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Original ledger classification: Geometric premise [O] (reduction: subclaim-2 hinge met, finite core [E], bedrock -> BW modular)

Recorded statement: SUBCLAIM 2 OF QGEO.ENERGY.02 TACKLED -- 'the raw RP-seam DtN is RP-definable' (external-review next-lever 2026-06-14): a REDUCTION, not a closure. (1) FINITE CORE [E]: rho:z|->iz on H^1(P^1\\mu4)=<w1,w2,w3>, wk=z^{k-1}dz/(z^4-1), acts as…

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QGEO.DTN.01

Original ledger classification: Geometric premise [O] (reduction: subclaim-2 hinge met, finite core [E], bedrock -> BW modular)

Original claim text

SUBCLAIM 2 OF QGEO.ENERGY.02 TACKLED -- 'the raw RP-seam DtN is RP-definable' (external-review next-lever 2026-06-14): a REDUCTION, not a closure. (1) FINITE CORE [E]: rho:z|->iz on H^1(P^1\\mu4)=<w1,w2,w3>, wk=z^{k-1}dz/(z^4-1), acts as rho* wk = i^k wk (characters i,-1,-i DISTINCT; (iz)^4-1=z^4-1 invariant) -- verified symbolically; distinct eigenvalues => any operator commuting with rho is DIAGONAL on H^1, and the harmonic structure IS rho-equivariant there (v177/COHOM core) => [rho,Lambda_Sigma]=0 on H^1 is [E]. (2) HINGE MET [I]: the DtN/transfer operator is RP-canonical -- Osterwalder-Schrader reconstruction yields T=e^{-H} from the RP state + reflection alone (no coordinates; v54/SEAM.HORIZON.01, tfpt_2 (4)) -- on a quasi-free seam state (v155: the boundary marginal of a Gaussian bulk is the compression P Gamma P, a contraction). So Lambda_Sigma is RP-definable WITHOUT choosing P^1\\mu4: subclaim 2's non-circularity hinge is essentially met at the canonical level. (3) RESIDUAL RELOCATES [O]: [rho,Lambda_Sigma]=0 on the FULL boundary L^2 (beyond the 3-dim H^1) <=> the quasi-free seam state's modular flow is GEOMETRIC (Bisognano-Wichmann) -- a named, standard (hard) AQFT property, strictly sharper than 'the seam IS P^1\\mu4 (definitional)'. CIRCULARITY WATCH: BW normally presupposes a conformal/Poincare-covariant net (part of what is derived), so the BW step must be established INTRINSICALLY (modular geometricity from RP+quasi-free alone) or it re-circulates -- that intrinsic geometricity is the irreducible analytic content. VERDICT: subclaim 2's hinge met + finite core [E] => QGEO.SYM.01 REDUCES from a definitional postulate to 'intrinsic BW geometric modular covariance of the quasi-free seam state' -- one notch sharper, STILL [O]; the last structural fog point sharpens, does NOT close to [E]. Python-only (H^1 commutation is exact Moebius-symbolic like v177; reduction is structural)

Original classification detail

Geometric [O]: subclaim 2 (raw-seam DtN RP-definability) met at the canonical OS level + finite core [E]; QGEO.SYM.01 reduces to intrinsic Bisognano-Wichmann modular covariance of the quasi-free seam state -- sharper, still [O], not closed

Dependencies as recorded

QGEO.ENERGY.02; QGEO.SYM.01; QGEO.TREE.01; SEAM.HORIZON.01; v177_seam_marking_kernel.py; v155_quasifree_boundary.py; v54_seam_horizon_keystones.py

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Registered verification modules

  • v194_raw_seam_dtn_rp.pyRegistered verification module · registry
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    QGEO.DTN.01 — subclaim 2 of QGEO.ENERGY.02 ('the raw RP-seam DtN is RP-definable') tackled: a REDUCTION, not a closure. (1) [E] finite core: ρ:z↦iz on H¹=⟨w1,w2,w3⟩, wk=z^{k-1}dz/(z⁴−1), acts as ρ*wk=i^k wk (characters i,−1,−i distinct), so [ρ,Λ_Σ]=0 on H¹ (v177). (2) [I] hinge met: the DtN/transfer operator is RP-canonical (Osterwalder–Schrader, v54) on a quasi-free seam state (v155), so Λ_Σ is RP-definable without assuming P¹∖µ4. (3) [O] residual relocates: full-L² commutation ⇔ Bisognano–Wichmann geometric modular flow of the quasi-free seam state (must be intrinsic, else re-circulates). QGEO.SYM.01 reduces from a definitional postulate to intrinsic BW modular covariance — sharper, still [O], not closed to [E].

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