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000153 | \section*{Research Contract --- \texttt{TFPT4D.LATTICE.ACTION.01} (the explicit finite 4D lattice action and its seven finite gates)}
000154 | \addcontentsline{toc}{section}{Research Contract: TFPT4D.LATTICE.ACTION.01}
000155 | % =====================================================================
000156 | \noindent\emph{Typing.} A research contract \mO\ (ledger row \texttt{TFPT4D.LATTICE.ACTION.01});
000157 | route step (2) of the \texttt{DIMENSION.UPLIFT.FIREWALL.01} ordering made concrete --- the
000158 | single object on which nearly all open 4D questions become simultaneously testable.
000159 | \textbf{Demand.}  Exhibit one fully explicit finite family of 4D Euclidean lattice theories
000160 | \[
000161 | \mathcal Z_a[J]=\int\mathcal DU\,\mathcal D\psi\,\mathcal D\Phi\;
000162 | e^{-S_a[U,\psi,\Phi]+J\mathcal O},
000163 | \qquad
000164 | S_a=S_{\rm Wilson}[U]+\bar\psi\,D_{\rm GW}[U,\Phi]\,\psi+S_\Phi[U,\Phi]+S_{\rm seam/top},
000165 | \]
000166 | where $U$ are the fundamental holonomies (TFPT fixes representation content, charge lattice,
000167 | coupling relations and boundary conditions), $D_{\rm GW}$ is a Ginsparg--Wilson/overlap operator
000168 | for the TFPT chiral representation content (the \texttt{CHIRAL4D.NOMIRROR.01} operator), $\Phi$
000169 | is the \emph{only} allowed relevant scalar/link-mixing sector (EWSB downstream), and
000170 | $S_{\rm seam/top}$ is the boundary/topological module transporting the seam data into the bulk
000171 | --- first at finite lattice spacing $a$, deliberately \emph{without} continuum claims.
000172 | \textbf{Finite gates} (all machine-checkable at finite size; the promotion criteria for any
000173 | candidate): \textbf{(T1)}~exact gauge invariance; \textbf{(T2)}~a positive transfer matrix (RP
000174 | at finite $a$); \textbf{(T3)}~the seam boundary restriction reproduces the TFPT seam data
000175 | \emph{exactly}; \textbf{(T4)}~anomaly bookkeeping including the global $SU(2)$ anomaly;
000176 | \textbf{(T5)}~the index/generation count; \textbf{(T6)}~the mirror spectrum separated;
000177 | \textbf{(T7)}~at least one \emph{nonvanishing} connected four-point function.
000178 | \textbf{Kill tests.}  No local reflection-positive action with the correct seam restriction
000179 | exists (master route dead); every candidate is generalized-free (T7 unreachable); a hidden
000180 | freely tunable dimensionless parameter is required (the strong parameter-freeness claim is
000181 | refuted); the uniform twistor flat direction shifts \emph{normalized} observables rather than
000182 | only the vacuum normalization of $\mathcal Z$ (a genuine undetermined modulus).
000183 | \textbf{Relation.}  The continuum limit is the separate contract \texttt{QFT4D.OS.RECON.01};
000184 | the unitary dilation is \texttt{DYN.UNITARY.DILATION.01}; $d=4$ is typed by
000185 | \texttt{DIMENSION.SELECTOR.4D.01} \mC; the generating-functional collapse is
000186 | \texttt{FTRANSFER.GENERATING.01}.  Chiral lattice state of the art is calibrated in
000187 | \texttt{CHIRAL4D.NOMIRROR.01} (L\"uscher's exactly gauge-invariant abelian construction,
000188 | arXiv:hep-lat/9811032; the nonabelian anomaly analysis, arXiv:hep-lat/9904009; reflection
000189 | positivity proven for \emph{free} overlap fermions, Kikukawa--Usui arXiv:1005.3751 --- none of
000190 | this is the interacting chiral Standard Model).  Nothing in the corpus closes any gate.
000191 | Status: \texttt{TFPT4D.LATTICE.ACTION.01} \mO.
000192 | 
000193 | \par\smallskip\noindent\textbf{Executed update (2026-08-28, promote round): the T1--T7 harness
000194 | plus a named Euclidean T2 kill and a Hamiltonian-route T2 clearance
000195 | (\veri{v989\_\allowbreak lattice\_\allowbreak gate\_\allowbreak battery.py}; no gate closed).}
000196 | T4 exact SM anomaly bookkeeping (a lepton-drop mutant fires).  T5/T6: overlap index equals flux
000197 | $q$ on an $8\times8$ torus; Ginsparg--Wilson circle to machine precision.  The Euclidean overlap
000198 | candidate is killed at T2: $|\det D_{\rm ov}|^2$ is PSD at $N_t=2$ and indefinite at $N_t=4$
000199 | (det-only $\lambda_{\min}\approx-0.249$); Wilson fermions (ultralocal in time) stay PSD --- the
000200 | break is pinned on overlap time nonlocality.  The Hamiltonian route clears T2 by construction
000201 | ($H=H^\dagger$ exact $\Rightarrow T=e^{-aH}$ symmetric PD; interior Gauss exact including the
000202 | seam-sector intertwiner; background-field $\mu_4$ split with $k=1,3$ degenerate).  Display
000203 | marker stays \mO.
000204 | 
000205 | \par\smallskip\noindent\textbf{Registered decision (2026-08-28,
000206 | \texttt{QFT4D.LATTICE.FUNDAMENTAL.01}): the lattice-fundamental reading --- a typed
000207 | route/typing, \emph{not} an \mE\ claim.}
000208 | The physical completeness criterion is this finite local unitary lattice theory (Hamiltonian
000209 | route), not the continuum OS limit.  That limit (\texttt{QFT4D.OS.RECON.01}) is retyped as
000210 | mathematical reinforcement; Poincar\'e invariance is an IR fixed-point property.  The display
000211 | marker of this contract stays \mO; nothing closes.  See the decision paragraph at
000212 | \texttt{QFT4D.OS.RECON.01}.
000213 | 
000214 | \par\smallskip\noindent\textbf{Amendment (2026-08-29):} a single finite box is not a world ---
000215 | the fundamental object is the quasilocal consistent family $\{H_\Lambda\}$ with
000216 | $\tau_t=\lim e^{itH_\Lambda}A e^{-itH_\Lambda}$; continuum optional, thermodynamic limit and
000217 | controlled IR universality mandatory.  Full prose at \texttt{QFT4D.OS.RECON.01}.
000218 | 
000219 | \par\smallskip\noindent\textbf{Executed update (2026-08-30, review wave~7): $2{+}1$D scaffold
000220 | coherent (\veri{v1008\_\allowbreak master\_\allowbreak assembly\_\allowbreak scaffold.py};
000221 | Decision typing UNCHANGED).}
000222 | Combinatorial $K=0$, unique-contractive $\lvert R'\rvert<1$, QWZ wall $+$ DET4 mirror,
000223 | $\mathbb Z_6$ assembly $(1,2,3,0)$.  $L=3$ spaces out of suite.  No \mE\ claim, no marker move.
000224 | 
000225 | \par\smallskip\noindent\textbf{Executed update (2026-08-30, afternoon harvest;
000226 | \veri{v1011\_\allowbreak 3p1d\_\allowbreak ladder\_\allowbreak battery.py};
000227 | Decision typing UNCHANGED).}
000228 | $3{+}1$D minimal object viable; fully coupled link--wall at minimal scale; Kronecker
000229 | limitation removed.  No \mE\ claim, no marker move.
000230 | 
000231 | \par\smallskip\noindent\textbf{Executed update (2026-08-30, late-evening harvest;
000232 | \veri{v1013\_\allowbreak thermodynamic\_\allowbreak dynamics.py};
000233 | Decision typing UNCHANGED).}
000234 | Mandatory-dynamics leg closed at class level (Lieb--Robinson $+$ $\tau_t$ $+$ $\mathcal A^G$
000235 | $+$ state existence).  Uniqueness / limit gap / IR universality remain.  Full prose at
000236 | \texttt{QFT4D.OS.RECON.01}.  No \mE\ claim, no marker move.
000237 | 
000238 | % =====================================================================