TFPT scientific source excerpt
Original repository path: documents/origin_theory/sections/001-section-the-whole-integer-skeleton-from-one-pair-gcar-nf.tex
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000038 | TFPT runs on two operational inputs: the seam constant $\cthree=\tfrac1{8\pi}$ (P1) and the carrier rank
000039 | $\gcar=5$ (P2). The boundary $\mu_4=\{1,i,-1,-i\}$ gives the four-punctured line $\mathbb P^1\setminus\mu_4$
000040 | with $H_1=\Z^3$, hence the family count $\Nfam=3$ ($A_3$). We show the rest of the discrete data is
000041 | generated by the single pair $(\gcar,\Nfam)=(5,3)$.
000042 | \par\smallskip
000043 | \noindent\emph{One object, two canonical invariants.} The pair $(\gcar,\Nfam)$ is read off the
000044 | \emph{same} four-point divisor $D=\mu_4$ (degree $4$) by the two canonical functors on $\PP^1$: the cycle
000045 | side $\operatorname{rank}H_1(\PP^1\setminus\mu_4)=\deg D-1=3=\Nfam$ and the function side (Riemann--Roch)
000046 | $h^0(\PP^1,\mathcal O(\mu_4))=\deg D+1=5=\gcar$, the latter matched by $\operatorname{rank}(D_5)=5$ (the
000047 | carrier is the $\mathfrak{so}(10)$ half-spinor). Reading the carrier as this meromorphic seam-mode space
000048 | (at most simple poles on $\mu_4$) is geometrically natural \mE\ in its arithmetic, though the physical
000049 | identification stays \mC\ and does not displace the over-determined $\gcar=5$ (Pascal/bootstrap/Lean)
000050 | \veri{v189\_riemann\_roch\_carrier.py}.
000051 | \par\smallskip
000052 | \noindent\emph{And the function side generates $D_5$ itself, not just $\gcar$.} The even Clifford algebra of
000053 | the $5$-dimensional mode space closes the chain $\mu_4\to\gcar\to D_5$: $\Lambda^{\mathrm{even}}(C_{\mathrm{car}})
000054 | =\binom50+\binom52+\binom54=1+10+5=16=2^{\gcar-1}=\dim S^+$, which is exactly the $\mathfrak{so}(10)=D_5$
000055 | half-spinor. So the carrier is not merely ``rank $5$ like $D_5$'' --- it \emph{is} the Clifford spinor of the
000056 | Riemann--Roch mode space \mE, with the physical identification again \mC\ \veri{v197\_rr\_carrier\_clifford\_d5.py}.
000057 | 
000058 | \begin{keybox}[The $(5,3)$ generator and the Pythagorean mass volume \mE\ (\veri{v53\_compiler\_core.py})]
000059 | By sum, difference and halving,
000060 | \[
000061 |   \operatorname{rank}E_8=\gcar{+}\Nfam=8,\qquad |\Z_2|=\gcar{-}\Nfam=2,\qquad |\mu_4|=\tfrac{\gcar+\Nfam}{2}=4,
000062 | \]
000063 | and $(\Nfam,|\mu_4|,\gcar)=(3,4,5)$ is \emph{the} Pythagorean triple. This is load-bearing: the
000064 | grand-mass-volume exponent $\Delta_Y=\gcar^2=25$ ($\det M_{\mathrm{SM}}\sim(\phiz)^{25}$) splits as a
000065 | \emph{difference of squares},
000066 | \[
000067 |   \boxed{\ \Delta_Y=\gcar^2=\underbrace{\Nfam^2}_{\text{down}=9}
000068 |   +\underbrace{(\gcar{-}\Nfam)(\gcar{+}\Nfam)}_{=\,|\Z_2|\cdot\operatorname{rank}E_8=\dim S^+=16}
000069 |   =9+16=25\ }
000070 | \]
000071 | (the $K$-row sums $(6,9,10)$: down carries $\Nfam^2$, up$+$lepton carry $|\Z_2|\!\cdot\!\operatorname{rank}E_8$).
000072 | The \emph{anchor} $a=(1,1,2)$ is the unique $3$-multiset whose elementary symmetric functions are the
000073 | compiler atoms $(|\mu_4|,\gcar,|\Z_2|)=(4,5,2)$; equivalently $\chi_a(t)=(t{-}1)^2(t{-}2)=t^3-|\mu_4|t^2+\gcar t-|\Z_2|$.
000074 | Sharper (\veri{v491\_p2\_partition\_corollary.py}): only $e_1{=}4$ is needed --- three positive integers
000075 | summing to $4$ already force $\{1,1,2\}$, so $\gcar{=}5$ and $|\Z_2|{=}2$ are corollaries of the four
000076 | marks given the weight typing; \texttt{AX.P2.01} stays the declared axiom (the residual is the typing).
000077 | And the typing is itself hardened (\veri{v499\_p2\_weights\_deligne\_bg.py}): Deligne $+$
000078 | Birkhoff--Grothendieck $+$ Mehta--Seshadri stability derive integrality, positivity ($h^0{=}0$) and the
000079 | sum $4$ from the four marks, rank $3$, the cusp class and (U) --- the \mC\ residual is the module
000080 | identification plus (U) itself.