TFPT scientific source excerpt
Original repository path: documents/tfpt_3_e8_audit_bootstrap/sections/013-section-honest-status-bootstrap-not-creation-from-nothin.tex
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000316 | \item \textbf{What is genuinely shown \mE:} $\gcar=5$ is an \emph{overdetermined} fixed point of the
000317 |   $E_8$ closure (two independent Lie-theory conditions); the integer ``$8$'' in $\cthree$ is the carrier
000318 |   Coxeter $=$ $E_8$ rank; the discrete data closes a loop. The earlier landscape scan confirms
000319 |   $\gcar=5$ is the \emph{unique} value at which all hierarchies/family/gauge close --- so the fixed point
000320 |   is unique, not one of many. \textbf{This uniqueness is now machine-formalised \mE\ in Lean 4} (Theorem A,
000321 |   \texttt{lean4-carrier-rigidity}): the division-free Pascal condition $2^{g}=g^2+g+2$ has the \emph{unique}
000322 |   solution $g=5$ (\texttt{carrier\_rank\_pascal\_unique}; growth lemma for $g\ge6$; $\Nfam=(2^{4}{-}1)/5=3$,
000323 |   $g{+}\Nfam=8$), audited to use only the standard kernel axioms. The 2026-06-11 hardening round adds two
000324 |   formal modules: \texttt{AnchorLadder.lean} (the anchor power-sum law $p_n(a)=2+2^n$ for \emph{all} $n$,
000325 |   the ladder atoms $(3,4,6,10)$, $240/8/248$, and the binary ladder $p_{n+1}{-}p_n=2^n$ as general
000326 |   theorems) and \texttt{PascalLadder.lean} (the odd-midpoint ladder identity
000327 |   $\sum_{k\le K}\binom{2K+1}k=2^{2K}$ for \emph{all} $K$ via Mathlib, plus the kernel-decidable bounded
000328 |   uniqueness of the $K{=}2$ carrier case) --- the solves-side of the Pascal Ladder Theorem
000329 |   (\veri{v108\_pascal\_ladder.py}) is thereby \mE.
000330 | \item \textbf{What this is \emph{not}:} a derivation from zero inputs. A self-consistency loop is
000331 |   consistent \emph{at} a value; uniqueness is what makes it predictive (and that is supplied by the
000332 |   landscape scan). The continuous $\pi$ is not produced by the loop --- it is the Möbius/boundary
000333 |   geometry, the one irreducible primitive.
000334 | \item \textbf{Conceptual gain:} the theory's foundation tightens from ``two free axioms'' to ``\emph{the}
000335 |   $E_8$-closure fixed point $+$ the geometric $\pi$.'' That is the Möbius self-consistency you intuited:
000336 |   the seed is the start of the cascade, and the cascade returns the seed's discrete content.
000337 | \item \textbf{The current reduction state (2026-07-22; text consolidation, markers unchanged):} the
000338 |   compiler's dimensionless inputs reduce to \emph{four marks} (derived from P1-side topology,
000339 |   \vref{v216}), \emph{one discrete symmetry-lift bit} (flag transitivity $\iff\tau=i$; equivalently:
000340 |   the clock / the CM-fixed half-period / the non-split extension class on the free seam circle;
000341 |   \vref{v499}/\vref{v506}/\vref{v507}/\vref{v510}/\vref{v512}), and $\pi$ --- with the \mC\
000342 |   identifications R1/R2
000343 |   of \vref{v499} as the remaining conditional layer. \texttt{AX.P2.01} stays a declared axiom and
000344 |   \texttt{QGEO.REALIZE.01} stays \mC/\mO: this line describes the \emph{reduction state}, not an
000345 |   abolition.