TFPT scientific source excerpt
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000304 | \section*{Research Contract --- \texttt{CHIRAL4D.MIRROR.DET16.01} (DET16 rank-one projector, finite cluster)}
000305 | \addcontentsline{toc}{section}{Research Contract: CHIRAL4D.MIRROR.DET16.01}
000306 | % =====================================================================
000307 | \noindent\emph{Typing.} A candidate \mC\ (ledger row \texttt{CHIRAL4D.MIRROR.DET16.01}),
000308 | registered 2026-08-29 from review wave~5.
000309 | (\veri{v1002\_\allowbreak spin10\_\allowbreak mirror\_\allowbreak projector.py}.)
000310 | \textbf{Content.}  Exact finite-cluster gapping on the full chiral $2^{16}$ Fock space via the
000311 | number-violating determinant vertex, as executed above.  \textbf{Open.}  Dynamical gauge fields,
000312 | domain-wall geometry, and a $4$D volume theorem --- this is not a $3{+}1$D closure.
000313 | \textbf{Relation.}  Does \emph{not} upgrade \texttt{CHIRAL4D.NOMIRROR.01}; the Casimir-projector
000314 | kill is recorded on that parent.  Status: Candidate \mC.
000315 | 
000316 | \par\smallskip\noindent\textbf{Executed update (2026-08-30, review wave~7): T1/T2 proved, T3
000317 | Michalakis--Zwolak cited-verified, dynamical [N], $1/192$ and BDL boxed
000318 | (\veri{v1007\_\allowbreak det16\_\allowbreak stability\_\allowbreak battery.py};
000319 | the candidate stays \mC).}
000320 | The $12$-page memorandum
000321 | \texttt{articles/2026-08-30/\allowbreak det16\_\allowbreak mirror\_\allowbreak gap\_\allowbreak theorem\_\allowbreak en.tex}
000322 | proves T1/T2 and verifies every Michalakis--Zwolak hypothesis (Local-TQO, $t_*>0$, perturbed
000323 | gap at least $1/2$).  Ordinary hopping is \emph{not} block-diagonal
000324 | ($\lVert(1-P)KP\rVert\neq0$), so $t_{0,\mathrm{count}}=1/192$ remains the uncertified counting
000325 | threshold, not a theorem.  Dynamical $\mathbb Z_2/\mathbb Z_4$ links on the four-mode analogue
000326 | are uniquely gapped on the compressed grid (full-probe minimum $0.728854416$ pinned);
000327 | second-order Schrieffer--Wolff is $1-\alpha t^2+O(t^4)$ with $c_{\mathrm{geo}}/L$ bounded.
000328 | BDL supplies a ground-energy truncation error with unspecified constants, not a volume-uniform
000329 | gap remainder --- boxed.  Display marker stays \mC.
000330 | 
000331 | \par\smallskip\noindent\textbf{Executed update (2026-08-31, Monday-morning harvest;
000332 | \veri{v1016\_\allowbreak state\_\allowbreak gap\_\allowbreak batteries.py};
000333 | the candidate stays \mC).}
000334 | Fröhlich--Pizzo hypotheses cited-verified for $\mathbb Z_2/\mathbb Z_4$ DET open chains:
000335 | volume-uniform spectral gap $\ge\Delta/2$ in the parametric window $\tau_{\mathrm{FP}}$
000336 | ($a=0.0233$, $a/16=0.00146$).  The dynamical-link leg is closed at that finite-group
000337 | level.  Remaining: rotor links, wall mixing, $3{+}1$D placement.  Display marker stays \mC.
000338 | 
000339 | \par\smallskip\noindent\textbf{Executed update (2026-09-05, round-3 integration):
000340 | derivative-filtered wall block
000341 | (\texttt{CHIRAL4D.MIRROR.DFILTER.BLOCK.01};
000342 | \veri{v1023\_\allowbreak derivative\_\allowbreak filtered\_\allowbreak wall.py}; no parent
000343 | marker move).}
000344 | For
000345 | \[
000346 |  H_{\rm df}=\Delta b^\dagger b+
000347 |  \lambda\bigl(b+gD_{\rm GW}p\bigr)^\dagger
000348 |               \bigl(b+gD_{\rm GW}p\bigr)
000349 | \]
000350 | each singular-value sector is the exact $2{\times}2$ block with determinant
000351 | $\lambda g^2\Delta s^2$.  It preserves $\ker D_{\rm GW}$ on the physical summand,
000352 | the adiabatically mirror-continuated high branch is at least $\Delta+\lambda$, and
000353 | \(
000354 |  \mu_-(s)=\lambda g^2\Delta(\Delta+\lambda)^{-1}s^2+O(s^4).
000355 | \)
000356 | The finite covariance and SVD-reduction defects are
000357 | $2.408{\times}10^{-15}$ and $1.776{\times}10^{-15}$.  This is only a finite
000358 | quasi-free block lemma.  For $s\ne0$ the bare $b$ subspace is not invariant and has
000359 | low-band weight, so no bare-mirror response gap is proved; the low branch has
000360 | $z=2$, not a Weyl cone.  No canonical DET16/DET32 realization, interacting parent,
000361 | admissibility/quasilocality theorem, measure/anomaly/index control, universality or
000362 | $3{+}1$D continuum limit follows.  The derivative-filtered witness therefore does
000363 | not upgrade \texttt{CHIRAL4D.MIRROR.DET16.01}~\mC,
000364 | \texttt{CHIRAL4D.NOMIRROR.01}~\mO, T4, or TOE.
000365 | 
000366 | % =====================================================================