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Original ledger classification: Mechanism exhibit [E-num]/[N]/[C]/[O] (bedrock omega o rho = omega manifest at the 4-interval state level; multi-interval modular theory cited; raw-seam premise open)

Recorded statement: THE FOUR mu4 MARKS ARE A FOUR-INTERVAL FREE-FERMION GEOMETRY (2026-07-14, bird's-eye round): the seam circle minus the four Gauss-Bonnet marks (v216) IS the n = 4 disjoint-interval region of a chiral free fermion -- the class whose…

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QGEO.MULTILOCAL.01

Original ledger classification: Mechanism exhibit [E-num]/[N]/[C]/[O] (bedrock omega o rho = omega manifest at the 4-interval state level; multi-interval modular theory cited; raw-seam premise open)

Original claim text

THE FOUR mu4 MARKS ARE A FOUR-INTERVAL FREE-FERMION GEOMETRY (2026-07-14, bird's-eye round): the seam circle minus the four Gauss-Bonnet marks (v216) IS the n = 4 disjoint-interval region of a chiral free fermion -- the class whose multi-interval modular theory is EXACTLY KNOWN (Casini-Huerta 2009 mixing term; Longo-Xu multi-interval subfactors CMP 251 (2004); Rehren-Tedesco multilocal fermions LMP 104 (2014); KLM CMP 219 (2001): holomorphic <=> trivial multi-interval inclusion, mu = 1 -- the SAME one bit as det K = 1). Until now the suite used only the single-ball CHM boost (v358). On the exactly solvable antiperiodic critical ring (N = 256, Q = 64, ell = 48): [E-num] BINARY CLOCK: the quarter-turn clock is a signed permutation with rho^4 = -1 EXACTLY -- the fermionic mu4 clock is the order-8 double-cover lift (the same Z2-lift pattern as 2I over I). [E-num] EXACT SECTOR DECOUPLING: the primitive-8th-root character projectors resolve the identity and commute with the reduced covariance at 1e-15 (the Rehren-Tedesco diagonal 4-fold-cover isomorphism at covariance level). [E-num] COVER/TWIST: each sector kernel is the single-interval quarter-ring kernel with boundary twist lambda_k; sector spectra union = full spectrum at 1e-14. [E-num] CLOCK INVARIANCE MANIFEST: rho C rho^-1 = C at 1e-14 => [rho, K] = 0 by functional calculus -- omega o rho = omega HOLDS at the 4-interval state level; NEGATIVE CONTROL: displacing ONE interval by one site re-couples the sectors at 1e-2. [N] CONJUGATE-POINT MIXING: every cross-interval K row peaks on the conjugate diagonal |s'-s| <= 2 at both sizes -- the lattice shadow of the Casini-Huerta mixing term (band energy spreads with size: long-range conformal tails, honestly noted). [C] the continuum multi-interval theory is CITED, not re-derived. [O] the raw-seam premise (QGEO.SYM.01 / SEAM.EQUIV.01) stays OPEN -- this exhibits the mechanism in the exactly solvable class the seam is argued to reach, it does not prove the seam reaches it. Also the natural continuum frame for the AP2 leg (v476/v478). No gate closes, no marker moves.

Original classification detail

Mechanism exhibit [E]/[N]/[C]/[O]: the mu4 marks = exactly solvable 4-interval modular geometry; binary clock rho^4 = -1, exact sector decoupling, omega o rho = omega manifest + configuration-forced; QGEO.SYM.01 premise stays [O]

Dependencies as recorded

QGEO.SYM.01; SEAM.EQUIV.01; SEAM.RIGIDITY.01; GRAV.ENTROPIC.COMPRESS.01; GRAV.ENTROPIC.LEGS.01; v216; v323; v358; v398; v426; v442; v476; v478

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Registered verification modules

  • v480_multilocal_four_interval.pyRegistered verification module · frontier
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    QGEO.MULTILOCAL.01: the four mu4 marks make the seam circle a FOUR-INTERVAL free-fermion geometry -- the class whose multi-interval modular theory is exactly known (Casini-Huerta mixing term; Longo-Xu; Rehren-Tedesco multilocal fermions; KLM: holomorphic <=> trivial multi-interval inclusion, mu=1 -- the same bit as det K=1). On the exactly solvable antiperiodic ring: [E-num] binary clock rho^4 = -1 exactly (the fermionic mu4 clock is the order-8 double-cover lift, the same Z2-lift pattern as 2I over I); [E-num] the primitive-8th-root character projectors decouple the 4-interval covariance at 1e-15 (the multilocal diagonal-cover isomorphism); [E-num] each sector = single-interval quarter-ring kernel with twist lambda_k, spectra match at 1e-14; [E-num] rho C rho^-1 = C at 1e-14 => omega o rho = omega MANIFEST at the state level; [E-num] negative control: one displaced interval re-couples sectors at 1e-2; [N] cross-interval K peaks on the conjugate diagonal (Casini-Huerta mixing shadow). [C] continuum theory cited; [O] the raw-seam premise stays open -- mechanism exhibited, not closed.

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