TFPT scientific source excerpt
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000053 | \par\smallskip\noindent\textbf{Why \emph{this} function: the boundary $U(1)$ Ward reading (Theorem C) \mE/\mC.}
000054 | $F_{U(1)}$ is not a chosen ansatz but the reduced Ward identity of the boundary $U(1)$ partition function
000055 | $Z_{\partial}=Z_{\mathrm{Maxwell}}\,Z_{\mathrm{Calder\acute on}}\,Z_{\mathrm{transport}}$, with three terms
000056 | whose coefficients are compiler atoms, not fits:
000057 | \[
000058 |   \underbrace{\alpha^3}_{\text{Maxwell moment}}\;
000059 |   \underbrace{-\,2\cthree^3\alpha^2}_{\text{Calder\'on sheet}\,(2=|\Z_2|)}\;
000060 |   \underbrace{-\,8b_1\cthree^6\log\tfrac1{\phiseam}}_{\text{$C_6$ transport det.}}=0,\qquad
000061 |   8b_1=\tfrac45 M=\tfrac45(\textstyle\sum L+N_\Phi)=\tfrac{164}{5}.
000062 | \]
000063 | The Calder\'on factor $2=|\Z_2|$ (sheet), the transport power $\cthree^6$ is the $C_6$ hexagon, and the
000064 | seam exponent $-\tfrac54=-q(D_5)$ is the carrier discriminant norm. \textbf{Typing:} the three-term
000065 | decomposition and the coefficient identities ($8b_1=\tfrac45 M$, $-\tfrac54=q(D_5)$) are exact \mE; the
000066 | \emph{physical origin} as a boundary Ward closure (the lemmas $\partial_\tau\log Z_{\mathrm{Maxwell}}=\alpha^3$,
000067 | $\partial_\tau\log Z_{\mathrm{Calder\acute on}}=-2\cthree^3\alpha^2$, $\partial_\tau\log\det\nolimits_\zeta(1-\mathcal T_{C_6,U(1)})=-8b_1\cthree^6\log\tfrac1{\phiseam}$)
000068 | is a conjectured \mC\ interpretation, not yet machine-proven. \veri{v48\_em\_ward.py}
000069 | \par\smallskip
000070 | \noindent\textbf{The determinant-line (Quillen) form.} (\veri{v341\_alpha\_quillen.py})
000071 | The three terms assemble into a single \emph{stationarity} statement for the boundary $U(1)$ determinant line,
000072 | \[
000073 |   \underbrace{\bigl[\alpha^3-2\cthree^3\alpha^2\bigr]}_{\delta\log\det\Delta_{U(1)}}
000074 |   \;=\;\underbrace{\bigl[8b_1\cthree^6\bigr]}_{\text{ctr.}}\;
000075 |   \underbrace{\bigl[\log\tfrac1{\phiseam}\bigr]}_{\text{seam resp.}},
000076 | \]
000077 | so $F_{U(1)}=0$ is $\delta\log\det\Delta_{U(1)}+\text{counterterm}+\text{seam response}=0$. Every factor is a
000078 | \emph{named} object, not a knob: the counterterm coefficient is $8b_1\cthree^6=(\operatorname{rank}E_8)\,b_1\,\cthree^6$
000079 | (index $\times$ heat-kernel), $\cthree=1/(8\pi)$ is the Gauss--Bonnet boundary coefficient, the exponent
000080 | $-\tfrac54=-q(D_5)$ is one carrier discriminant norm and the seam base $\phibase=1/(6\pi)=\tfrac43\cthree$ carries
000081 | the \emph{other}, $\cthree/\phibase=\tfrac34=q(A_3)$ --- both glue forms appear, and the split is verified at the
000082 | root. \textbf{Typing (and: no new gate).} The determinant-line assembly and all coefficient identities are exact
000083 | \mE; $\ainv$ itself stays closed \mE\ (unique root $+$ ablation). This adds \emph{no} open item --- it
000084 | \emph{sharpens} the pre-existing \mO/physical-origin obligation \texttt{EM.WARD.01} (the ``why \emph{this}
000085 | function'' residual, already conjectural) by naming every coefficient. The one still-open part \emph{is} that
000086 | residual, now \emph{named} as its own tracked target \texttt{ALPHA.QUILLEN.EXACT.01}
000087 | (\veri{v382\_alpha\_quillen\_exact.py}): the from-first-principles AQFT/$\zeta$ proof that
000088 | $\delta_\tau(\log\det\nolimits_\zeta\Delta_{U(1)}+8b_1\cthree^6\log\varphi_{\mathrm{seam}})=0$ --- i.e.\ that
000089 | $F_{U(1)}$ is the \emph{exact} Quillen functional of the seam $U(1)$, the variational principle \emph{derived}, not
000090 | assembled. It is a \emph{face} of \texttt{SEAM.EQUIV.01} (\vref{v336}) --- the seam $U(1)$ determinant line lives on
000091 | the same holomorphic net, so once the net is $(E_8)_1$ its $U(1)$ sub-determinant is fixed --- hence no second
000092 | independent gate, just the sharpest remaining EM-Ward obligation, now citable on its own.
000093 | \emph{Honest progress towards it (\veri{v391\_alpha\_quillen\_progress.py}), without closing it:} a solvable
000094 | model $\det_\zeta(-\partial_\tau^2{+}m^2)$ on a circle $=(2\sinh(\beta m/2))^2$ has the closed $m$-variation