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000163 | \section*{Externalization Contract --- \texttt{ALPHA.QUILLEN.EXACT.01} (the determinant-line variation theorem, packaged for external specialists)}
000164 | \addcontentsline{toc}{section}{Externalization Contract: ALPHA.QUILLEN.EXACT.01}
000165 | % =====================================================================
000166 | \noindent\emph{Typing.} The second hand-off document: the one remaining EM obligation, packaged
000167 | for an outside specialist in determinant lines / $\eta$-invariants / index theory. It
000168 | \emph{documents}, it does not close: \texttt{ALPHA.QUILLEN.EXACT.01} stays \mO\ (external math,
000169 | type A, \vref{v384}), and it is a \emph{face} of \texttt{SEAM.EQUIV.01} (\vref{v382}) --- not a
000170 | second independent gate. The \emph{value} $\alpha^{-1}$ is closed \mE\ regardless (unique root
000171 | $+$ ablation, \vref{v3}); open is only ``why \emph{this} functional''. Nothing here moves a
000172 | marker.
000173 | 
000174 | \par\smallskip\noindent\textbf{Relation (2026-08-28).}  The nonabelian upgrade of this $U(1)$
000175 | face is the new contract \texttt{GAUGE.DETLINE.FIXPOINT.01} (all three gauge couplings as unique
000176 | stationary points of one zeta-det/inflow functional; honest: $\alpha_s(M_Z)$ is currently an
000177 | external input).  This row stays the $U(1)$ face; that row is the three-coupling demand.
000178 | 
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000186 | 
000187 | \par\smallskip\noindent\textbf{(1) Minimal theorem statement.} Two steps, both over the moduli
000188 | of flat $U(1)$ connections on the $\mu_4$ seam (the twist torus):
000189 | \begin{itemize}
000190 | \item \textbf{(T1) Occupied Berry line and continuum determinant-line comparison.}
000191 |   \emph{Finite object.} Over the same twist torus $B$, specify the gapped collar
000192 |   Hamiltonians $H_L(b)$, their gauge identifications and
000193 |   $P_L(b)=\mathbf{1}_{(-\infty,0)}(H_L(b))$. The gap defines the smooth occupied
000194 |   bundle and its Berry line $L_{\mathrm{occ},L}=\det\operatorname{Ran}P_L$.
000195 |   Its FHS integer is $C_{\mathrm{FHS}}=+1$ within numerical tolerance at
000196 |   $L=4,6$ (\vref{v472}). For $\nabla=d+A_{\mathrm{occ}}$ with
000197 |   $A_{\mathrm{occ}}=\operatorname{tr}(U^\dagger dU)$ and $c_1=iF_{\mathrm{occ}}/(2\pi)$,
000198 |   the continuum Berry convention is $C_{\mathrm{Berry}}=\int_B F_{\mathrm{occ}}/(2\pi i)$,
000199 |   hence $c_1(L_{\mathrm{occ},L})[B]=-C_{\mathrm{Berry}}$.
000200 |   Identify the finite-mesh integer with this continuum value by a controlled
000201 |   mesh limit or an independent topological calculation; check duality before
000202 |   comparing its sign with a continuum theta-line convention.
000203 |   The gap check establishes no global nonvanishing section: such a section of
000204 |   this same line would force $c_1=0$, contradicting the nonzero Chern class.
000205 |   A global projector or ground-state ray is not a globally chosen ground-state vector.
000206 | 
000207 |   \emph{Continuum object.} Define independently a chiral elliptic Fredholm family
000208 |   $D_b^+$, including domains, coupling data and gauge descent, with Quillen line
000209 |   $L_Q=(\det\ker D^+)^{-1}\otimes\det\operatorname{coker}D^+$, Quillen metric
000210 |   and Bismut--Freed connection. Fix the chirality/duality convention in the comparison.
000211 |   The null-mode locus $\Theta\subset B$ is allowed and must be specified; a globally
000212 |   invertible $D^+$ has a trivial Fredholm determinant line and cannot supply $|c_1[B]|=1$.
000213 | 
000214 |   \emph{Comparison demand.} Construct the comparison between $L_{\mathrm{occ},L}$
000215 |   and $L_Q$ (or its explicitly specified dual) over the common base and prove
000216 |   convergence at the level of Hermitian metrics and connections after a fixed
000217 |   relative renormalisation. Norm-resolvent convergence and a Fermi gap alone are
000218 |   insufficient: uniform control of the high-frequency/heat-kernel contributions
000219 |   and their parameter variations is required. Match the curvature forms and the
000220 |   holonomies on the two independent twist cycles; equal integrated Chern numbers
000221 |   do not determine the connection. The reference $A_0$ fixes the remaining constant
000222 |   normalisation. On $B\setminus\Theta$ the logarithmic determinant variation is
000223 |   ordinary; its extension across $\Theta$ must retain the kernel and divisor data.
000224 | 
000225 |   \emph{Dai--Freed data.} A Dai--Freed vector requires an actual family of fillings
000226 |   and takes values in the pulled-back inverse boundary line. A global nonvanishing
000227 |   section may be asserted only on that pullback or on a specified anomaly-cancelled
000228 |   product, not on the uncompensated line with nonzero $c_1$. Alternatively, cylinders and
000229 |   mapping tori may define transport and holonomy without a global trivialisation.
000230 |   All continuum comparison statements remain \mO; the finite Berry/gap results
000231 |   of \vref{v472} are unchanged.
000232 | \item \textbf{(T2) Variation $=$ Bismut--Freed curvature $=$ inflow response, with the exact
000233 |   corpus coefficients.} On the continuum line of T1,
000234 |   \[
000235 |     \delta_\tau\Bigl(\log\det\nolimits_\zeta\Delta_{U(1)}\;+\;8\,b_1\,\cthree^{6}\,
000236 |     \log\varphi_{\mathrm{seam}}\Bigr)=0 ,
000237 |   \]
000238 |   i.e.\ the EM closure $F_{U(1)}(\alpha)=k_0\,\alpha^3-2\,\cthree^{3}\alpha^2-8\,b_1\,\cthree^{6}
000239 |   \ln(1/\varphi_{\mathrm{seam}})=0$ \emph{is} the stationarity of the exact
000240 |   $\zeta$-regularised Quillen functional, with every coefficient as it stands in the corpus:
000241 |   the Chern level $k_0=|C|=1$ (the computed bulk invariant of the S3 collar, normalised by the
000242 |   embedding index $k_Y=\operatorname{tr}(Y^2)/\operatorname{tr}(T_3^2)=5/3$, \vref{v470}); the
000243 |   Gauss--Bonnet boundary coefficient $\cthree=1/(8\pi)$ with heat-kernel powers
000244 |   $\cthree^{3},\cthree^{6}$ (\vref{v216}/\vref{v342}); the $U(1)_Y$ $a_4$/index coefficient
000245 |   $b_1=41/10$ ($8b_1=(4/5)\cdot41$, \vref{v434}); and the discriminant exponent
000246 |   $-\tfrac54=-q(D_5)$ (\vref{v341}). The derived stationarity must then have
000247 |   $\alpha^{-1}=137.0359992168$ as its unique physical root (\vref{v3}).
000248 | \end{itemize}
000249 | The open content splits into two named faces (one target, per \vref{v434}/\vref{v435}): the
000250 | \emph{analytic} face --- T1 plus the identification of $\delta\log\det_\zeta$ with the seam
000251 | Seeley--DeWitt data, i.e.\ the diagonal $\zeta$-renormalisation at the marks and the
000252 | multiplicity matching $41=10\,b_1$ (\vref{v484}/\vref{v485}) --- and the \emph{topological}
000253 | face, the \texttt{EM1} lemma: the from-first-principles proof that
000254 | $\partial_\tau\log Z_{\mathrm{Maxwell}}=k_0\,\alpha^3$ with $k_0$ the bulk Chern level
000255 | (\vref{v48}/\vref{v435}).
000256 | \emph{Both named faces now carry computed witnesses (2026-08-27,
000257 | \texttt{ALPHA.QUILLEN.FACES.01} \mE\ new; \veri{v974\_alpha\_faces\_computed.py};
000258 | \texttt{ALPHA.QUILLEN.EXACT.01} stays \mO):} the diagonal-$\zeta$ (analytic) face is executed
000259 | as a BFK gluing --- the gluing constant is exactly $2^{-N}$, the jump matrix
000260 | $16\,\cthree\,\operatorname{circ}(2,-1,0,-1)$ \emph{is} the inverse mark Green matrix, and the
000261 | \vref{v485} scale channel is reproduced via the KMS unit, while against the \vref{v484} mark
000262 | matrix the comparison is channel-wise exact but with the kernels in \emph{swapped} $\mu_4$
000263 | channels, so the analytic face is \emph{sharpened} to the channel-swap question; the
000264 | topological (\texttt{EM1}) face is computed --- pumped charge $=$ spectral flow $=$
000265 | St\v{r}eda $=$ Chern $=1$ on the \vref{v367} collar; and the finite Quillen identity is
000266 | verified \emph{two-sided} (Berry integral $=$ Poincar\'e--Lelong divisor integral $=+1$, the
000267 | difference an exact form). The remaining open content is typed: the rigidity premise, the
000268 | channel-swap question, and the continuum Bismut--Freed identification;
000269 | $\alpha^{-1}=137.0359992168$ is re-anchored with split residual $6.4\times10^{-58}$.
000270 | \emph{Channel-swap update (2026-08-28, external master-object review wave 2;