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WOIT.BETA1.GSO.01: the WOIT-beta1 milestone of the OS twistor bridge WOIT.OS.TWISTOR.01 executed -- 'the clock is time-like, GSO is the gauge datum' (typed result: verdict UNDECIDED per the frozen preregistration, mechanism identified exactly, gauge-fixed RP established under the corrected GSO typing; 16 checks, ~5 s). [E] HERMITICITY WITNESS: 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 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. [E] TIME-LIKE TYPING: ALL 16 dihedral reflection axes of the NS seam circle invert the clock lift (R S R^-1 = S^-1), NONE commutes -- the mu4 clock IS Woit's euclidean rotation (the time translation itself); the gaugeable (theta-compatible) 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). [E] 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 Gram kills all odd rows/columns exactly, even block (8,0,0) PD (3.35e-3), no degree truncation; the site-cut defect SURVIVES gauge fixing ((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, Theta_Fock restricts with Theta^2 = +1; Ramond control: the wrap +1 'true Z4' lift fails state invariance -- the NS/Z8 tower is forced. [E] KILL-TEST BOOKKEEPING: kill test (1) stays discharged also gauge-invariantly (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. [E] WRONG-PROJECTION CONTROLS: invariant-dimension chain parity 120 > deck 64 > clock 32 strict, but Hermiticity singles out PARITY alone (deck and clock averages non-Hermitian): the finer 'gauge' projections are structurally ill-posed pre-quotient. 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 quadruples + 4 halved doublets = 32) -- documented in full; verdict by the frozen [C-3] logic (Hermiticity failure => structural-failure branch => UNDECIDED). [C] CONTRACT PRECISION (iii), prose only: '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 via beta2; WOIT.OS.TWISTOR.01 stays [O]; no marker moves.

Linked claims

WOIT.BETA1.GSO.01

Paper sections

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