A registered result in its source context

WOIT.BETA1.GSO.01

Original ledger classification: UNDECIDED per the frozen preregistration ([E] exact Hermiticity witness + censuses, time-like reflection census, GSO 2-torsion identification, 40-digit gap-certified GSO-fixed RP certificates, exact Cl(16) Kramers restriction, Ramond/wrong-projection/site-cut/family-A controls; [C] contract precision (iii) -- prose only; [O] beta1 proper via beta2, kill tests live on A_hol); NO marker moves

Original claim text

THE WOIT-BETA1 MILESTONE EXECUTED (2026-07-22; one module, 16 checks, ~5 s; typed result, verdict UNDECIDED per the frozen preregistration -- 'the clock is time-like, GSO is the gauge datum'): contract clause beta_1 ('Theta on the gauge-invariant subalgebra + gauge-fixed RP on the equivariant SDYM(E8) sector'; kill test (2): 'RP fails after gauge fixing') executed on the free/equivariant seam skeleton. FINDING: the preregistered [C-1] clock-tower reading of 'gauge' is structurally the WRONG operationalisation. (1) HERMITICITY WITNESS (the mechanism): the mu4 clock average violates Hermiticity EXACTLY -- the one-step insertion T_1 = <theta(e_a), alpha_S(e_b)> has entry -i/(8 sin(5 pi/16)) where the conjugate transpose demands +i/(8 sin(5 pi/16)); the pairing is complex-SYMMETRIC (745 entries Hermitian-matching, 96 anti-matching, 0 violations): 'OS-symmetric' is STRICTLY WEAKER than Hermitian -- 'positivity after gauge fixing' is not well-posed for the clock reading. (2) TIME-LIKE TYPING: ALL 16 dihedral reflection axes of the NS seam circle INVERT the clock lift (R S R^-1 = S^-1 exactly), NONE commutes -- the mu4 clock IS Woit's euclidean rotation (the time translation itself); the theta-compatible (gaugeable) part of the Z8 tower is EXACTLY its 2-torsion {1, (-1)^F} (Hermiticity census over all powers; alpha^4 = the full wrap = the 2 pi rotation of the seam = fermion parity, lattice spin-statistics). (3) POSITIVE PART -- GSO-FIXED RP HOLDS EXACTLY: N = 16 even deg <= 2 parity-projected bond-cut Gram (29, 0, 0) PD (min eigenvalue 1.78e-6 at 40 digits); N = 8 complete half algebra: the projected 16x16 Gram kills all odd rows/columns exactly and its even block is (8, 0, 0) PD (3.35e-3) with NO degree truncation; the site-cut defect SURVIVES gauge fixing (inertia (7, 9, 6)) -- the RP 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 = id on A^G (the (-1)^F class does not penetrate the even subalgebra), Theta_Fock restricts with Theta^2 = +1; Ramond control: the wrap +1 'true Z4' lift fails state invariance -- the NS/Z8 tower is forced. (4) KILL-TEST BOOKKEEPING: kill test (1) stays discharged also gauge-invariantly (Theta^2 = +1 on A^G, Kramers class trivial; RP separates the families on the invariant sector); kill test (2)'s free shadow does NOT fire (no legitimate Hermitian gauge-fixed form has a negative direction); both stay formally live on the interacting algebra A_hol. (5) WRONG-PROJECTION CONTROLS: the invariant-dimension chain parity 120 > deck 64 > clock 32 is strict, but Hermiticity of the projected form singles out PARITY alone (deck and clock averages non-Hermitian): projecting by the finer 'gauge' groups is structurally ill-posed pre-quotient. TRANSPARENCY (documented in full): the first frozen run scored 6/13 -- the preregistered clock-reading checks failed exactly as the mechanism predicts, and the preregistered deg-2 clock-invariant dimension 30 was a combinatorial error (correct orbit census: 28 free shift-4 quadruples + 4 halved antipodal doublets = 32); the preregistration text is unchanged, the verdict follows the frozen [C-3] logic (Hermiticity failure => structural-failure branch => UNDECIDED). [C] CONTRACT PRECISION (iii) (prose, no marker): 'gauge-invariant subalgebra' at the free level means the GSO/fermion-parity Z2 -- the only theta-compatible part of the mu4 tower (the free-fermion shadow of the E8 glue: (E8)_1 = even NS of SO(16)_1 + R spinor); the clock is the euclidean rotation itself (timelike), its equivariant statement moves to beta2 (OS quotient first, then the clock as reconstructed transfer operator). [O] beta1 PROPER stays open (the clock-equivariant statement needs the beta2 OS quotient); both kill tests live on A_hol; WOIT.OS.TWISTOR.01 stays [O]. Discovery provenance: woit_os_beta1_gauge_invariant_theta_probe.py (16/16) in experiments/tfpt-discovery/ (2026-07-22). NO marker moves.

Original classification detail

UNDECIDED (typed, per the frozen preregistration): the [C-1] clock-tower reading of 'gauge' fails Hermiticity structurally (the mu4 clock is euclidean-time-like); the gaugeable part is exactly the GSO/fermion-parity Z2, under which gauge-fixed RP HOLDS ((29,0,0)/(8,0,0) PD); kill tests (1) and (2) do NOT fire at the free level; contract precision (iii) added (prose); beta1 proper re-routed via beta2; WOIT.OS.TWISTOR.01 stays Open

Dependencies as recorded

WOIT.OS.TWISTOR.01; WOIT.THETA.FREE.01; SEAM.BIT.RPBLIND.01; SEAM.CLOCK.RIGIDITY.01; v510; v519