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Original ledger classification: Clock-invariance derived [E]/[C]: [rho,Lambda]=0 is a CONSEQUENCE of mu4-symmetry of the RP data (the four marks) via OS reconstruction (v54) + the modular formula (v258) -- the OS canonical transfer of mu4-symmetric data inherits the clock (swept N, iff-sharp neg control), is a contraction, and its modular Hamiltonian commutes with the clock. With v445 this reduces the rigidity residual from an operator commutation to 'the RP data is mu4-symmetric' (QGEO.SYM.01); SEAM.EQUIV.01 stays [O]

Recorded statement: DERIVING clock-invariance ('transfer in the commutant') from RP-symmetry (2026-06-30; closes the single residual of v445/SEAM.RIGIDITY.FORCING.01 one notch further) -- v445 reduced rigidity to the OPERATOR statement [rho,Lambda_Sigma]=0…

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SEAM.RIGIDITY.CLOCKINV.01

Original ledger classification: Clock-invariance derived [E]/[C]: [rho,Lambda]=0 is a CONSEQUENCE of mu4-symmetry of the RP data (the four marks) via OS reconstruction (v54) + the modular formula (v258) -- the OS canonical transfer of mu4-symmetric data inherits the clock (swept N, iff-sharp neg control), is a contraction, and its modular Hamiltonian commutes with the clock. With v445 this reduces the rigidity residual from an operator commutation to 'the RP data is mu4-symmetric' (QGEO.SYM.01); SEAM.EQUIV.01 stays [O]

Original claim text

DERIVING clock-invariance ('transfer in the commutant') from RP-symmetry (2026-06-30; closes the single residual of v445/SEAM.RIGIDITY.FORCING.01 one notch further) -- v445 reduced rigidity to the OPERATOR statement [rho,Lambda_Sigma]=0 (clock-invariance); this module shows that commutation is itself a CONSEQUENCE of a STRUCTURAL symmetry (mu4-invariance of the seam RP data, the four Gauss-Bonnet marks) via the OS canonical reconstruction (v54) and the free modular formula (v258), NOT an independent assumption. [E] OS RECONSTRUCTION INHERITS THE SYMMETRY: for a mu4-invariant RP covariance ([rho,C(t)]=0), the OS canonical one-step transfer Lambda=C(1)C(0)^-1=e^{-h} (the transfer fixed by RP+reflection, v54) satisfies [rho,Lambda]=0 -- swept N=4..32 over random block-diagonal h. [E] SHARPNESS/NEG CONTROL: a mu4-BREAKING covariance gives [rho,Lambda]!=0 -- iff-sharp, so the inheritance is non-vacuous; mu4-symmetry of the RP data is NECESSARY. [E] CONTRACTION/OS-VALIDITY: the reconstructed Lambda has spectrum in (0,1] (a positive contraction) -- the physical canonical transfer, not an arbitrary commuting matrix. [E] MODULAR IMPLEMENTATION (Tomita-Takesaki): the free modular Hamiltonian K=log((1-C)C^-1) (the v258 covariance->Dirac formula) of the mu4-invariant state satisfies [rho,K]=0 (the clock commutes with modular flow), while the symmetry-breaking control gives [rho,K]!=0. [C] SYNTHESIS: [rho,Lambda_Sigma]=0 is a CONSEQUENCE of mu4-invariance of the seam RP data (the four marks, v216/v323) via OS reconstruction (v54) + the modular fact (v258), not an extra assumption; with v445 (commute <=> block-diagonal), RP-definability PLUS the marks FORCE the block-diagonal Lambda_Sigma. [C]/[O] VERDICT: v445's residual is reduced from an OPERATOR commutation to the STRUCTURAL statement 'the seam RP data is mu4-symmetric' (QGEO.SYM.01); the continuum Tomita-Takesaki implementation on the full L^2 is the cited backbone. SEAM.RIGIDITY.01/SEAM.EQUIV.01 stays [O].

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Dependencies as recorded

SEAM.RIGIDITY.FORCING.01; SEAM.RIGIDITY.01; SEAM.EQUIV.01; QGEO.SYM.01; v445; v442; v54; v258; v216; v323

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  • v446_seam_clock_invariance.pyRegistered verification module · frontier
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    SEAM.RIGIDITY.CLOCKINV.01: DERIVING clock-invariance ('transfer in the commutant') from RP-symmetry, closing v445's single residual one notch further. [E] OS RECONSTRUCTION INHERITS THE SYMMETRY: for a mu4-invariant RP covariance ([rho,C(t)]=0), the OS canonical transfer Lambda=C(1)C(0)^-1=e^{-h} (fixed by RP+reflection, v54) has [rho,Lambda]=0 -- swept N=4..32 over random block-diagonal h. [E] SHARPNESS/NEG CONTROL: a mu4-BREAKING covariance gives [rho,Lambda]!=0 -- iff-sharp. [E] CONTRACTION/OS-VALIDITY: Lambda has spectrum in (0,1]. [E] MODULAR (Tomita-Takesaki): the free modular Hamiltonian K=log((1-C)C^-1) (v258) of the invariant state has [rho,K]=0, broken control [rho,K]!=0. [C] SYNTHESIS: [rho,Lambda_Sigma]=0 is a CONSEQUENCE of mu4-invariance of the seam RP data (the four marks, v216/v323) via OS reconstruction (v54) + the modular fact (v258), not an extra assumption. [C]/[O] the residual reduces from an operator commutation to 'the RP data is mu4-symmetric' (QGEO.SYM.01); SEAM.EQUIV.01 stays [O]. Python-only (numpy/scipy).

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