A registered result in its source context
Original ledger classification: Reduction [E] (flat => commutator all orders) + [C] (tau=i forced) + [O] (one geometric postulate)
Recorded statement: FLAT PILLOWCASE CLOSES THE MODULAR COMMUTATOR TO ALL ORDERS (2026-06-18): collapses the bedrock QGEO.SYM.01 to ONE geometric postulate (does NOT prove the premise). For a quasi-free seam state QGEO.SYM.01 = (omega o rho = omega) <=>…
Statement, assumptions and sources
Paper context: tfpt research contracts
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QGEO.SYM.03
Original ledger classification: Reduction [E] (flat => commutator all orders) + [C] (tau=i forced) + [O] (one geometric postulate)
Original claim text
FLAT PILLOWCASE CLOSES THE MODULAR COMMUTATOR TO ALL ORDERS (2026-06-18): collapses the bedrock QGEO.SYM.01 to ONE geometric postulate (does NOT prove the premise). For a quasi-free seam state QGEO.SYM.01 = (omega o rho = omega) <=> [rho,C]=0 <=> [rho,H]=0 (C=(1+e^H)^-1, v258). [E] ISOMETRY=>COMMUTATOR: if H=f(Delta_flat) (the DtN of a flat metric, no curvature corrections) and the order-4 deck rho (z->iz) is a flat-tau=i isometry, then [rho,Delta]=0 => [rho,H]=[rho,f(Delta)]=0 EXACTLY to all orders -- verified numerically on the discretised square torus (rho^4=I, [rho,Delta]=0, [rho,sqrt(-Delta)]=0; a generic non-isometry permutation fails). [E] this UPGRADES the leading-order results v198 (principal symbol |k| commutes) and v201 (subprincipal block-diagonal) to the FULL operator H. [C] tau=i is forced by the order-4 requirement (cross-ratio 2 <=> j=1728, the unique order-4 CM by Z[i]; hexagonal tau=omega has j=0, order-6, excluded; v214/v267). [O] QGEO.SYM.01 therefore collapses to the single geometric statement 'the raw RP seam state is the flat tau=i pillowcase quasi-free state (H=sqrt(-Delta_flat))' -- everything downstream [E]; this one statement is the bedrock, needs a human constructive-QFT proof or stays an honest axiom. Sharpest non-circular form.
Original classification detail
[E] the isometry=>commutator closure (numerically verified, all orders) + the all-orders upgrade of v198/v201; [C] tau=i forced by order-4; [O] the one geometric postulate (raw seam = flat tau=i pillowcase) stays open. Narrows QGEO.SYM.01 to a single flat-geometry statement; does NOT close it.
Dependencies as recorded
QGEO.SYM.01; QGEO.SYM.02; QGEO.MODULAR.01; QGEO.SUBPRIN.01; QGEO.PILLOW.01; PS.DIRAC.03
Open the original ledger source, row 370Location recorded in the ledger
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- Research contracts
Sections that explicitly cite this claim
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- 2026-06-18 (QGEO.SYM.01 narrowed: the flat pillowcase closes the commutator to all orders, v276)Research changelog; TFPT — Introduction and reading guide
- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportResearch contracts
Navigation through a shared verification script
These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.
- The closure ledger: what is done, and what genuinely remainsTFPT — Introduction and reading guide
- Front matter / supportResearch contracts
- 2026-06-18 (the two proof routes decomposed: Route (i) and Route (ii) into lemma chains, v284/ v285)Research changelog; TFPT — Introduction and reading guide
- 2026-06-18 (the Gr\"al: QGEO.SYM.01 as one precise lemma [formula in source] the all-orders step Lean-formalised, v279/ FORM.QGEO.04)Research changelog; TFPT — Introduction and reading guide
- 2026-06-18 (closing sprint: the Seam Equivalence Theorem named + both routes attacked, v286– v288)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportFrontier and open physical interfaces
- Appendix: computational verification (reproducibility)TFPT — Introduction and reading guide
- 2026-06-18 (QG.AMB.01 Tier-B made concrete: the seam-Calder\'on [formula in source] match, v277)Research changelog; TFPT — Introduction and reading guide
Registered verification modules
- v276_qgeo_flat_closes_commutator.pyRegistered verification module · registry
Read the description recorded in the module registry
QGEO.SYM.03 — the flat tau=i pillowcase premise CLOSES the modular-commutator chain to all orders, collapsing the bedrock QGEO.SYM.01 to ONE geometric postulate (it does NOT prove the premise). For a quasi-free seam state QGEO.SYM.01 = (omega o rho = omega) <=> [rho,C]=0 <=> [rho,H]=0. [E] ISOMETRY=>COMMUTATOR: if H=f(Delta_flat) and the order-4 deck rho (z->iz) is a flat-tau=i isometry, then [rho,Delta]=0 => [rho,H]=0 EXACTLY (all orders), verified on the discretised square torus (rho^4=I, [rho,Delta]=0, [rho,sqrt(-Delta)]=0; generic non-isometry fails); [E] this UPGRADES v198 (principal symbol) + v201 (subprincipal) from leading orders to the full H (flatness removes curvature terms); [C] tau=i is forced by the order-4 requirement (j=1728; hexagonal j=0 is order-6, excluded; v214/v267); [O] QGEO.SYM.01 collapses to the single statement 'the raw seam IS the flat tau=i pillowcase quasi-free state (H=sqrt(-Delta_flat))' — needs a human constructive-QFT proof or stays an honest axiom (like c=const). Sharpest non-circular form.
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