TFPT scientific source excerpt
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000001 | % Authored reconciliation of existing results, 2026-11-05. No ledger promotion.
000002 | \section{Shared origin of the seam normalization and carrier data}
000003 | \label{sec:shared-p1-p2-origin}
000004 | 
000005 | \noindent\textbf{What is being explained.} P1 supplies a primitive reflection-positive
000006 | kernel on an oriented seam with unit winding. The number $c_3$ is its normalization.
000007 | P2 declares the five-slot $3+2$ readout; its algebraic consequences and its
000008 | conditional source reductions are already available. Calling both entries
000009 | ``axioms'' records these interfaces and does not erase those reductions.
000010 | 
000011 | \paragraph{One marked geometry, several typed readouts.}
000012 | In the flat spherical $\mathbb Z_2$ branch class, each branch point has deficit
000013 | $\pi$. Gauss--Bonnet gives $n\pi=4\pi$, hence four marks
000014 | (\veri{v216\_marks\_gauss\_bonnet.py}). For the common divisor
000015 | $D=\mu_4\subset\mathbb P^1$,
000016 | \[
000017 |  \operatorname{rank}H_1(\mathbb P^1\setminus D)=3,
000018 |  \qquad h^0(\mathcal O(D))=5.
000019 | \]
000020 | The functions and cycles belong to the same marked geometry
000021 | (\veri{v189\_riemann\_roch\_carrier.py},
000022 | \veri{v197\_rr\_carrier\_clifford\_d5.py}). They are different spaces:
000023 | meromorphic functions have four independent principal parts, while
000024 | meromorphic differentials have the residue-sum constraint and dimension three.
000025 | These dimensions alone do not identify a physical colour/weak $3+2$ split.
000026 | The square position of the marks is additional to their count; in the
000027 | flag-symmetric branch it is equivalent to $\tau=i$
000028 | (\veri{v512\_seam\_tau\_flag.py}).
000029 | 
000030 | \paragraph{The anchor and the two numerical readouts.}
000031 | The typed parabolic splitting $a=(1,1,2)$ gives
000032 | \[
000033 |  e_1(a)=4,\quad e_2(a)=5,\quad e_3(a)=2,
000034 |  \qquad c_3=\frac{1}{2\pi e_1(a)}=\frac1{8\pi},
000035 |  \qquad g_{\rm car}=e_2(a)=5.
000036 | \]
000037 | The uniqueness of three positive integer weights summing to four is exact;
000038 | the Deligne extension, Birkhoff--Grothendieck splitting and stability route
000039 | explain the typing for the specified connection
000040 | (\veri{v491\_p2\_partition\_corollary.py},
000041 | \veri{v499\_p2\_weights\_deligne\_bg.py}). Later analytic descent of
000042 | the actual $A_0^\star$ system gives finite $S_4$ monodromy, so a positive
000043 | invariant Hermitian form exists by finite-group averaging. Historical
000044 | analytic-monodromy and unitarizability obligations are therefore not a
000045 | current independent numerical gap. Identifying that mathematical module
000046 | and its frame with the actual physical source retains its stated scope.\footnote{TFPT Gesamtsynthese, 28 September 2026, the fourfold cover section; recovered source proof in \texttt{REC20261105C3}.}