A registered result in its source context

WOIT.OS.TWISTOR.01

Original ledger classification: Open (research contract)

Original claim text

THE OSTERWALDER-SCHRADER TWISTOR BRIDGE (2026-07-22; the new central research contract after the external review of the celestial/twistor branch -- THIS IS THE ACTUAL BRIDGE FROM COMPILER TO PHYSICS; everything in CELEST.SEAM.01 is preparation for it). INPUT: X_+ = (PT/Gamma)_+ (Gamma = Gamma_ALE = <Deck> ~ Z4 in SU(2), normalised by the clock rho = diag(i,1) with rho^2 = Deck); A_hol = the INTERACTING open+closed twistorial algebra (SDYM(E8) + BCOV); rho = the order-4 clock; Theta = the anti-linear real structure induced by the seam reflection (the free covering deck, v510); mu_BCOV+SDYM = the interacting functional (global BCOV+SDYM quantisation on PT/Gamma). TARGET THEOREM: Theta^2 = 1, Theta rho Theta = rho^-1; the gauge-invariant algebra is reflection-positive NAMED TOY THREAT (v529 SEAM.INT.FKTOY.01, NO status change): on the first genuinely interacting 16-Majorana Fidkowski-Kitaev seam toy, Kill-Test 1 does NOT fire (Theta exists with Theta^2=+1 and clock-tower inversion) but Kill-Test 2 FIRES AT TOY LEVEL -- RP breaks for every g>0 by an interference mechanism, with the STRADDLE LAW (RP fails exactly on quartet-straddled cuts) as a concrete hurdle for beta3/gamma; fence: one toy / one interaction class / [C] flat-band parent -- narrows the route (every A_hol must protect RP against exactly this mechanism), does not close it.; the OS quotient produces (H, Omega, U(P^up_+), A_Mink) with: positive Hilbert metric, positive energy, local causality, ONE chiral fermion generation WITHOUT mirror doubling, the TFPT charge lattice, an internal SU(3) x SU(2) x U(1) action, and the (E8)_1 seam net as an ACTUAL boundary net (not only as a character). KILL TESTS (all seven, preregistered): (1) Theta incompatible with the clock; (2) RP fails after gauge fixing; (3) the reconstruction produces a vector-like mirror generation; (4) the Penrose transform reaches only free/self-dual states; (5) the reconstructed net has the right character but the wrong OPE / fails net equivalence; (6) the internal SU(2) remains a spacetime factor; (7) the four mu4 marks are not incidence-compatibly extendable over spacetime. SCOPE FENCE (explicit non-claims of the celestial/twistor branch until this closes): both helicities, generic amplitudes, local matter, full Einstein dynamics, EWSB, confinement. EXTERNAL FRAME: Woit's Euclidean Twistor Unification (arXiv:2104.05099 -- chiral Wick rotation problematic for purely chiral theories; Euclidean-holomorphic theory on PT, Lorentzian theory via conjugation structure/boundary values) is a NAMED reference frame, never a confirmation: until Theta + interacting RP exist, the Woit connection is a strong analogy, not a mathematical integration. ALPHA STAGE EXECUTED (2026-07-22) via v519: Theta exists with Theta^2 = +1 and Theta rho Theta = rho^-1 on all four levels (sphere, C^4, Z8 spin, Cl(16) Fock), free RP holds on the bond cut and selects the SAME family as the clock-inversion relation; kill test 1 does NOT fire at the free level and stays live on the interacting algebra; beta/gamma milestones named (beta-1 Theta on the gauge-invariant subalgebra + gauge-fixed RP on the equivariant SDYM(E8) sector; beta-2 the OS quotient of the free system explicit; beta-3 the PT <-> PT* duality vs sigma_std typed; gamma the chirality theorem + the mark incidence). TWO CONTRACT PRECISIONS (prose, no marker): (i) 'Theta induced by the seam reflection' means the seam-circle REFLECTION, not the deck -- the deck furnishes the (-1)^F Kramers class (the v510 bridge); (ii) the marks sit RP-side on the BOND MIDPOINTS of the 16-Majorana circle. BETA-1 EXECUTED (2026-07-22) via v522: verdict UNDECIDED per the frozen preregistration -- the clock-tower reading of 'gauge' fails Hermiticity structurally (the mu4 clock is euclidean-time-like: all 16 mirror axes invert it); the gaugeable part of the Z8 tower is exactly the GSO/fermion-parity Z2, under which gauge-fixed RP HOLDS ((29,0,0) PD at N = 16, (8,0,0) PD full N = 8 algebra); kill test (2)'s free shadow does NOT fire, kill test (1) stays discharged also gauge-invariantly; the clock-equivariant statement is re-routed to beta-2 (OS quotient first, then the clock as reconstructed transfer operator). THIRD CONTRACT PRECISION (prose, no marker): (iii) 'gauge-invariant subalgebra' at the free level means the GSO/fermion-parity Z2 -- the only theta-compatible part of the mu4 tower; the clock is the euclidean rotation itself (timelike), its equivariant statement moves to beta2. BETA-2 EXECUTED (2026-07-23) via v524: verdict SUCCESS, [C]-typed per precision (iii) -- the OS quotient of the free system is EXPLICIT: H_phys nondegenerate and PD at both levels (dim 37 = 29+8 at N = 16 deg <= 2; dim 16 = 4^2 at N = 8 complete half algebra -- compact euclidean time reconstructs a thermal/KMS representation), the Klein-Landau local transfer semigroup is exactly Hermitian with the site/bond dichotomy as its positivity pattern (even steps PSD via T(2j) = A*A, the one-step transfer NOT positive -- the chirality datum), the mu4 clock = T^{N/4} is a positive self-adjoint transfer step with spectral calculus (N = 8 spectrum exactly {1, sqrt(2)-1} = {1, 1/delta_Silver}) and a reconstructed unitary rotation group U(s) = exp(isH); the v522 non-Hermiticity is resolved as EXACTLY the domain/wrap artifact; the pre-declared KMS deviation (no contraction on the compact circle) carries exactly the silver witnesses; Theta_phys^2 = +1 on every sector (Kramers-free); kill tests (1)/(2) strengthened, (3) shadow sharpened, none fires -- all seven stay live on A_hol; beta-3 is EXECUTED (v565, WOIT.BETA3.DUALITY.01, 2026-07-30, verdict SUCCESS): the PT <-> PT* duality typed against sigma_std -- the compatible class is the two-block Hermitian family (clock-forced), the rho_tw pairing splits it into an Euclidean branch (sigma_std's home, no OS cut possible) and a Minkowski branch whose null locus meets the seam line exactly in the seam circle (the OS cut forced) with total signature (2,2) by the mirror theorem; the contract kill branch (family A) is EMPTY BY ALGEBRA (rho4^2 not scalar), and every seam-preserving Minkowski member induces the unique conjugation z -> -conj(z) = theta_perp = the v524 Theta_phys carrier (regressions re-verified; role separation: sigma_std ~ the OS cut/metric, the Phi-twisted duality ~ the descending mirror); gamma (the chirality theorem + the mark incidence) is the next milestone. [O] throughout; no marker moves anywhere. [2026-08-01 XXVI: the gamma stage gains its FREE SLICE -- v608: mirror modes are strictly OS-negative sector-exactly (kill test 3 cannot fire at the free level), the chiral generation is multiplicity-free ({1, sqrt2-1}), and the mark transport exists (tau_2 = mark-A -> mark-B, PD; kill test 7 does not fire free); gamma itself stays open on A_hol; see WOIT.GAMMA.FREE.01.] UPDATED 2026-08-27 (v972): the kill-test-2 counterweight NARROWED -- on the same v529 toy the quartet-AVOIDING cut family of the clock-invariant m = 4 member generates the full even algebra (rank 15, dim 32768) and carries a POSITIVE OS/GNS structure at g = 1 (min +4.4338e-5, GNS 37/37) while the straddled control genuinely fails (SEAM.INT.OSAVOID.01); the straddle law itself refines to the odd-split law (SEAM.INT.ODDSPLIT.01). The threat paragraph STANDS: every A_hol must still protect RP against the odd-split mechanism -- but the toy now EXHIBITS an OS-viable cut geometry selected by its own mu4 clock. Contract stays Open; no marker move.

Original classification detail

Open (research contract): the OS twistor bridge -- Theta real structure + interacting RP + OS reconstruction to (H, Omega, U(P^up_+), A_Mink) with one chiral generation, the TFPT charge lattice and the (E8)_1 net as an actual boundary net; seven preregistered kill tests; the actual bridge from compiler to physics; alpha stage executed via v519 (Theta + free RP, kill test 1 discharged on the free level only -- stays live on the interacting algebra); beta-1 executed via v522 (UNDECIDED: the clock reading was the wrong operationalisation, the clock is time-like, GSO-fixed RP holds, kill test 2 does not fire; contract precision (iii) added); beta-2 executed via v524 (SUCCESS, [C]-typed per precision (iii): H_phys explicit and PD at both levels, the clock a positive transfer step with spectral calculus -- N = 8 spectrum exactly {1, sqrt(2)-1} -- and a reconstructed unitary rotation group; the pre-declared KMS deviation carries exactly the silver witnesses; no kill fires); beta-3/gamma milestones next

Dependencies as recorded

CELEST.SEAM.01; SEAM.EQUIV.TWISTOR.01; SEAM.EQUIV.01; SEAM.BIT.FREEDOM.01; SEAM.CLOCK.RIGIDITY.01