Publication history: 2026-07-30

Details

• Discovery probe woit_os_beta3_pt_dual_probe.py (25/25, 2.4 s, verdict SUCCESS per the frozen preregistration; contract wording verbatim: success = Woit's Minkowski conjugation identified with the family-D conjugation on the quotient, kill = the duality forces a clock-centralising (family A) structure). The class (block theorem): a \mathbb{PT}\leftrightarrow\mathbb{PT}^* duality is a nondegenerate Hermitian form \Phi on \mathbb{C}^4; clock compatibility has entrywise residual (\bar\lambda_j\lambda_k-1)\Phi_{jk} and forces the four cross-eigenspace entries to zero: two 2\times2 Hermitian blocks, 16\to8 real parameters (the deck adds nothing; the wrong clock lift breaks the split). The dichotomy and the forced (2,2): the \rho_{\mathrm{tw}} pairing M^\dagger\Phi M=\varepsilon\bar\Phi splits the class exactly in two — the Euclidean branch (\sigma_{\mathrm{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 \Phi_1=-\bar\Phi_i exactly (charpoly mirrored), so the total signature is (2,2) for every nondegenerate member — Woit's Minkowski signature is a theorem of clock + \rho_{\mathrm{tw}} pairing, not an input. The kill is empty by algebra: every compatible duality inverts clock and deck (family D); family A would need \rho_4^2= scalar, but \rho_4^2=\operatorname{diag}(-1,1,-1,1) — no solution, decided not scanned; s_\Phi^2>0: the Kramers-free class. The identification: every seam-preserving Minkowski member induces the same conjugation z\to-\bar z (\mu=-1, fixed points \pm i), perpendicular to \sigma_{\mathrm{std}}'s z\to+\bar z; under the declared RP-side placement (exact rational-\pi arithmetic, offsets \mp\pi/2) it is \theta_\perp — the mirror whose descent v524 certified: \Theta_{\mathrm{phys}} is the quotient shadow of the duality (regressions re-verified: Gram PD (16,0,0), anti-unitary with \eta_\perp=+i, \Theta^2=+1 per sector, time reversal, torsor closure). Role separation complete: \sigma_{\mathrm{std}}\sim the OS cut (the metric), the \Phi-twisted duality \sim the descending mirror. Free/quotient level only; \mathcal A_{\mathrm{hol}} untouched; all seven kill tests stay live; no marker moves.
• Promotion v565_woit_beta3_pt_duality.py (WOIT.BETA3.DUALITY.01, 22 checks, {\sim}2 s, sympy-exact + two 40-digit inertia certificates). New ledger row; WOIT.OS.TWISTOR.01 updated (\beta_3 executed, \gamma next); suite 558\to559 scripts.
• Sync. tfpt_research_contracts: title line (\alpha/\beta_1/\beta_2/\beta_3 executed, \gamma next), the \beta_3 executed block, the roadmap item, the status paragraph, the straddle-hurdle wording (\gamma and beyond); tfpt_3: new Step 32, story counters thirty-one \to thirty-two (title, intro, preamble), the Woit-bridge paragraph gains the \beta_3-stage status; diary entry in experiments/next.txt; website: papers.ts WOIT milestone list + v565, suite.ts 558\to559, ScriptIndex/changelog mirrors regenerated; README/zenodo counters 559.