TFPT scientific source excerpt
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000352 | \subsection*{\interlinepenalty=150\relax 2026-07-22 (XII) (\texttt{v522} --- the WOIT-$\beta_1$ milestone of the OS twistor bridge \texttt{WOIT.OS.TWISTOR.01} executed and promoted as the ledger row \texttt{WOIT.BETA1.GSO.01}: ``the clock is time-like, GSO is the gauge datum'' --- a TYPED result (verdict UNDECIDED per the frozen preregistration, the mechanism identified exactly, gauge-fixed RP established under the corrected typing; the celestial/twistor story in \texttt{tfpt\_3} gains Step 23) --- (1) the HERMITICITY WITNESS \mE: the $\mu_4$ clock average violates Hermiticity EXACTLY (the one-step insertion $T_1=\langle\theta(e_a),\alpha_S(e_b)\rangle$ has entry $-i/(8\sin(5\pi/16))$ where the conjugate transpose demands $+i/(8\sin(5\pi/16))$; entrywise census $745$ Hermitian-matching, $96$ ANTI-matching, $0$ violations --- the pairing is complex-SYMMETRIC, strictly weaker than Hermitian: ``positivity after gauge fixing'' is not well-posed for the clock reading); (2) TIME-LIKE TYPING \mE: ALL $16$ dihedral reflection axes of the NS seam circle invert the clock lift ($RSR^{-1}=S^{-1}$), NONE commutes --- the $\mu_4$ clock IS Woit's euclidean rotation; the gaugeable ($\theta$-compatible) part of the $\Z_8$ Fock tower is EXACTLY its $2$-torsion $\{1,(-1)^F\}$ $=$ the GSO/fermion-parity $\Z_2$ (Hermiticity census over all powers; $\alpha^4=$ the full wrap $=$ the $2\pi$ rotation of the seam $=$ fermion parity); (3) the POSITIVE PART \mE: GSO-fixed RP holds EXACTLY --- $N=16$ even $\deg\le2$ parity-projected bond-cut Gram $(29,0,0)$ PD (min eigenvalue $1.78\times10^{-6}$ at $40$ digits), $N=8$ complete half algebra: all odd rows/columns killed exactly, even block $(8,0,0)$ PD ($3.35\times10^{-3}$), no degree truncation; the site-cut defect SURVIVES gauge fixing ($(7,9,6)$ --- the bond placement is gauge-invariant information); family A keeps failing ($(17,12,0)$ indefinite vs family D's $(29,0,0)$); Kramers: $\Theta_t^2=\mathrm{id}$ on $\mathcal A^G$, $\Theta_{\mathrm{Fock}}$ restricts with $\Theta^2=+1$; Ramond control: the wrap $+1$ ``true $\Z_4$'' lift fails state invariance --- the NS/$\Z_8$ tower is forced; (4) KILL-TEST BOOKKEEPING: kill test (2)'s free shadow does NOT fire (no legitimate Hermitian gauge-fixed form has a negative direction), kill test (1) stays discharged also gauge-invariantly; both stay formally live on $\mathcal A_{\mathrm{hol}}$; (5) TRANSPARENCY: the first frozen run scored $6/13$ (the clock-reading checks failed exactly as the mechanism predicts) and the preregistered deg-2 invariant dimension $30$ was a combinatorial error (correct census $28+4=32$) --- documented in full, the verdict by the frozen [C-3] logic (Hermiticity failure $\Rightarrow$ UNDECIDED); CONTRACT PRECISION (iii) (prose, no marker): ``gauge-invariant subalgebra'' at the free level means the GSO/fermion-parity $\Z_2$ --- the only $\theta$-compatible part of the $\mu_4$ tower; the clock is the euclidean rotation itself (timelike), its equivariant statement moves to $\beta_2$ (OS quotient first, then the clock as reconstructed transfer operator); Wolfram extension (with v520/v521) $522\to540$ (verified engine run); \emph{NO marker moves --- \texttt{WOIT.OS.TWISTOR.01} stays Open, $\beta_1$ proper re-routed via $\beta_2$})}
000353 | \begin{itemize}[leftmargin=1.5em]
000354 | \item \textbf{\texttt{v522\_woit\_beta1\_gso\_gauge.py} (\texttt{WOIT.BETA1.GSO.01}, 16 checks,
000355 | $\sim$5 s; exact sympy Pfaffian--Wick $+$ exact $\mathrm{Cl}(16)$ Fractions; definiteness via
000356 | $40$-digit spectra, tolerance $10^{-25}$; deterministic).} G1 the gauge datum (Z8/NS tower,
000357 | Ramond control, invariant dimensions $120>64>32$); G2 $\Theta$ normalises the tower and
000358 | restricts as an involution; G3 the $(-1)^F$ class does not penetrate $\mathcal A^G$; R1 the
000359 | tower average is OS-symmetric, the odd sector dies; M1 the time-like census; M2 the exact
000360 | Hermiticity witness; M3 the GSO $2$-torsion census; R2 GSO-fixed RP ($N=16$/$N=8$); C1--C3
000361 | site-cut/wrong-projection/family-A controls; V1 the frozen verdict. Registered in
000362 | \texttt{run\_all.py} (suite $516$ modules, green), \texttt{script\_registry.csv} (cluster
000363 | \emph{frontier}), ledger row \texttt{WOIT.BETA1.GSO.01} (new).
000364 | \item \textbf{Contract updated (markers unchanged).} \texttt{WOIT.OS.TWISTOR.01}: the $\beta_1$
000365 | stage moved to \emph{done} (typed UNDECIDED) with its own block, precision (iii) added, the
000366 | $\beta_1$ roadmap bullet annotated \emph{executed}, the closing status line carries the
000367 | $\beta_1$ state; ledger \texttt{WOIT.OS.TWISTOR.01} text extended (beta-1 executed via v522).
000368 | \item \textbf{Paper integration.} \texttt{tfpt\_3} (new Step 23 ``the clock is time-like, GSO
000369 | is the gauge datum''; the Woit-bridge paragraph extended by the $\beta_1$ status);
000370 | \texttt{veri} citations in \texttt{tfpt\_research\_contracts} and \texttt{tfpt\_3}.
000371 | \item \textbf{Wolfram.} Six exact mirrors (the Z8/NS tower $S^4=-1$ $+$ the Ramond witness
000372 | $1/(4\sin(5\pi/16))$; the invariant-dimension chain by exact ranks; the time-like reflection
000373 | census; the exact Hermiticity witness with the full $745/96/0$ census via exact Pfaffian--Wick;
000374 | the GSO $2$-torsion census; the family-A tower-normalisation failure) added to
000375 | \texttt{tfpt\_readouts\_extension.wl}; the GSO-fixed RP inertia certificates stay Python-only,