A registered result in its source context

WOIT.BETA2.OS.01

Original ledger classification: SUCCESS per the frozen preregistration, [C]-typed per contract precision (iii) ([E] exact Hermiticity/semigroup/square identities, exact clock spectrum {1, sqrt(2)-1}, exact wrap-artifact resolution and censuses, 40-digit gap-certified inertia/spectral certificates; [C] the precision-(iii) operationalisation of 'the clock acting unitarily'; [O] beta3 next, all seven kill tests live on A_hol); NO marker moves

Original claim text

THE WOIT-BETA2 MILESTONE EXECUTED (2026-07-23; one module, 20 checks, ~5 s; verdict SUCCESS per the frozen preregistration, [C]-typed per contract precision (iii)): contract clause beta_2 ('the OS quotient of the free system made explicit -- success: (H, Omega) constructed from the v519 bond-cut form with the clock acting unitarily; kill: the quotient degenerates (zero or indefinite completion)') executed. (1) H_phys EXPLICIT: N = 16 (deg <= 2): the 37x37 bond-cut OS Gram is exactly Hermitian, parity-block-diagonal, inertia (37, 0, 0) PD (min eigenvalue 1.7801e-6 at 40 digits), null space {0}, dim 37 = 29 (+) 8; N = 8 (complete half algebra, no degree truncation): (16, 0, 0) PD (min 3.3471e-3), dim 16 = 8 (+) 8 = 4^2 -- compact euclidean time reconstructs a THERMAL (KMS) representation; exact certificate sin^2(3pi/8) - sin(pi/8) sin(5pi/8) = 1/2. (2) TRANSFER SEMIGROUP (Klein-Landau): tau_k = omega(theta(a) alpha_k(b)) exactly Hermitian on every shrinking domain D_k (k = 1..7 at N = 16, dims 29, 22, 16, 11, 7, 4, 2), chain identities exact, vacuum fixed exactly; the positivity pattern IS the v519 site/bond dichotomy: even k PSD via the exact square identity T(2j) = A_j*A_j ((22,0,0)/(11,0,0)/(4,0,0)), odd k indefinite with exactly zero one-particle diagonal ((10,12,7)/(5,6,5)/(2,2,3)) -- the one-step transfer is NOT positive (the free chirality/staggering datum, kill-test-3 shadow sharpened), its even powers ARE. (3) THE CLOCK = T^{N/4}: N = 16: tau_4 PD (11, 0, 0) (min 6.6821e-5), compressed spectrum [0.0194517, 1.64144] with vacuum eigenvalue exactly 1, spectral projections certified ~1e-40; N = 8: compressed spectrum EXACTLY {1, sqrt(2)-1} = {1, 1/delta_Silver} in both parity sectors -- the silver-midpoint axes of the v519 mu4 torsor return as the exact clock-transfer eigenvalue; the reconstructed rotation group U(s) = exp(isH) is unitary with the group law (~1e-40) -- per precision (iii) the [C]-operationalisation of 'the clock acting unitarily'. (4) v522 RESOLUTION: the clock-insertion census (745, 96, 0) reproduced; EVERY anti-matching entry is a wrap overlap and every overlap-free entry matches -- the pre-quotient non-Hermiticity is EXACTLY the domain/wrap artifact; restricted to the quotient domain the same insertion IS tau_4 entrywise: what dissolves in beta2 is the ill-posedness of clock positivity (the clock now has spectral calculus); what remains time-like is the euclidean-rotation character itself. (5) KMS DEVIATION (pre-declared, exactly the declared witnesses): NO contraction on the compact circle -- the edge mode expands by C(1)/C(3) = 1 + sqrt(2) = delta_S exactly (N = 8), det(G - tau_4) < 0 on the odd one-particle block (N = 16, lambda_max = 1.6414 > 1): a typed KMS finding, not a beta2 failure. (6) GSO/THETA: fermion parity descends G-unitarily (H_even (+) H_odd exact); theta_cut and the deck candidate do NOT act on H_phys (support censuses -- theta_cut IS the metric); the PERPENDICULAR torsor mirror descends anti-unitarily (eta_perp in {+-i}), Theta_phys^2 = +1 on EVERY sector incl. odd (the quotient is KRAMERS-FREE), time-reverses the semigroup, and theta_cut o theta_perp = alpha_{N/2} exactly (the torsor closes onto the deck). (7) CONTROLS: site cut indefinite ((3,3,1)/(2,2,4)) -- the contract kill branch fires for the site placement; family A (4,4,0) no quotient; anti-chiral state (8,8,0) -- quotient existence itself selects the chiral orientation; wrong semigroup direction non-Hermitian off-domain (census (58, 6, 0), all mismatches wrap-overlap), Hermitian on the mirrored domain. KILL-TEST BOOKKEEPING: (1), (2) strengthened, (3) shadow sharpened, none fires; (4)-(7) untouched; all seven stay formally live on A_hol. TRANSPARENCY: run 1 scored 18/20 -- a sympy simplify failure on the TRUE product formula sin(7pi/16) - sin(5pi/16) = 2 cos(3pi/8) sin(pi/16), fixed by a Laurent-polynomial certificate; no criterion, tolerance or expectation changed. [C] FENCE: 'the clock acting unitarily' operationalised per precision (iii) as the reconstructed transfer/rotation structure. [O] beta3 (the PT <-> PT* duality vs sigma_std) is the next milestone; WOIT.OS.TWISTOR.01 stays [O]. Discovery provenance: woit_os_beta2_os_quotient_probe.py (20/20) in experiments/tfpt-discovery/ (2026-07-23). NO marker moves. [2026-08-01 XXVI: the reconstructed net of this module now carries the free gamma slice -- v608 (mirror exclusion sector-exact, multiplicity-free generation, mark transport); see WOIT.GAMMA.FREE.01.]

Original classification detail

SUCCESS ([C]-typed per precision (iii)): the OS quotient of the free system is explicit -- H_phys PD and nondegenerate at both levels (37 = 29+8; 16 = 4^2 thermal/KMS), the Klein-Landau transfer semigroup exactly Hermitian with the site/bond dichotomy as its positivity pattern, the mu4 clock = T^{N/4} positive with spectral calculus (N = 8 spectrum exactly {1, sqrt(2)-1}) and a reconstructed unitary rotation group; the v522 non-Hermiticity is exactly the wrap artifact; the pre-declared KMS deviation carries exactly the silver witnesses; Theta_phys^2 = +1 Kramers-free; no kill fires, all seven stay live on A_hol; beta3 next; WOIT.OS.TWISTOR.01 stays Open

Dependencies as recorded

WOIT.OS.TWISTOR.01; WOIT.THETA.FREE.01; WOIT.BETA1.GSO.01; SEAM.BIT.RPBLIND.01; v510; v519; v522