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A registered result in its source context

Original ledger classification: Numerical (NCG/modular) [E]/[C]

Recorded statement: DIRAC AS MODULAR/COVARIANCE INDUCTION (2026-06-17): the finite Dirac operator D_F (v252) is the modular/covariance induction of the boundary seam quasi-free (KMS, beta=1) state restricted to the carrier, not an independent posit -- closes…

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Paper context: Research contracts

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PS.DIRAC.03

Original ledger classification: Numerical (NCG/modular) [E]/[C]

Original claim text

DIRAC AS MODULAR/COVARIANCE INDUCTION (2026-06-17): the finite Dirac operator D_F (v252) is the modular/covariance induction of the boundary seam quasi-free (KMS, beta=1) state restricted to the carrier, not an independent posit -- closes the 'posited Yukawa vs derived operator' gap of v250/v252 (proposal Q2: D_F as Kasparov product, covariance formula as the finite implementation). Quasi-free/modular dictionary: one-particle modular Hamiltonian H=log((1-C)C^-1) (Araki; Casini-Huerta), inverse to the KMS occupation C=(1+e^H)^-1. The v18 Yukawas are the readout of C_Sigma=P_F C_Sigma P_F, not an extra input.

Original classification detail

[E] inversion identity log((1-C)C^-1)=H exactly + C_F a positive contraction + chirality-odd (Dirac-type) + unitary-equivalent to v252 (same mass spectrum) + Majorana=off-diagonal covariance (seesaw m_D^2/M_R); [C] C_F=P_F C_Sigma P_F so D_F is the KK shadow of the boundary geometry (rests on the open seam realisation).

Dependencies as recorded

PS.DIRAC.02; DYN.KMS.01; CONTRACT.QFT4D.DIRAC.01

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Registered verification modules

  • v258_dirac_covariance_induction.pyRegistered verification module · registry
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    PS.DIRAC.03 — D_F is the modular/covariance INDUCTION of the boundary seam state, not an independent posit; closes the 'posited Yukawa vs derived operator' gap of v250/v252 (Q2: Dirac as Kasparov product + covariance formula). Quasi-free/modular dictionary: a Gaussian state is fixed by its covariance C (contraction, spec in (0,1)) with one-particle modular Hamiltonian H=log((1-C)C^-1) (Araki/Casini-Huerta), inverse to the KMS occupation C=(1+e^H)^-1 (beta=1, v239). [E] inversion identity log((1-C)C^-1)=H exactly (random Hermitian + a chirality-graded Yukawa block); [E] C_F=(1+e^{D/mu})^-1 a positive contraction; [E] chirality-odd (anticommutes with gamma_F); [E] unitary-equivalent to v252 (inducing C_F from the 9 Yukawas returns the same spectrum); [E] Majorana = off-diagonal covariance (seesaw m_light=m_D^2/M_R); [C] C_F=P_F C_Sigma P_F, so D_F is the KK shadow [D_Sigma](x)_{(E8)_1}[K_car] and the v18 Yukawas are the readout of C_Sigma.

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