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Original ledger classification: Formal

Recorded statement: LEAN: QGEO.SYM.01 FROM SEAM.EQUIV.01 VIA BISOGNANO-WICHMANN (2026-06-22). Lean 4 formalisation (TfptCarrier/BWKeystone.lean) of the v323 linkage: the nodes RotationInvariantVacuum -> GeometricModularFlow -> Mu4ModularSymmetry ->…

Statement, assumptions and sources

Paper context: Origin Theory

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FORM.QGEO.BW.01

Original ledger classification: Formal

Original claim text

LEAN: QGEO.SYM.01 FROM SEAM.EQUIV.01 VIA BISOGNANO-WICHMANN (2026-06-22). Lean 4 formalisation (TfptCarrier/BWKeystone.lean) of the v323 linkage: the nodes RotationInvariantVacuum -> GeometricModularFlow -> Mu4ModularSymmetry -> StateInvariance (=QGEO.SYM.01) assembled with the cited Bisognano-Wichmann/Hislop-Longo steps as named axioms, composed over the v308 chain (seamEquivChain discharges SeamIsE8). The theorem bedrockReducesToRotationInvariance : RotationInvariantVacuum -> StateInvariance shows the WHOLE bedrock QGEO.SYM.01 reduces to the SINGLE shared open premise RotationInvariantVacuum (the two open bedrock items collapse to one). #print axioms confirms the residual = the v308 cited steps + recovery_gap + the named BW steps + RotationInvariantVacuum (no hidden assumption). Plus a decide-proven discriminator bw_strictly_stronger: (1|4) and not(4|1) -- the BW geometric flow (period 1) is strictly stronger than the v309 clock invariance (period 4). Built end-to-end (lake build) on Lean v4.29.1 + Mathlib. The VALUE is the machine-checked collapse of the two bedrock items to one shared premise, NOT a from-scratch proof -- it does NOT close QGEO.SYM.01.

Original classification detail

[E] Lean-built: the composition bedrockReducesToRotationInvariance + the axiom audit (residual = chain steps + recovery_gap + named BW steps + the ONE shared premise) + the decide-proven geometric-vs-clock discriminator. Machine-pins the v323 collapse (two bedrock items -> one shared premise RotationInvariantVacuum); plus qgeoSymIsCorollary (2026-06-22): with the cited conformal-net vacuum axiom (conformal_net_vacuum_rotation_invariant), StateInvariance (=QGEO.SYM.01) follows from SeamIsE8 ALONE, so QGEO.SYM.01 is a COROLLARY of SEAM.EQUIV.01 with no independent open premise. Does NOT close SEAM.EQUIV.01.

Dependencies as recorded

QGEO.BW.01; QGEO.SYM.01; SEAM.EQUIV.CHAIN.01; FORM.SEAMEQUIV.01; v323; v308

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Registered verification modules

  • v323_bw_geometric_modular.pyRegistered verification module · frontier
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    The keystone pushed one step: Bisognano-Wichmann makes the seam modular flow GEOMETRIC, so the mu4 deck invariance is not an independent axiom. The clock rho=diag(i^n)=exp(i(pi/2)L) is a geometric rotation (L=diag(n) the rotation generator, rho^4=I, mu4 in U(1)_rot). For a rotation-covariant covariance C=f(L) the modular Hamiltonian K=log((1-C)/C)=g(L), so [K,L]=0 (the modular flow commutes with all rotations) and [rho,K]=[rho,C]=0 -- the clock is a modular symmetry for free. Discriminator vs v309: a period-4 curvature preserves the clock but its modular flow is NOT geometric ([K,L]!=0); BW is strictly stronger. Linked bedrock: GIVEN the seam is the (E8)_1 chiral net (v308), BW/Hislop-Longo make the vacuum modular flow geometric => QGEO.SYM.01 (omega o rho=omega) is downstream of SEAM.EQUIV.01 + a rotation-invariant vacuum. Residual: the raw collar vacuum is rotation-invariant (cleaner than v309, shared with v308)

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