TFPT scientific source excerpt
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000232 | \subsection*{2026-07-21 (XIII) (\texttt{v503} --- the \texttt{v215} deferred emergence test (lever 7) executed \emph{light} and promoted as \texttt{QGEO.EMERGE.LIGHT.01}: NO dynamical selection of the square modulus --- the free field's global $\det{}'$ winner is the \emph{hexagonal} torus, $\tau{=}i$ is a saddle, criticality at the square is Palais symmetry; the square is pinned by \emph{rigidity} alone (the raw DtN is mod-$4$ block-diagonal \emph{iff} $\tau{=}i$, an isolated zero); and the residual \emph{converges}: QGEO-R1 $\equiv$ P2-R1 $=$ one common rest, ``why does the raw seam carry the order-$4$ clock''; Wolfram extension $420\to424$ (verified engine run); \emph{NO marker moves --- \texttt{QGEO.REALIZE.01} stays \mC/\mO, \texttt{AX.P2.01} stays a declared axiom})}
000233 | \begin{itemize}[leftmargin=1.5em]
000234 | \item \textbf{\texttt{v503\_qgeo\_emergence\_light.py} (\texttt{QGEO.EMERGE.LIGHT.01}, 17 checks;
000235 | mpmath $40$ digits $+$ \texttt{v280}-style lattice DtN $+$ sympy exact).} The \texttt{v215} lever 7
000236 | (``the decisive kill is deferred: compute the raw seam-collar DtN \emph{without} inputting the marks
000237 | and check whether four square marks \emph{emerge}'') executed in its first honest \emph{light}
000238 | version. \emph{No dynamical selection}: $\log F=\log(\tau_2|\eta|^4)$ (Osgood--Phillips--Sarnak) is
000239 | exactly $S$-invariant (dev $<10^{-35}$), so $t{=}1$ is symmetry-forced critical on the rectangular
000240 | family $\tau=it$; there $g''(1)=-1.298<0$ (strict maximum) but the $\tau_1$-direction has
000241 | $h''(0)=+0.298>0$: $\tau{=}i$ is a \emph{saddle} of the full moduli space; the only critical points
000242 | are the CM points ($E_2^*(i)=E_2^*(\rho)=0$ at $40$ digits, generic $0.163\neq0$); the \emph{global}
000243 | $\det{}'$ winner at fixed area is the \emph{hexagonal} torus ($\log F(\rho)=-1.03352>\log
000244 | F(i)=-1.05469$, diff $0.021169$); criticality at the square is Palais symmetric criticality (the
000245 | independent heat trace is equally critical). \emph{Sharp rigidity}: the DtN mod-$4$ off-character
000246 | indicator (the \texttt{v215}-K3 lever as a \emph{modulus function}) vanishes to $8\times10^{-15}$
000247 | exactly at $\tau{=}i$ and rises strictly monotonically both ways (min $8.4\times10^{-2}$ away) ---
000248 | an isolated zero; $R^2=\Z_2$ exact as permutation matrices (the \texttt{v492} $\Z_8$ bridge on the
000249 | lattice); $j(it)\ge1728$ with equality only at $t{=}1$. \emph{The $\Z_2$ chain is modulus-blind}:
000250 | $4$ marks $+$ $[\Z_2,\Delta_t]=0$ on \emph{every} torus (\texttt{v216} confirmed without mark
000251 | input); pillowcase $\det{}'=\tfrac12$ torus $\det{}'$ exactly ($\pm$-pair structure) --- only the
000252 | order-$4$ clock distinguishes $\tau{=}i$; $\mathrm{Aut}(E_\rho)=\Z_6$ has \emph{no} order-$4$
000253 | element (the dynamical winner cannot carry the clock); given the clock the modulus freezes
000254 | automatically ($j=1728$ identically, \texttt{v493} mini). \emph{Residual convergence (the core
000255 | finding)}: the order-$4$ element on $H^1$ is a Coxeter element of $W(A_3)$ (char poly
000256 | $\lambda^3{+}\lambda^2{+}\lambda{+}1$, order $4=|\mu_4|=\Nfam{+}1$) $=$ the P2 cusp-class carrier
000257 | datum of \texttt{v499} R1 $\Rightarrow$ \emph{QGEO-R1 $\equiv$ P2-R1, one common residual}; chain:
000258 | marks derived (\texttt{v216}), square derived \emph{given} the clock (\texttt{v493}), clock $=$
000259 | carrier input [\mC], rest $=$ its dynamical realisation \mO; route classification (\texttt{v347}
000260 | analogue): B-rigidity in the sharpened form ``RP seam data $\Rightarrow$ order-$4$ clock'' is the
000261 | live route. \texttt{v215} lever 7 thereby moves from ``deferred, not executable'' to ``light version
000262 | executed: rigidity, not dynamical selection'' --- documented in \texttt{v503}, \texttt{v215} itself
000263 | unchanged.
000264 | \item \textbf{Ledger $+$ papers $+$ Wolfram $+$ website.} Ledger: new row
000265 | \texttt{QGEO.EMERGE.LIGHT.01}; \texttt{QGEO.REALIZE.01} text \emph{precised} (rest $=$ the order-$4$
000266 | clock, identical with the P2 typing rest R1; square $+$ mark count derived) at \emph{unchanged}
000267 | marker \mC/\mO; cross-reference added in \texttt{P2.TYPING.01} (convergence theorem). Papers:
000268 | contracts kill-test paragraph (lever-7 light result $+$ the residual-convergence sentence, both with
000269 | \texttt{v503}), \texttt{tfpt\_3} Step 2 (light-converse sentence), \texttt{origin\_theory}
000270 | necessity paragraph, \texttt{introduction}/\texttt{tfpt\_1}/\texttt{tfpt\_2} one-residual
000271 | cross-references. Wolfram: four mirrors (CM-point identities $E_2(i)=3/\pi$, $E_2(\rho)=2\sqrt3/\pi$
000272 | $\Rightarrow$ $E_2^*=0$ exact, $j$/DedekindEta special values; the no-dynamical-selection witnesses;
000273 | the pillowcase halving as an exact algebraic identity; clock $\Rightarrow$ square $+$ Coxeter)
000274 | verified with the active engine the same day ($420\to424$, \texttt{ALL WOLFRAM EXTENSION CHECKS
000275 | PASSED}); the lattice DtN curves stay Python-only (flagged). \texttt{VerificationDag} masses node
000276 | gains \texttt{v503}; registry $+$ \texttt{run\_all} (497 modules, suite green). No gate closes, no
000277 | marker moves.
000278 | \end{itemize}
000279 |