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When modular flow has a geometric interpretation

Original ledger classification: Reduction [E]/[C]/[O] (BW geometric modular flow)

Recorded statement: THE KEYSTONE PUSHED ONE STEP -- BISOGNANO-WICHMANN MAKES THE SEAM MODULAR FLOW GEOMETRIC (2026-06-22): the mu4 deck invariance is NOT an independent axiom. [E] CLOCK = GEOMETRIC ROTATION: rho=diag(i^n)=exp(i(pi/2)L), L=diag(n) the seam…

Statement, assumptions and sources

Paper context: Origin Theory

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QGEO.BW.01

Original ledger classification: Reduction [E]/[C]/[O] (BW geometric modular flow)

Original claim text

THE KEYSTONE PUSHED ONE STEP -- BISOGNANO-WICHMANN MAKES THE SEAM MODULAR FLOW GEOMETRIC (2026-06-22): the mu4 deck invariance is NOT an independent axiom. [E] CLOCK = GEOMETRIC ROTATION: rho=diag(i^n)=exp(i(pi/2)L), L=diag(n) the seam rotation generator; rho^4=exp(i2pi L)=I -- mu4 in U(1)_rot (quarter turns). [E] BW: a rotation-covariant covariance C=f(L) gives 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 mu4 clock is a modular symmetry FOR FREE (no fine-tuned curvature, unlike v309). [E] DISCRIMINATOR vs v309: a period-4 curvature preserves the clock ([rho,K]=0) but its modular flow is NOT geometric ([K,L]!=0) -- BW is strictly stronger (pins the whole rotation group, not just mu4). [E] NEG: a non-rotation-covariant C breaks [rho,K]!=0. [C] LINKED BEDROCK: GIVEN the seam is the (E8)_1 chiral net (v308/SEAM.EQUIV.01), Bisognano-Wichmann/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, NOT an independent axiom (two open items collapse toward one). [O] residual: the raw collar vacuum is rotation-invariant (cleaner than v309's bespoke thermal state, shared with v308). Sharpens v309/v198; does NOT close QGEO.SYM.01.

Original classification detail

[E] the finite-model BW construction (rotation-covariant C => geometric modular flow + a free clock symmetry) + the v309 discriminator (period-4 preserves the clock but is not geometric); [C] the linkage to v308 via the cited BW/Hislop-Longo theorem (QGEO.SYM.01 downstream of SEAM.EQUIV.01); [O] rotation-invariance of the raw seam. Sharpens v309; does NOT close the bedrock.

Dependencies as recorded

QGEO.SYM.01; QGEO.MODHAM.01; SEAM.EQUIV.CHAIN.01; v308; v309; v198

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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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