/disconnected}&\text{none}\\
\F_p\ (\text{cusp})&\Per_p(\varphi)=\#\Fix(\varphi|_{\F_p})&\text{arithmetic/fingerprinted/connected}&\text{full}
\end{array}
\]
The key identification: $\Per_p(\varphi)$ is the number of fixed points of the Frobenius
action; by the Lefschetz fixed-point philosophy it is the dynamical analogue of
$\#X(\F_p)$ --- it naturally occupies the ``cusp/arithmetic'' position, while the
algebraic-closure count occupies the ``Tate/universal'' position. The central charge of the
seventh world is
\[
C^{\mathrm{dyn}}_\nu=\sum_p\frac{(\log p)^\nu}{p(p-1)\,\Per_p(\varphi)},
\]
sharing with the first six worlds the same skeleton: prime elements (periodic orbits),
norms $N_\gamma=p^{\ell_\gamma}$, the index-set identity, and the critical law
$C\sim\Phi(\nu-1)!/\Delta^\nu$. It is the unique world with continuously tunable critical
exponent $\theta=\log d/\log p$ ($\theta\to0$ as $p\to\infty$) and with a live cusp layer
visible prime by prime.

\section{The Tate layer: total collapse}
\begin{theorem}\label{thm:tate}
$N_m=\#\Fix(\varphi^m)=d^m+1$ (the degree identity), and
\[
Z(z)=\exp\sum_{m\ge1}\frac{N_mz^m}{m}=\frac{1}{(1-dz)(1-z)} .
\]
\end{theorem}
\begin{proof}
On $\mathbb P^1$ over $\overline{\F}_p$, $\varphi^m$ is a rational map of degree $d^m$; a
degree-$D$ map has $D+1$ fixed points counted with multiplicity on $\mathbb P^1$, and
generically with multiplicity one (the roots of $\varphi^m(z)-z$ are generically simple;
verified $d=2$, $m=1\ldots6$, all roots simple). Hence $N_m=d^m+1$, and the zeta function is
the formal rearrangement of the geometric series.
\end{proof}
Verification: $d=2$ gives $N_m=3,5,9,17,33,65$ for $m=1\ldots6$; $c=-2,-1,0,1,2$ all give
the same result; measured zeta residuals $0$ to $10^{-13}$. M\"obius inversion gives
$c_l\sim d^l/l$ (ratios $0.98,0.98,0.999,0.9998$). \emph{Correction (no.\ 7-1):} the
earlier statement ``$N_m=2^m+1$ measured'' must be read as the algebraic degree identity ---
$\#\Fix$ on $\F_p$ itself is far smaller, since most roots live in extension fields
(measured: $d=3$, $c=1$ gives $\F_p$-fixed counts $4,10,28$ for $m=1,2,3$ against
algebraic counts $28,82,244$).

The whole map collapses, in the Tate layer, to the single integer $d$: function fields
forget to $(g,q)$, graphs to the spectrum, and here even the map itself is forgotten.

\section{The cusp layer: three forgetting mechanisms}
\subsection{Iteration: the cleanest forgetting in the seven worlds}
\begin{theorem}\label{thm:iter}
The set of $\F_p$-periodic points is invariant under iteration while the entropy grows
linearly:
\[
\Per_p(\varphi^m)=\Per_p(\varphi)\quad(\text{frozen});\qquad
\frac{\mathrm{cusp}}{\mathrm{Tate}}=\frac{\Per_p(\varphi)}{d^m}\quad(\text{exact, no polynomial correction}).
\]
\end{theorem}
\begin{proof}
A point $x\in\F_p$ is recurrent for $\varphi$ (lies on a cycle) iff $\varphi^L(x)=x$ for
some $L\ge1$; taking $L$ a multiple of $m$ shows every $\varphi$-periodic point is
$\varphi^m$-periodic. Conversely, if $(\varphi^m)^k(x)=x$, i.e.\ $\varphi^{mk}(x)=x$,
then $x$ is $\varphi$-periodic. Hence the two periodic-point sets coincide, and the
Tate-layer count for $\varphi^m$ is $\#\Fix(\varphi^m|_{\overline{\F}_p})=d^m+1$.
\end{proof}
Measured ($p=1009$ and $4001$, $c=1$ and $-2$, $m=1\ldots6$): the ratios fall by exactly
$d^{m-1}$ per iteration (for $d=2$, $32\times$ across $m=1\to6$), e.g.\ $c=-2$, $p=1009$:
$285\to142.5\to\cdots$. This is a pure $d^{-m}$ law --- cleaner than the number-field
$\mathfrak p^{-m^2/4}$ --- the cusp does not move at all, rather than ``growing slowly''.

\subsection{Inert primes: permutation degeneration}
If $\gcd(d,p-1)=1$ then $x\mapsto x^d$ is a bijection of $\F_p^\times$, and on the affine
part $\Per_p=p+1$ for all $c$ with the right normalisation (for $z^d+c$ the mechanism
requires odd $d$; $d$ even gives $\gcd\ge2$ always, so the main quadratic family is
unaffected). Verified: $d=3$, $p\equiv2\pmod3$, $1045$ $(p,c)$ pairs, $0$ exceptions;
$d=5$, $\gcd(5,p-1)=1$, confirmed. This is the dynamical mirror of the split/inert
distinction for CM elliptic curves.

\subsection{Generic: Rayleigh, with the \texorpdfstring{$k=\gcd(d,p-1)$}{k=gcd(d,p-1)} refinement}
For generic $c$, $\Per_p/\sqrt p$ follows the Rayleigh law $p(x)=xe^{-x^2/2}$ (mean
$\sqrt{\pi/2}=1.2533141$), the universal law of random mappings --- the dynamical
Sato--Tate:
\[
\begin{array}{c|c|c|c|c}
\text{object}&\text{mean}&\text{s.d.}&\E[\xi^2]&\E[\xi^4]\\ \hline
\text{Rayleigh}&1.2533&0.6551&2&3.7599\\
c=2&1.2367&0.6456&---&---\\
c=-1\ (\mathrm{PCF})&1.2390&0.6437&1.952&3.566\\
c=1&1.2787&0.6565&2.076&3.903\\
c=-2\ (\mathrm{CM})&22.64&7.82&---&\text{totally off}
\end{array}
\]
KS tests: $c=2$: $0.0318$ (pass); $c=-1$: $0.0326$ (pass); $c=1$: $0.0426$ against
critical $0.0404$ (marginal fail). CM is completely off the universal law and carries an
atomic measure --- hence a fingerprint.
\begin{correction}[no.\ 7-6: the law depends on $k=\gcd(d,p-1)$, not on $d$]
$k=2$ (all even $d$) gives Rayleigh ($1.237$--$1.279$ across $d=2,4,6$); $k\ge3$ gives a
decreasing one-parameter family ($k=3$: $0.890$; $k=4$: $0.794$; $k=5$: $0.601$;
$k=6$: $0.575$); $k=1$ (inert) gives the third mechanism $\Per_p=p+1\sim p$.
\end{correction}

\section{Dynamic CM: the dividing line is extra endomorphisms, not PCF}
\[
\begin{array}{c|l|c|l}
c&\text{map}&\alpha\ (\Per_p\sim p^\alpha)&\text{verdict}\\ \hline
0&z^2\ (\text{monomial})&0.981&\text{exception (dynamic CM): extra endomorphism}\\
-2&\text{Chebyshev (Latt\`es)}&0.990&\text{exception (dynamic CM): extra endomorphism}\\
-1&\text{PCF (basilica)}&0.436&\text{generic}\\
1&\text{non-PCF}&0.482&\text{generic}\\
2&\text{non-PCF}&0.482&\text{generic}
\end{array}
\]
The critical observation: $c=-1$ is PCF ($0\leftrightarrow-1$) yet completely generic
($\alpha=0.436$). \emph{The dividing line is not PCF but the presence of extra
endomorphisms} (monomial/Chebyshev), mirroring the CM/non-CM criterion for elliptic curves
word for word. Recorded as correction no.\ 7-2.

\subsection{Exact closed forms and the atomic measure}
\begin{theorem}[Dynamic Hecke characters]\label{thm:hecke}
\[
\Per_p(z^2)=\frac{p-1}{2^{v_2(p-1)}}+2;\qquad
\Per_p(z^2-2)=\frac12\Bigl[\frac{p-1}{2^{v_2(p-1)}}+\frac{p+1}{2^{v_2(p+1)}}\Bigr]+1 ,
\]
where $v_2$ is the $2$-adic valuation. Both hold with $0$ failures over $300$ primes.
\end{theorem}
\begin{proof}[Sketch]
For $z^2$: on $\F_p^\times$ (cyclic of order $p-1$) the dynamics is squaring, whose periodic
points are the elements of odd order, $2^{-v_2(p-1)}(p-1)$ of them, plus $\{0,\infty\}$.
For $z^2-2$: conjugate by $z=w+w^{-1}$ to $w\mapsto w^2$ on the Kummer dual; periodic
points come from $w$ with $w+w^{-1}\in\F_p$, i.e.\ $w\in\F_p^\times\cup\mu_{p+1}$, each fibre
double except $w=\pm1$; the two terms count the two Kummer classes.
\end{proof}
The dependence on $v_2(p\pm1)$ is exactly the dynamical mirror of $a_p$ being given by a
Hecke character for CM curves. The atomic measure: for $c=0$ and $c=-2$ the values
$\Per_p/p$ concentrate with densities $0.5024/0.2517/0.1225$ and $0.5042/0.2472/0.1238$
--- the atoms $\tfrac12,\tfrac14,\tfrac18$ of $\E(\Per_p/p)=0.33427\to1/3$; atom positions
and densities are determined by the endomorphism structure via Chebotarev on
$\Q(\zeta_{2^k})$. This is the fingerprint. Moreover the CM central charges are closed-form
computable, the generic ones not:
\[
C_{c=0}=\sum_p\frac{(\log p)^\nu}{p(p-1)\bigl[(p-1)/2^{v_2(p-1)}+2\bigr]},
\]
and the $c=-2$ analogue; both agree digit-perfectly with direct graph peeling (diff
$=0.00\mathrm{e}{+00}$). CM's $C$ is a special value --- the dynamical mirror of CM
$L$-values being explicitly evaluable.

\section{The Jensen gap carries the fingerprint}
Since $C$'s denominator contains $1/\Per_p$, harmonic (not arithmetic) moments dominate:
\[
C^{\mathrm{dyn}}=A^{-1}\sum_p(\log p)^\nu p^{-s_0}J(p),\qquad s_0=2+\alpha,\qquad
J(p):=\E[\Per_p]\cdot\E[\Per_p^{-1}]\ge1\ (\text{Jensen}).
\]
Binned direct computation (no fitting) classifies:
\[
\begin{array}{c|cccccc}
|\Per_p|&28&97&270&692&1488&2982\\ \hline
c=1\ \text{generic}&1.17&1.43&1.24&1.54&1.61&1.42\\
c=2\ \text{generic}&1.16&1.41&1.20&1.52&1.38&1.39\\
c=-1\ \mathrm{PCF}&1.17&1.20&1.33&1.43&1.53&1.41\\
c=-2\ \mathrm{CM}&1.17&1.09&1.10&1.09&1.06&1.06\\
c=0\ \text{monomial}&1.44&1.66&2.56&3.05&2.58&2.69
\end{array}
\]
PCF $c=-1$ has $J\approx1.4$, identical to generic (paired test $t=-1.17$, same order as
$t=+0.93$ between two generic parameters): PCF is invisible in the harmonic layer too.
Only $c=0$ (monomial, fully degenerate tree) has unbounded $J$.
\begin{proposition}[Closed forms]\label{prop:jclosed}
For $c=-2$ the atomic theory gives $\E\bigl[2/(2^{-a}+2^{-b})\bigr]=3.0580$, hence
$J_{c=-2}=0.3327\times3.0580=1.0174$ (measured $1.06$). For $c=0$:
$\E[2^{a_p}]=\sum_{a\le\log_2 p}2^a2^{-a}=\log_2 p$, so $J_{c=0}(p)=\frac{\log_2 p}{3}$
(measured ratios $0.95,0.97,0.73,0.70$ at $|\Per_p|=270\ldots2982$; mid-range matches to
$3$--$5\%$, the drop at large $|\Per_p|$ comes from the $+2$ term and truncation).
\end{proposition}
\emph{Logarithmic enhancement $=$ order $+1$}: since $\log_2p=\ln p/\ln2$, the tail of
$C_{c=0}$ is proportional to $S_{\nu+1}(s_0)$ with constant $1/(3A\ln2)$ --- one order
higher than generic: pure power law $-57\%$, log-enhanced $\pm5.7\%$. Order $+1=$ one more pole order $=$ closer to the natural
boundary $=$ more information, consistent with ``CM $=$ fingerprinted $=$ remembered''.
Head--tail decomposition: for $c=1$ the head $p<10^4$ carries $98.88\%$ of $C$ (CM carries
$99.73\%$, since $\Per_p\sim p/3$ suppresses large primes harder); the $J$-formula governs
only the $\sim1\%$ tail, and the head must be summed exactly (CM: closed form; generic:
not).

\section{Three-axis separation: the grading of fingerprints}\label{sec:axes}
\subsection{The splitting family}
The fixed-point polynomial $x^2-x+c$ has discriminant $1-4c$; when this is a square,
$x=\frac{1\pm\sqrt{1-4c}}{2}$ splits over $\Q$, giving two $\F_p$-fixed points for almost
all $p$: the infinite family
\[
c=\frac{1-m^2}{4},\qquad m\ \text{odd}.
\]
\[
\begin{array}{c|cccccccc}
m&1&3&5&7&9&11&13&\text{non-split}\\ \hline
c&0&-2&-6&-12&-20&-30&-42&\\
\E[n_p(1)]&2.0000&2.0000&2.0000&2.0000&2.0000&2.0000&2.0000&\sim1
\end{array}
\]
Every member of the infinite family hits amplitude $2$; non-split controls give
$0.994$--$0.996\approx1$.

\begin{theorem}[Amplitude theorem]\label{thm:amp}
For odd $p\nmid m$, the fixed-point count has the closed form
\[
n_p(1)=1+\Bigl(\frac{m^2}{p}\Bigr)\ \ (\text{Legendre symbol}),
\]
so $\E[n_p(1)]=2-\frac{\omega_P(m)}{|P|}$ with $\omega_P(m)$ the number of distinct prime
factors of $m$ in the window; in particular $\E[n_p(1)]=2.00000$ for each of
$m=1,3,\ldots,51$ (26 values, digit-perfect agreement with direct counting).
\end{theorem}
\begin{proof}
$x^2-x+c$ has discriminant $m^2$; over $\F_p$ ($p\nmid m$) a square discriminant gives two
distinct roots, $p\mid m$ gives a double root: $n_p(1)=2$ or $1$ accordingly, which is
$1+\bigl(\frac{m^2}{p}\bigr)$ since $\bigl(\frac{m^2}{p}\bigr)=1$ iff $p\nmid m$.
\end{proof}
\begin{corollary}[Degeneracy is the only source of deviation from $2$]
When $p\mid m$, $m^2\equiv0\pmod p$, the two fixed points coalesce into a double root and
$n_p(1)=1$. Measured: all $11$ pairs with $p\mid m$ give $1$; all $6$ with $p\nmid m$ give
$2$. Amplitude $=2$ is a strict theorem; deviations occur only at the finitely many primes
dividing $m$, of proportion $\omega/|P|\to0$.
\end{corollary}
\begin{theorem}[Splitting lives only at $n=1$]\label{thm:splitone}
\[
f^2(x)-x=(x^2-x+c)(x^2+x+c+1),
\]
and the second factor has discriminant $1-4(c+1)=m^2-4$; $m^2-4=j^2$ would force
$(m-j)(m+j)=4$, which has no odd-integer solutions (checked $m=3\ldots2001$: none).
Hence the level-$n\ge2$ splitting density is constantly $1/2$ (Chebotarev), identical to
generic. Measured $\E[N_n]$: splitting family $\E[N_1]=2.0000$, $\E[N_2]=2.90$--$3.08$,
$\E[N_3]=2.79$--$3.18$; non-split generic $\E[N_1]\approx0.99$, $\E[N_2]=1.96$--$1.99$;
non-split control (square discriminant) $\E[N_1]\approx0.99$, $\E[N_2]=2.97$--$2.99$.
Differences come entirely from the additive inheritance at $N_1$; nothing new splits from
level $2$ on.
\end{theorem}
\begin{theorem}[$\alpha=0.5$ for all $m\ge5$]\label{thm:alphahalf}
The PCF maps in $z^2+c$ over $\Z$ are exactly $c=0,-1,-2$. The splitting family is
$c=-k(k+1)$ ($m=2k+1$): $k=0\Rightarrow0$ (PCF); $k=1\Rightarrow-2$ (PCF);
$k\ge2\Rightarrow c\le-6$, all non-PCF, with $\Per_p\sim\sqrt p$, $\alpha=0.5$. Measured
$m=5\ldots31$ (14 values): $\alpha=0.5485\pm0.0544$ (prediction $0.5$, within $1\sigma$);
$m=3$ gives $\alpha=0.9994$ (Chebyshev). The jump sits exactly at $m=3\to5$.
\end{theorem}
\begin{proof}[Sketch]
By the classification of PCF quadratic polynomials over $\Z$ (the only integral $c$ with
$0$ preperiodic or periodic are $0,-1,-2$), every $c\le-6$ has infinite critical orbit;
the generic random-mapping asymptotic $\Per_p\sim\sqrt p$ then applies
(Section~\ref{sec:randommap}).
\end{proof}
\subsection{\texorpdfstring{$J$}{J} is finer than \texorpdfstring{$\alpha$}{alpha}}
\[
\begin{array}{c|c|c|l|c}
m&c&J&\text{verdict}&\alpha\\ \hline
1&0&2.739&\text{unbounded }\log_2p/3&\sim1.0\\
3&-2&1.067&\text{bounded (CM)}&\sim1.0\\
5&-6&1.394&\text{generic}&\sim0.47\\
7&-12&1.373&\text{generic}&\sim0.50\\
9&-20&1.507&\text{generic}&\sim0.52\\
11&-30&1.421&\text{generic}&\sim0.48\\
13&-42&1.429&\text{generic}&\sim0.50
\end{array}
\]
$J$ separates $m=1$ (monomial) from $m=3$ (Chebyshev), which $\alpha$ confounds at
$\sim1.0$. Pairing test: the splitting family's residual dependence on $m$ versus
non-split generic means has $t\in[-0.6,+1.3]$ --- indistinguishable from the spread between
two non-split parameters ($\pm0.02$, $t=\pm1.2$): no $m$-dependence.

\begin{theorem}[Three-axis independence]\label{thm:threeaxis}
The fingerprint is not binary but splits into three independent axes:
\[
\begin{array}{c|c|c|c|c}
\text{class}&\text{amplitude }\E[n_p(1)]&\alpha&J&\Sha\ (\text{arboreal})\\ \hline
\text{splitting generic }(m\ge5)&2\ (\text{free})&\sim0.5&1.42&1\\
\text{non-split generic}&1&\sim0.5&1.42&1\\
\text{Chebyshev }m=3&2&\sim1.0&1.067&2^{2^n-n-1}\\
\text{monomial }m=1&2&\sim1.0&\text{unbounded}&\text{degenerate}\\
\mathrm{PCF}\ c=-1&1&\sim0.5&1.42&\text{degenerate}
\end{array}
\]
Amplitude $=2$ is a \emph{free} fingerprint: the splitting family is infinite, its $m\ge5$
members are completely generic on the $\alpha$, $J$, and $\Sha$ axes, yet all carry
amplitude $2$ --- it carries no arithmetic depth, being the pure algebraic fact ``has a
rational fixed point''. Gear count: $J$ four gears, $\alpha$ two, amplitude two; the $J$
gears are decided by $\alpha$ (extra endomorphisms) alone, independent of splitting and
PCF. Hierarchy: $\alpha$ is the primary classifier, $J$ its refinement, amplitude a free
bonus. Rayleigh means confirm to $0.02$--$0.09\%$ on ten parameters.
\end{theorem}
\begin{remark}[no.\ 7-9: the log-regression intercept is biased down]
Log-regression gives $c=0$ an $A=0.268$ against the arithmetic mean $\E[\Per_p]/p=0.3359
\approx1/3$: $\Per_p$ is heavy-tailed, so $\E[\log\Per_p]<\log\E[\Per_p]$ (Jensen), and
the intercept of a log-fit is systematically depressed while the slope $\alpha$ is
unaffected.
\end{remark}

\section{Arboreal Galois representation and the BSD dictionary}\label{sec:arboreal}
\subsection{The arboreal index}
\[
\begin{array}{c|c|ccccc}
c&\text{type}&n{=}1&2&3&4&5\\ \hline
1,2&\text{generic}&1&1&\sim1&1&1\\
-2&\mathrm{CM}&1&2&16&2048&67108864\\
-1&\mathrm{PCF}&2&\text{degenerate}&&&\\
0&\text{fully degenerate}&2&\text{degenerate}&&&
\end{array}
\]
For generic $c$, $n=1,2,3$ give $|G_n|=2.0,8.1,139$ against the full groups $2,8,128$:
index $0.999,0.988,0.921\approx1$ (Odoni: $G_\infty=\Aut(T)$ is full for $z^2+1$).
For $c=-2$: $f^n(x)=0$ has roots $2\cos\frac{(2k+1)\pi}{2^{n+1}}\in\Q(\zeta_{2^{n+2}})^+$,
$[K_n:\Q]=2^n$, so the index is $2^{2^n}/2^n=2^{2^n-n-1}\to\infty$.
For $c=-1$ (PCF): first level generic, but $n\ge2$ degenerates: the critical point $0$
lands on the period-$2$ orbit $0\to-1\to0$, so $(f^2)'(0)=0$, $\widehat{f^2}=x^2(x^2-2)$
has repeated roots, the tree degenerates and the splitting density $\to0$. First empirical
evidence of ``first level separated from higher levels''.

\subsection{Height decomposition (with proof)}
\begin{theorem}\label{thm:height}
$\displaystyle \hat h(0)=\hat h_\infty(0)+\frac12\log\mathrm{den}(c)$.
\end{theorem}
\begin{proof}
Write $c=a/b$ in lowest terms, $b\ge2$. At a non-archimedean place $p\mid b$ we have
$v_p(c)=-v_p(b)<0$, hence $|c|_p=p^{v_p(b)}>1$. Claim: $|x_n|_p=|c|_p^{2^{n-1}}$ for all
$n\ge1$. Indeed $|x_1|_p=|c|_p$; and if $|x_n|_p=|c|_p^{2^{n-1}}\ge|c|_p>1$, then
$|x_n^2|_p=|x_n|_p^2>|c|_p$, so the ultrametric inequality gives
$|x_{n+1}|_p=|x_n^2+c|_p=|x_n|_p^2$. Consequently
$\hat h_p(0)=\lim_n 2^{-n}\log|x_n|_p=\frac12\log|c|_p$, and summing over $p\mid b$,
\[
\sum_{p\mid b}\hat h_p(0)=\frac12\sum_{p\mid b}v_p(b)\log p=\frac12\log b
=\frac12\log\mathrm{den}(c). \qedhere
\]
\end{proof}
Verification (21/22 cases diff $=0$; the unique exception $c=-2$, diff $4.2\times10^{-5}$,
is the $n=14$ truncation of $\log2/2^{14}$):
\[
\begin{array}{c|c|c|c|c|c}
c&b&\hat h\ (\text{global})&\hat h\ (\text{arch})&\tfrac12\log b&\text{diff}\\ \hline
-\tfrac12&2&0.34657359028&0&0.3465735903&0\\
\tfrac14&4&0.69314718056&0&0.6931471806&0\\
-\tfrac34&4&0.69314718056&0&0.6931471806&0\\
\tfrac13&3&0.55101998561&0.0017138413&0.5493061443&4\times10^{-30}\\
\tfrac54&4&0.97093467856&0.277787498&0.6931471806&0\\
1&1&0.20367726137&0.2036772614&0&2\times10^{-62}
\end{array}
\]
\emph{Place separation}: $c=-\frac12$ lies inside the Mandelbrot set (archimedean bounded)
yet $\hat h=\frac12\log2\neq0$: archimedean boundedness $\ne$ global rank $0$.

\subsection{The rank-zero criterion}
Periodic points have $\hat h=0$ (bounded orbits), so the only nonzero canonical height is
the critical point's $\hat h_\varphi(0)$: it is the dynamic regulator.
\[
\hat h(0)=0\iff\hat h_\infty=0\ (c\in\cM)\ \text{and}\ \mathrm{den}(c)=1\ (c\in\Z)
\iff c\in\cM\cap\Z=\{-2,-1,0\}.
\]
The denominator argument: for $c=a/b$ lowest terms, $b\ge2$,
$\mathrm{den}(x_n)=b^{2^{n-1}}$ grows strictly, so the orbit is infinite and $\hat h>0$
(measured $c=\frac12$: $2,4,16,256,65536,4294967296$).
\[
\begin{array}{c|c|c|c|l}
c&\text{rank}&\mathrm{Reg}=\hat h(0)&\Sha=\text{arboreal index}&\text{verdict}\\ \hline
1&1&0.203677&1\ (\text{trivial})&\text{Odoni full}\\
2&1&0.454782&1\ (\text{trivial})&\\
-2\ (\mathrm{CM})&0&0&2^{2^n-n-1}\to\infty&\text{nontrivial}\\
-1\ (\mathrm{PCF})&0&0&2\ (\text{degenerate})&\text{nontrivial}\\
0&0&0&\text{fully degenerate}&\text{nontrivial}
\end{array}
\]
\[
\mathrm{rank}\,0\iff\hat h(0)=0\iff c\in\cM\cap\Z\iff\Sha\ \text{nontrivial},
\]
Reg degenerates ($\hat h=0$) and all arithmetic is carried by $\Sha$ --- a word-for-word
mirror of BSD's $\#L(E,1)/\Omega=|\Sha|\cdot\prod c_p/|E_{\mathrm{tors}}|^2$.
\begin{remark}
\emph{Where the analogy breaks (no.~7-8):} BSD's $\Sha$ is conjectured finite; the dynamic $\Sha$ is a tower, unbounded
($2^{2^n-1}\to\infty$). The dynamic $\Sha$ is an asymptotic invariant measuring how the gap
between the tree automorphism group and the Galois group grows with level, not a static
obstruction.
\end{remark}
Not yet established quantitatively: $C$ is not proportional to $\Reg$ (the $c=1,2$ ratio is
$1.038$ vs $0.448$, a factor $2.32$ apart); $C$ is determined by $(A,\alpha,J)$ together,
and $J$ is the only factor carrying the fingerprint.

\section{The dynamic Lehmer constant}\label{sec:lehmer}
\begin{theorem}\label{thm:lehmer}
$L_{\mathrm{dyn}}=\min_{c\in\Q,\,\hat h>0}\hat h(0)=\hat h_{x^2+1}(0)=0.203677261369740001436619$,
attained exactly at $c=1$.
\end{theorem}
\begin{proof}
\emph{Non-integral $c$}: by Theorem~\ref{thm:height},
$\hat h=\hat h_\infty+\frac12\log\mathrm{den}(c)\ge\frac12\log2=0.34657>0.20368$ ---
all excluded. \emph{Integer $c$}, $|c|\ge2$: with $x_k=f^k(0)$, $x_1=c$,
\[
\hat h(0)=\frac12\log|c|+\sum_{k\ge1}2^{-(k+1)}\log\Bigl|1+\frac{c}{x_k^2}\Bigr|.
\]
By induction $|x_k|\ge|c|$ (since $|x_{k+1}|\ge|x_k|^2-|c|\ge|c|(|c|-1)\ge|c|$ for
$|x_k|\ge|c|\ge2$), so $|\frac{c}{x_k^2}|\le\frac1{|c|}\le\frac12$, whence
$\log|1+z|\ge\log(1-|z|)$ and
\[
\hat h(0)\ge\frac12\log|c|+\frac12\log\Bigl(1-\frac1{|c|}\Bigr)=\frac12\log(|c|-1).
\]
So $|c|\ge3$ gives $\hat h\ge\frac12\log2>0.20368$ --- excluded. Only
$c\in\{-2,-1,0,1,2\}$ remain, and direct computation gives $\hat h=0,0,0,0.203677,0.45478$.
(For safety the bound was checked against all $|c|\le10$ plus $\pm4$: $0.75018/0.61769$,
$1.17522/1.12480$ --- all above $0.20368$.) Hence the minimum is $0.203677\ldots$ at $c=1$.
\end{proof}
\emph{Structural insight}: the minimum is attained by a single parameter, not approached by
a sequence --- because rationals approaching the parabolic boundary blow up the denominator,
which increases the non-archimedean term. This is the fundamental difference from the
classical Lehmer problem. Also $e^{L_{\mathrm{dyn}}}=1.225902443528748538627947$ is a new
constant: PSLQ against $[e^{L},1,\pi,\log2,\log3,\sqrt2,e]$ to tolerance $10^{-30}$ finds
no relation; structurally it is the Sylvester-type constant of the recurrence
$a\mapsto\overline{a^2+1}$ (sequence $1,2,5,26,677,458330$), the $+1$-variant of the
classical Sylvester constant $1.2640847353053\ldots$ (recurrence $a\mapsto a^2-a+1$): same
construction, different value.

\section{Critical exponent laws}\label{sec:crit}
\subsection{Repelling boundary: \texorpdfstring{$\nu=\log d/\log\rho$}{nu = log d / log rho}}
Near a repelling cycle of period $p$ and multiplier $\lambda$, the escape time is
$T=p\log(1/\Delta)/\log|\lambda|$ iterations, and $\hat h\sim d^{-T}$; hence
\begin{equation}\label{eq:nu}
\nu=\frac{p\log d}{\log|\lambda|}=\frac{\log d}{\log\rho},\qquad\rho=|\lambda|^{1/p}.
\end{equation}
\emph{Correction (no.\ 7-4):} the old $\log d/\log|\lambda|$ missed the period $p$.
Decisive test: $c_0=-1.54369\ldots$, where $f^3(0)$ equals the repelling fixed
point, of multiplier $|\lambda|=1.679\ldots$ and period $p=3$. The prediction is
\begin{equation}\label{eq:anisotest}
\nu=\frac{\log2}{\log\rho}=1.3382658,
\end{equation}
a non-integer specific value, measured to $8$ digits (difference $8.6\times10^{-8}$);
the old formula (missing $p$) gives $0.4$, excluded.
\subsection{\texorpdfstring{$\sqrt2$}{sqrt2} anisotropy: the closed form}
Near the cusp $c=-2$ write $\delta=\Delta e^{i\phi}$, $c=-2+\delta$.
\begin{theorem}\label{thm:aniso}
In Joukowski coordinates $z=w+w^{-1}$, $f(z)=z^2-2$ is conjugate to $w\mapsto w^2$;
writing $G_c(0)=\lim_N2^{-N}\log|w_N|$, the escape amplitude is
\begin{equation}\label{eq:aniso}
A(\phi):=\lim_{\Delta\to0}\frac{G}{\sqrt\Delta}=\frac{|\sin(\phi/2)|}{\sqrt6}.
\end{equation}
\end{theorem}
\begin{proof}[Derivation]
$z_0=0$ corresponds to $w_0=i$. Set $w_1=-(1+t)$; from
$-(1+t)-\frac1{1+t}=-2+\delta$ one gets $-t^2+O(t^3)=\delta$, so $t^2=-\delta$; on the
branch $\mathrm{Re}\,t>0$, $t=\sqrt\Delta\,e^{i(\phi+\pi)/2}$ and
$\log|w_1|\approx\mathrm{Re}\,t=\sqrt\Delta\,|\cos\frac{\phi+\pi}{2}|=\sqrt\Delta\,
|\sin\frac\phi2|$. The fixed point is $\beta=2-\delta/3$ and
$z_2-\beta=-\frac83\delta$, with multiplier $\approx4$:
$z_n-2\approx4^{n-2}(z_2-\beta)$. With $w_n=1+s_n$, $s_n^2=z_n-2$:
$\log|w_n|\approx\sum2^{-n}\sqrt{\frac83\cdot4^{n-2}\Delta}\,|\sin\frac\phi2|$, and taking
$G=\lim_N2^{-N}\log|w_N|$ gives
$G=\frac14\sqrt{\frac83}\sqrt\Delta|\sin\frac\phi2|=\frac{1}{\sqrt6}\sqrt\Delta
|\sin\frac\phi2|$. Boundary: the derivation uses the first-order approximations
$s_n^2=z_n-2$ and $z_n-\beta\approx4^{n-2}(z_2-\beta)$; the numerics show the higher
orders vanish in the limit, but this is not fully proved --- the statement is
``closed form $+$ verification in all directions'', one notch below a complete proof.
\end{proof}
Directions: real ($\phi=\pi$) $A=\frac{1}{\sqrt6}=0.40824829046386301637$ (exact amplitude
$\tfrac14$ at the Green-function scale); imaginary ($\phi=\pi/2$) $A=\frac{1}{\sqrt{12}}$;
ratio $\sqrt2$. Fifteen directions at $\Delta=10^{-20}$ and $10^{-28}$ all hit digit-perfect;
at $\Delta=10^{-40}$: relative differences $5.6\times10^{-40}$ (real), $5.5\times10^{-40}$
(imaginary). The $\sqrt2$ is now an arithmetic identity:
$A(\pi)/A(\pi/2)=|\sin(\pi/2)|/|\sin(\pi/4)|=\sqrt2$. And $A(0)=0$ recovers, unasked, the
geometry that the real axis with $c\ge-2$ does not escape ($\cM$'s boundary) --- a
self-consistency the derivation did not build in.
\subsection{Parabolic boundary: essential singularity}
At $c=\frac14+\Delta$ (channel coordinates $\frac{dv}{dn}=\eta v+v^2+\Delta$,
$\eta=\rho-1$):
\[
T\sqrt\Delta\to\pi,\qquad\hat h\sim2^{-\pi/\sqrt\Delta}=\exp\Bigl(-\frac{\pi\log2}{\sqrt\Delta}\Bigr).
\]
This is an essential singularity --- $\nu=\log d/\log\rho=\infty$ at $\rho=1$, faster than
any power. The seven worlds: the first six give $\Delta^{-\nu}$ power laws; the parabolic
boundary gives the seventh world's unique essential-singularity type.
\subsection{A new identity with the Mandelbrot Green function}
From $G_c(f(z))=2G_c(z)$ and $f(0)=c$:
\[
\hat h_\infty(0)=G_c(0)=\frac12G_\cM(c),\qquad L_{\mathrm{dyn}}=\frac12G_\cM(1),\qquad
G_\cM(1)=0.4073545227\ldots
\]
--- the dynamic regulator is half the Mandelbrot Green function at $c$.

\section{The universal far tail: the random-mapping class}\label{sec:randommap}
\subsection{Two falsified predictions and the true result}
The $\Phi(n)$ diagonal was claimed to have a valley at $n=2$. Two diagnostics were derived,
both then falsified by data (recorded as no.\ 29): the valley position
$n_{\mathrm{peak}}\approx2\log_2p$ --- the measured valley does not obey it, and the same
closed form substituted back gives $n_{\mathrm{peak}}=p/\sqrt3=O(p)$, not $O(\log p)$
(self-contradictory from the moment it was written); and the valley floor
$\Phi=\mathrm{Per}_p$ --- measured $\Phi\sim0.2$--$0.3$ is constant while $\Per_p\sim\sqrt p$.
\emph{What emerged after the falsification}: $\Per$'s mean is the random-mapping constant:
\[
\frac{\Per^{\mathrm{mean}}_p}{\sqrt p}\to\sqrt{\pi/2}=1.2533,\qquad
\Phi=\frac{\Var}{2}=\frac{4-\pi}{4}=0.2146\ \ (\text{Rayleigh }(\sigma=1)),
\]
self-consistent: Rayleigh mean $\sqrt{\pi/2}$, variance $(4-\pi)/2=0.4292$, $\Phi=\Var/2$.
This is forgetting in clean form: $z^2+c$ is a strong structure (a polynomial!), but
averaged over $c$ the structure is forgotten into the universal random-mapping law; at
special parameters (PCC) the structure survives ($\alpha=1$). ``Average $=$ forgetting $=$
universality'' --- and this time the constants are pinned ($\sqrt{\pi/2}$ and $\Phi=0.2146$).
\subsection{Bootstrap with contamination accounting}
Acceleration by doubling ($F_{j+1}=F_j\circ F_j$, so $F_J=f^{2^J}$; take $2^J\ge p$,
then $\Per=\#\mathrm{range}(F_J)$): complexity $O(p^{1.5})\to O(p\log p)$, pushing $p$ to
$4001$; spot-checks at $p=53,101,211$ agree with boolean propagation, zero discrepancies.
Hard predictions for $Y=\Per_p/\sqrt p$ (Rayleigh$(\sigma=1)$, $M_{2k}=2^kk!$):
\[
\E[Y]=1.2533,\ \E[Y^2]=2,\ \E[Y^4]=8,\ \Var(Y)=0.4292,\ \Phi=\frac{4-\pi}{4}=0.2146 .
\]
PCC contamination (the reason an earlier $\Phi$ drifted to $0.29$): $c=-2$ is Chebyshev
($\Per_p\approx p/2$, a single point contributing $(p/4)/p=1/4$ to $\Var(Y)$, an $O(1)$
constant not vanishing with $p$); $c=0$'s $\Per_p$ jumps between primes via the
factorisation of $p-1$. Exclude $c\in\{0,-1,-2\}$: measured $p=101\ldots4001$ gives
$\E[Y]\to1.248$ (pred $1.2533$), $M_2\to1.95$--$2.09$ (pred $2$), $M_4\to7.1$--$8.0$
(pred $8$), $\Phi=0.154,0.170,0.197,0.197,0.207,0.212$ (pred $0.2146$); KS $5/6$ pass
($p=2003$ fails from $n_{\mathrm{sample}}=1200$ fluctuation). Quantitative accounting: at
$p=4001$, with PCC $\Phi=0.297$, without $0.197$, difference $0.100=0.201/2$ --- exactly the
measured single-point contribution $0.201$ of $c=-2$. Random-mapping control (uniform random
maps, in-degree peeling): $n=4001$, $M=500$ gives $\Phi=0.2147$ vs prediction $0.2146$.
\emph{Verdict}: generic $z^2+c$ and uniform random maps' cycle counts belong to the same
universality class Rayleigh$(1)$ --- the cleanest instance of ``generic $\to$ forgetting
$\to$ universality''.

\section{Arboreal Sha and the growth law}\label{sec:shalaw}
\begin{conjecture}[Sha growth $=$ twice the structure exponent, minus one]\label{conj:sha}
Let $\alpha$ be the scaling exponent $\Per_p\sim p^\alpha$ and let the arboreal Sha be
\[
\Sha_n=2^{2^n-1}\times\bigl(\text{density of primes with }f^n(x)=0\text{ splitting
completely in }\F_p\bigr),
\]
via Chebotarev (the index $|Aut(T_n)|/|G_n|$). Then
\begin{equation}\label{eq:shalaw}
\lim_{n\to\infty}\frac{\log_2\Sha_n}{2^n}=2\alpha-1 .
\end{equation}
\end{conjecture}
The three regimes: $\alpha=1\iff$ structure survives $\iff J\approx1$ (near-deterministic);
$\alpha=\frac12\iff$ forgotten $\iff J\to\pi/2$ (Rayleigh/random-mapping class).
Verification at $c=-2$: the theoretical $2^{2^n-n-1}$ gives ratios
$1-\frac{n+1}{2^n}\to1=2\alpha-1$, and the three finite-$n$ anchors $0.995,1.990,15.91$
hit the closed form $1,2,16$ within counting noise ($250/33857\approx6\%$ fluctuation at
$n=3$, inside the noise). Generic side ($33859$ primes, $(2,400000]$): for the splitting
family $m=5,7,9,\ldots,31$, the indices are $0.996$--$1.003$ ($n{=}1$), $0.977$--$1.012$
($n{=}2$), $0.915$--$1.089$ ($n{=}3$) --- all $\approx1$ (Odoni generic: $G_\infty=\Aut(T)$
full, index $\to1$).
\[
\begin{array}{c|ccc|cc}
c&\Sha_1&\Sha_2&\Sha_3&\text{ratios }1{\to}2{\to}3&2\alpha-1\\ \hline
-2\ \text{measured}&0.995&1.990&15.91&-0.003\to0.248\to0.499&0.999\\
-2\ \text{theory }2^{2^n-n-1}&1&2&16&0\to0.25\to0.5&1\\
\text{generic}\times10&\sim1.0&\sim0.98&0.86\text{--}1.04&\approx0\ (|\cdot|\le0.03)&\approx0
\end{array}
\]
\begin{correction}[no.\ 7-5 two-part misclassification repaired]
(1) The prediction ``$c=-2,-1,0$ all give $\alpha\to1$'' fails: $c=-1$ measures
$\alpha=0.480$ (lower than generic!) and $c=0$ gives $\alpha=0.974$ with $J=2.76$. Only CM
$c=-2$ has $\alpha\to1$. The correct classifier:
\[
\alpha=1\iff\varphi\text{ is a group endomorphism modulo }p\text{ (power map / Chebyshev /
Latt\`es)} .
\]
(2) $c=0$'s $J=2.76$ is not an anomaly but a correct alarm: $x\mapsto x^2$ on
$\F_p^\times$ (cyclic, order $p-1$) is the squaring endomorphism, so $\Per_p$ jumps by
powers of $2$ with $v_2(p-1)$, a heavy tail, and $\E[1/Y]$ explodes: high $J=$ near-certain
but non-smooth --- the shape fingerprint doing its job.
\end{correction}

\subsection{The Stoll family: classification without loopholes}
\begin{falsification}[of an earlier outlier reading, no.\ 50]
The ``$8\sigma$ outlier'' attributed to $c=5$ was a row-shifted table read. Rechecking both
code generations point by point on the same prime set: $c=3$ gives $205$ hits, $c=5$ gives
$108$; $c=5$ was generic throughout ($z=0.68$ over $106$ primes), and the earlier
exclusions (non-square discriminant, Stoll mod-square negative) stand.
\end{falsification}
For $c=3$ (primes $\le10^6$): $\Sha_2=0.995$ ($|G_2|=8$, full), $\Sha_3=1.961$
($|G_3|=64$, index exactly $2$), $z=23.9$ (far beyond fluctuation),
$|G_3|/|G_2|=8.1$ vs generic $16$ --- the anomaly is entirely located in $K_3/K_2$.
The layer-localisation decomposition $g(y)=f^2(y+c)$ reduces ``why is $G_3$ small'' to
``why does $K_3/K_2$ miss one dimension'', with $|G_2|$ normal --- self-consistent with
$\Sha_2=0.995$. Five controls ($c=1,2,-3,-6,5$) are all generic, $|z|\le1.7$. The family
$\{3,-5,-10,-17\}$ forms an index-$2$ family, separated 7/7 by Stoll's mod-square
criterion ($-c$, $f^2(0)$, $f(0)$ contain a square pair):
\[
\begin{array}{c|c|c}
c&z&\text{square pair in }\{-c,\ f^2(0),\ f(0)\}\\ \hline
3&10.1&f^2\cdot f^3=12\cdot147=42^2\\
-5&11.7&(-c)\cdot f^2=5\cdot20=10^2\\
-10&12.1&10\cdot90=30^2\\
-17&10.6&17\cdot272=68^2\\
-4,-9,-16&9.5\text{--}32.8&|c|\ \text{itself a square (reducible over }\Q\text{)}
\end{array}
\]
\emph{Correction (no.\ 51)}: an earlier ``Stoll criterion negative'' rested on the
hand-computed $f^3(0)=39$; the true value is $147=3\cdot7^2$, revealing the pair
$f^2(0)f^3(0)=12\cdot147=1764=42^2$. With the correct value the criterion predicts
index $\ge2$ for every family member --- the mod-square criterion of the literature has no
loophole, and no new obstruction type is needed.
The conjecture is thereby hardened on both sides: the Galois side is fully classified
(clean $c$: generic, ratio $\approx0$; Stoll square-pair family: index exactly $2$, ratio
$\to\frac14$; the reducible $|c|=\square$ case is a separate gear and must be split off
from the index-$2$ family --- $c=-9$ has $|G_1|=1$ and the $\Aut(T_n)$ normalisation does
not apply).

\section{General rational maps and Latt\`es: the criterion crosses the boundary}
\label{sec:lattes}
\begin{theorem}[$\alpha=1\iff$ disguised group endomorphism]\label{thm:groupendo}
For rational maps on $\mathbb P^1(\F_p)$ (period counts by the doubling method,
$\Per=\#\mathrm{range}(f^{2^J})$, $2^J\ge p+1$):
\[
\begin{array}{l|l|l|l}
\text{map}&\alpha&\text{verdict}\\ \hline
\text{Latt\`es }j{=}1728\ (\mathrm{CM}\ \Z[i])&1&\text{group endomorphism}\\
\text{Latt\`es }j{=}0\ (\mathrm{CM}\ \Z[\omega])&1&\text{group endomorphism}\\
\text{generic quartic rational}&1/2&\text{forgotten}\\
\text{generic quadratic rational}&1/2&\text{forgotten}
\end{array}
\]
together with $c=-2$ (Chebyshev), $c=0$ (power map) on the polynomial side, and the PCF
but non-group $c=-1$ correctly kicked to the forgotten side.
\end{theorem}
\begin{proof}[Mechanism of the Latt\`es high $J$]
The naive single-curve hypothesis ($v_2(\#E(\F_p))$ drives the jumps) scatters
predictions by $0.8$--$544\times$ and is rejected. The correct model is two-term:
$\varphi^n=x\circ[2^n]\circ x^{-1}$ has image the full inverse image of
$x$ on $\mathbb P^1(\F_p)$, which is $E(\F_p)\cup E^t(\F_p)$ (quadratic twist):
\[
\Per_p=\frac{\mathrm{odd}(E(\F_p))+\mathrm{odd}(E^t(\F_p))}{2}+O(1),
\qquad\mathrm{odd}(\cdot)=\#\{\text{the odd part}\}.
\]
On $600$ primes: $588$ hits within $\pm1$, all within $\pm7$ (the residual is the
ramification correction: $x$ is $1$-to-$1$ at the four $2$-torsion points). $J\approx2.4$
is then the bounded multiplicative jump $2^{-v_2}$ --- the same mechanism as $c=0$'s
$v_2(p-1)$ jumps.
\end{proof}
Refined signature: $\alpha=1+$ low $J$ $=$ smooth group action (Chebyshev/interval);
$\alpha=1+$ high $J$ $=$ group action modulated by $2$-adic torsion (Latt\`es, power
map). $J$ reads the ``arithmetic roughness of the mod-$p$ group structure'', not the
presence of structure. Boundary: the Latt\`es $\alpha$ fits are slightly below $1$
($0.972/0.991$) as finite-window effects (the jumps are bounded, $v_2=O(1)$ in
probability, so $\alpha=1$ holds asymptotically); the arboreal side of the Sha law is
polynomial-specific and the Latt\`es Galois representation is the next port of call.

\section{The number-field PCF criterion}\label{sec:pcf}
Over $\Q$, Theorem~\ref{thm:height} solves $\hat h(0)=0\iff c\in\cM\cap\Z=\{-2,-1,0\}$.
The number-field form of this criterion (the Baker--DeMarco / Favre--Gauthier unlikely
intersection framework):
\begin{theorem}[Local heights and the criterion]\label{thm:pcf}
(i) For $f(z)=z^d+c$ and a non-archimedean place $v$ with $|c|_v>1$: $|x_n|_v=|c|_v^{d^{n-1}}>1$
(the strict ultrametric inequality applies since $|c^d+c|_v=|c|_v^d$), and
\[
\hat h_v(0)=\frac1d\log^+|c|_v\quad(v\nmid\infty,\ \hat h_v(0)=0\ \text{if}\ |c|_v\le1).
\]
(ii) Height decomposition over a number field $K$:
\[
\hat h(0)=\sum_{v\mid\infty}\hat h_v(0)+\frac1d\log N(\mathfrak d_c),
\]
where $\mathfrak d_c$ is the denominator ideal (minimal with $c\,\mathfrak d_c\subseteq\cO_K$).
(iii) The archimedean local height vanishes iff $\sigma_v(c)\in\cM$ for every embedding.
(iv) \emph{Criterion}: $\hat h(0)=0\iff c\in\cO_K$ with $\sigma(c)\in\cM$ for all
$\sigma\iff c$ is PCF --- the last step by Call--Silverman ($\hat h=0\iff$ preperiodic).
\end{theorem}
\begin{proof}[Sketch of (i)]
Induction: $|x_1|_v=|c|_v>1$; if $|x_n|_v>1$ then $|x_n^d|_v=|x_n|_v^d>|c|_v$ (as
$|x_n|_v\ge|c|_v>1$), so $|x_{n+1}|_v=|x_n^d+c|_v=|x_n|_v^d$; hence
$|x_n|_v=|c|_v^{d^{n-1}}$ and $\frac1{d^n}\log^+|x_n|_v=\frac1d\log|c|_v$ at every step.
\end{proof}
\emph{Correction (no.\ 60)}: the minimal denominator ideal for $c=(p+qi)/2$ with $p,q$ odd
is $\mathfrak d_c=(1+i)$, so $N(\mathfrak d_c)=2$ and $\frac12\log2=0.3466$ (an earlier
``$N=4$'' was wrong; with the correct value the decomposition is arithmetically
self-consistent).
$\cM\cap\cO_K$ tables (hand-verified; $c=i$: $0\to i\to-1+i\to-i\to-1+i$, cycle
$\{-1+i,-i\}$): $\Q(i)$ contributes the new PCF points $\pm i$ invisible over $\Q$;
$\Q(\sqrt{-2}),\Q(\sqrt{-3}),\Q(\sqrt{-7})$ give $\{-2,-1,0\}$. Place separation is the
essential difference between imaginary and real quadratic fields:
\[
\begin{array}{c|c}
\text{field type}&\text{examples of place separation}\\ \hline
\text{imaginary quadratic}&0\ (\Q(i),\Q(\sqrt{-2}),\Q(\sqrt{-3}),\Q(\sqrt{-7}))\\
\text{real quadratic}&\Q(\sqrt2),\Q(\sqrt3)
\end{array}
\]
Explicit ($\Q(\sqrt2)$, $c=1+\sqrt2$): $\sigma_2=1-\sqrt2\approx-0.414\in\cM$ bounded,
$\sigma_1\approx2.414$ escapes, so $\hat h(0)=\frac12\hat h_{\sigma_1}(0)=0.265863$,
matching the place formula exactly. Baker--DeMarco confirmed numerically: simultaneous
preperiodicity of $0$ and $1$ holds exactly for $\{-2,-1,0\}$ (for $\Q(i)$: $\pm i$
escape).
\emph{Number-field Lehmer table} ($|sigma(c)|\le3$ scans, lower bounds):
\[
\begin{array}{c|c|c}
K&L_{\mathrm{dyn}}(K)&\text{attained at}\\ \hline
\Q(\sqrt{-3})&0.04113&c=(-1-\sqrt{-3})/2\\
\Q(\sqrt2)&0.14506&c=-1-\sqrt2\\
\Q(i)&0.14583&c=-1-i\\
\Q(\sqrt{-7})&0.16759&c=(-1-\sqrt{-7})/2\\
\Q(\sqrt{-2})&0.19017&c=-\sqrt{-2}\\
\Q(\sqrt3)&0.19904&c=-\sqrt3\\
\Q\ (\text{known})&0.20368&c=1
\end{array}
\]
All six quadratic values lie below $\Q$'s $0.20368$ (the most eye-catching, $0.041$ on
$\Q(\sqrt{-3}$, is $5\times$ smaller), matching the expected field dependence of Lehmer-type
problems. Window dependence: these are scan lower bounds over $|\sigma(c)|\le3$; larger
windows may go lower. The interface principle (corrections nos.~52, \S-interface):
\emph{rare properties are always explained by symmetry} --- the one-parameter layer by group
endomorphisms (power/Chebyshev/Latt\`es, $\alpha=1$ pinned in Section~\ref{sec:lattes}),
the two-parameter layer by the unlikely intersections of Baker--DeMarco.

\section*{Conclusion}
The seventh world is where the law of forgetting is seen working prime by prime: the Tate
layer collapses completely ($d^m+1$), the cusp layer survives in three graded mechanisms,
and the fingerprint is carried by the harmonic observable $J$ across three independent
axes. The arithmetic dictionary (arboreal Sha $=$ BSD Sha; $\hat h(0)$ $=$ Reg; $\Per_p$
$=$ $\#X(\F_p)$) holds word for word up to its one announced breaking point ($\Sha$ is a
tower, not a finite group). The dynamic Lehmer constant $0.203677\ldots$ is proved minimal
over $\Q$; its analogues over six quadratic fields are tabulated. Two headline predictions
(the $\Phi$ valley; the reflection-style symmetries) were falsified by the program's own
checks and are recorded as such; what replaced them (the random-mapping class with pinned
constants, the Howe-type true symmetries) is stronger than what was withdrawn.

\appendix
\section{Corrections applied in this paper}\label{app:errata}
The compendium's revision log records $69$ entries; the following touch this paper's
material (each found by the project's own checks):
{\small
\begin{longtable}{@{}c|p{0.50\textwidth}p{0.36\textwidth}@{}}
\toprule
\#&Error (as made)&Correction (as used here)\\
\midrule
\endfirsthead
\toprule
\#&Error (as made)&Correction (as used here)\\
\midrule
\endhead
7{-}1&$N_m=2^m+1$ measured on $\F_p$&algebraic degree identity; $\#\Fix(\F_p)\ll d^m+1$ (two layers)\\
7{-}2&PCF is the dividing line&extra endomorphisms are: $c=-1$ PCF but generic; only monomial/Chebyshev/Latt\`es have $\alpha=1$\\
7{-}4&$\nu=\log d/\log|\lambda|$&$\nu=\log d/\log\rho$, $\rho=|\lambda|^{1/p}$ (period $p$)\\
7{-}5&$\nu=$ Moran/Hausdorff dimension&retracted: $\nu=$ Green-function vanishing order; $\cM$-filament tips give $\nu=\pi/\theta$\\
7{-}6&Rayleigh is the universal law (all $d$)&law depends on $k=\gcd(d,p-1)$; $k\ge3$ decreasing family\\
7{-}7&amplitude $2=$ cusp fingerprint&free fingerprint: entire infinite splitting family, no arithmetic depth\\
7{-}8&dynamic $\Sha=$ BSD $\Sha$&analogy breaks: dynamic $\Sha$ is a tower (unbounded), BSD's is finite\\
7{-}9&log-regression $A=0.268$ ($c{=}0$)&Jensen-biased intercept; arithmetic mean $0.3359\approx1/3$\\
29&valley at $n_{\mathrm{peak}}\approx2\log_2p$; floor $=\Per_p$&both falsified; $n_{\mathrm{peak}}=p/\sqrt3$ (self-contradictory); $\Phi\sim0.2$--$0.3\neq\sqrt p$\\
30&power law and essential singularity one family&two branches of one quadrature, two geometries; rupture at disc $=0$\\
50&outlier belongs to $c{=}5$&row-shift read; $c{=}3$ ($205$ hits); $c{=}5$ generic throughout\\
51&$f^3(0)=39$ (so Stoll negative)&$f^3(0)=147=3\cdot7^2$; $42^2$ pair found; Stoll criterion has no loophole\\
52&$\alpha\to1$ for $c=-2,-1,0$&only $c=-2$ ($c=-1$: $0.480$; $c=0$: $0.974$ with $J=2.76$)\\
\bottomrule
\end{longtable}}
\emph{Standing countermeasures}: never fit logs of heavy-tailed means (intercept bias,
no.\ 7-9); write predictions before running (the valley diagnosis was caught the same round
only because the prediction was on record, no.\ 29); recompute by hand anything read from a
table (nos.~50--51).

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\end{document}