TFPT scientific source excerpt
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000068 | Everything downstream is a theorem \emph{given} two structural inputs that are not reduced further and
000069 | are honestly flagged as the theory's axioms:
000070 | \begin{description}[leftmargin=2.2em,style=nextline]
000071 |   \item[\textnormal{(P1) Boundary kernel.}] A single oriented seam $\Sigma$ carries a primitive,
000072 |   reflection-positive boundary kernel with unit winding $[u_\Sigma]=1$; its self-linking normalisation
000073 |   fixes $\cthree=1/(8\pi)$. \emph{Status:} primitive postulate. \emph{Organizing identity (2026-08-06,
000074 |   markers unchanged):} the closed normalization $I\cdot\beta_{\rm angle}\cdot\cthree=4\cdot2\pi\cdot
000075 |   \tfrac1{8\pi}=1$ holds exactly with the Jones index $I=4=|\mu_4|$ a $\cthree$-free theorem and
000076 |   $\beta_{\rm angle}=2\pi$ independently measured --- the identity \emph{explains} the value, it does
000077 |   not derive it; the open bridge is the even modular-response normalization
000078 |   (\texttt{P1.INDEX.KMS.01} \mO, \veri{v813\_p1\_index\_kms.py}). \emph{Hardenable by:} the Calder\'on
000079 |   projector / determinant-character chain (the P1 hardening) and the Lean carrier-rigidity development
000080 |   (\texttt{TfptCarrier.*}, boundary-polarization and Calder\'on interfaces).
000081 |   \item[\textnormal{(P2) Carrier interface.}] This splits into an \emph{axiom} and a \emph{theorem},
000082 |   which must not be conflated:
000083 |   \begin{itemize}\setlength\itemsep{1pt}
000084 |   \item \textbf{P2$_{\mathrm{interface}}$ \mO\ (axiom):} the seam is read out by a five-slot carrier
000085 |   $\gcar=5$ ($3$ colour $+\,2$ weak) --- a postulate (the readout interface).
000086 |   \item \textbf{P2$_{\mathrm{algebra}}$ \mE\ (theorem):} \emph{given} the carrier, hypercharge
000087 |   ($y_-=-\tfrac13$, $y_+=\tfrac12$), rigidity and the family count follow from the idempotent/lattice
000088 |   axioms --- machine-verified in Lean (\texttt{Hypercharge.lean}, \texttt{LatticeRigidityGeneral.lean},
000089 |   \texttt{Rigidity.lean}).
000090 |   \end{itemize}
000091 |   So P2 is an \emph{interface axiom} whose \emph{algebraic consequences are a formalised theorem} ---
000092 |   ``axiom or theorem?'' has the precise answer: interface axiom, algebra theorem.
000093 | \end{description}
000094 | These two are \emph{inputs}: they can be formalised and stress-tested (more Lean, more interface
000095 | theorems) but not computed away. Every other number in this note --- $\Nfam=3$, $\Oadm=48$,
000096 | $b_1=41/10$, $\ainv$, the flavor matrix, the $E_8$ glue, $A_s$ --- is a consequence of (P1),(P2).
000097 | \smallskip\\
000098 | \noindent\textbf{Anchor-first refinement.} The two inputs are themselves the elementary symmetric
000099 | polynomials of the single parabolic anchor $a=(1,1,2)$ (the exponents-at-infinity of the splitting type
000100 | $\mathcal O(-2)\oplus\mathcal O(-1)^2$): $e_1(a)=4=|\mu_4|$, $e_2(a)=5=\gcar$, $e_3(a)=2=|\Z_2|$, with
000101 | $\cthree=1/(2e_1(a)\pi)=1/(8\pi)$. The power sums $p_n(a)=2+2^n$ then give $|R(E_8)|=p_1p_2p_3=240$,
000102 | $\dim E_8=240+(p_4{-}p_3)=248$ and the binary ladder $p_{n+1}{-}p_n=2^n$. So the input set reduces to
000103 | $\{a=(1,1,2),\,\pi\}$. \veri{v23\_anchor\_generator.py}  The whole ladder is one affine map: $p_{n+1}=2p_n-2$
000104 | identically, with unique fixed point $2=|\Z_2|$ and the compiler quintet $(4,6,10,18,34)$ as the $T$-orbit
000105 | of $4$ --- the affine normal form of Theorem~8 (\veri{v832\_anchor\_flavor\_checksum.py}).
000106 | \par\smallskip\noindent\textbf{The parabolic self-code of the anchor
000107 | (\veri{v566\_parabolic\_anchor\_selfcode.py}).} The anchor is not only
000108 | generated --- it is \emph{coded twice} by one operator structure that
000109 | already lives in the compiler.  The sheet-diamond direction operators