Follow a result back to its evidence.

WOIT.BETA3.DUALITY.01: the WOIT-beta3 milestone of the OS twistor bridge executed — 'the PT ↔ PT* duality typed against sigma_std' (verdict SUCCESS per the frozen preregistration; the contract kill branch is EMPTY BY ALGEBRA; no marker moves): (1) the COMPATIBLE CLASS — a duality is a nondegenerate Hermitian form Phi on C^4, and clock compatibility (rho4 = diag(i,1,i,1)) forces the four cross-eigenspace entries to zero exactly: two 2×2 Hermitian blocks on {e1,e3} ⊕ {e2,e4}, 16 → 8 real parameters (deck adds nothing; the wrong clock lift breaks the split); (2) the DICHOTOMY — pairing against Woit's Euclidean structure rho_tw splits the class exactly in two: the EUCLIDEAN branch (sigma_std's home) has seam-line norm a(|z|²+1), no null point, no OS cut ever; the MINKOWSKI branch has norm a(|z|²−1) — the null locus meets the seam line EXACTLY in the seam circle for every member (the OS cut is FORCED, not imposed), and the second block is exactly minus the conjugate of the first (charpoly mirrored): the TOTAL SIGNATURE IS (2,2) for every nondegenerate member — Woit's Minkowski signature is a theorem of clock + rho_tw pairing, not an input; (3) the KILL IS EMPTY BY ALGEBRA — every compatible duality inverts clock AND deck (family D); a clock-centralising (family A) duality would need rho4² = scalar, but rho4² = diag(−1,1,−1,1): no solution, decided not scanned; s² = diag(a²,a²,c²,c²) > 0: Kramers-free; (4) the INDUCED MEMBER IS UNIQUE — seam preservation ⇔ b = 0, and then every member induces the SAME conjugation z → −conj(z) (mu = −1, fixed points ±i), perpendicular to sigma_std's z → +conj(z) (mu = +1, fixed points ±1), the clock exchanging exactly these two; (5) the QUOTIENT IDENTIFICATION — exact placement arithmetic and coordinate transport land the induced member on the k = 3 axis: it IS theta_perp, the mirror whose descent v524 certified (Theta_phys), while sigma_std's member transports to the CUT axis (the OS metric itself) — role separation complete; v524 regressions re-verified at N = 8 (Gram PD (16,0,0), anti-unitarity exact with eta_perp = +i, Theta² = +1 on both sectors, time reversal, torsor closure); controls: family A (4,4,0) no quotient, the Euclidean branch cannot cut, non-mu4 reflections not mark-anchored. Free/quotient level only; A_hol untouched; gamma is next (22 checks, ~2 s)

Linked claims

WOIT.BETA3.DUALITY.01

Paper sections

Source section 1Source section 2Source section 3Source section 4Source section 5Source section 6Source section 7Source section 8Source section 9Source section 10Source section 11Source section 12Source section 13Source section 14Source section 15Source section 16Source section 17Source section 18