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A registered result in its source context

Original ledger classification: Reduction [E]/[O] (modular bedrock)

Recorded statement: BEDROCK omega o rho = omega VIA AN EXPLICIT MODULAR HAMILTONIAN (2026-06-21, TOE attack point 2): the next concrete step on the open bedrock QGEO.SYM.01. [E] NON-CIRCULAR upstream: marks=2 chi(S^2)=4 (Gauss-Bonnet, v216) and the order-4…

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QGEO.MODHAM.01

Original ledger classification: Reduction [E]/[O] (modular bedrock)

Original claim text

BEDROCK omega o rho = omega VIA AN EXPLICIT MODULAR HAMILTONIAN (2026-06-21, TOE attack point 2): the next concrete step on the open bedrock QGEO.SYM.01. [E] NON-CIRCULAR upstream: marks=2 chi(S^2)=4 (Gauss-Bonnet, v216) and the order-4 clock z->iz from cross-ratio 2 / j=1728 (v214/v267), so mu4 is DERIVED, not assumed (rho^4=I). [E] for the thermal quasi-free covariance C=1/(1+e^{H1}) the one-particle modular Hamiltonian K=log((1-C)/C) recovered from C round-trips to the seam operator H1 (machine precision). [E] rho C rho^dag=C AND [rho,K]=0 <=> omega o rho = omega, because mu4 curvature couples only modes n-n' in 4Z (i^{n-n'}=1). [E] NEG: a non-mu4 (period-2/Z2) curvature breaks the invariance. [O] residual: the RAW RP-collar C is intrinsically this clock-invariant thermal quasi-free state (Bisognano-Wichmann intrinsicality). Sharpens v198/v210; does NOT close QGEO.SYM.01.

Original classification detail

[E] the non-circular marks+clock assembly + the explicit modular-Hamiltonian round-trip + [rho,K]=0 state invariance + the Z2 negative control; [O] the Bisognano-Wichmann intrinsicality of the raw collar. The sharpest single open theorem made explicit; NOT a closure.

Dependencies as recorded

QGEO.SYM.01; v198; v210; v214; v216

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Registered verification modules

  • v309_modular_bedrock.pyRegistered verification module · frontier
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    TOE attack 2 -- the bedrock omega o rho = omega via an explicit modular Hamiltonian, non-circular: marks=4 from Gauss-Bonnet (v216) + the order-4 clock from cross-ratio 2 (v214) => mu4 derived not assumed; the quasi-free modular Hamiltonian K=log((1-C)/C) round-trips to the seam operator and [rho,K]=0 <=> omega o rho = omega (a Z2 profile breaks it). Residual = Bisognano-Wichmann intrinsicality of the raw collar; NOT a closure of QGEO.SYM.01

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