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Original ledger classification: Identity [I] (signed cumulant inversion + Wick functoriality + PF dichotomy) + [P] (the scope premise: v56 acts on the full cone)

Recorded statement: SEAM GAUSSIANITY AS A CONDITIONAL FIXED-POINT THEOREM (the proposed closure of premise (A), formalised and honestly typed; the load relocates to ONE internal statement): (1) QUASI-FREE => kappa_{2n}=0 [I] - with the 2n-point moment =…

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GATE.METRIC.16

Original ledger classification: Identity [I] (signed cumulant inversion + Wick functoriality + PF dichotomy) + [P] (the scope premise: v56 acts on the full cone)

Original claim text

SEAM GAUSSIANITY AS A CONDITIONAL FIXED-POINT THEOREM (the proposed closure of premise (A), formalised and honestly typed; the load relocates to ONE internal statement): (1) QUASI-FREE => kappa_{2n}=0 [I] - with the 2n-point moment = Pf(Gamma submatrix), the SIGNED set-partition moment<->cumulant inversion gives kappa(|S|=2)=Gamma_ij and kappa(|S|=4)=kappa(|S|=6)=0 exactly (verified on a concrete pure Majorana covariance) => 'the Wick state is Gaussian' is an identity of the one 2-point kernel (the v113 one-kernel property at the cumulant level); (2) FIXED POINT BY FUNCTORIALITY [I] - a kernel-preserving transport O Gamma O^T = Gamma fixes every correlator (each a Pfaffian of a Gamma submatrix), so fixing the degree-2 datum K_Sigma fixes the whole quasi-free state; (3) PF DICHOTOMY [I] (toy cone) - a gapped primitive transport has a UNIQUE fixed ray (fixed-space dim 1); a degenerate variant with a second unit eigenvalue has dim 2 => a non-Gaussian fixed datum kappa_{2n} would be a SECOND fixed ray and break uniqueness, while a non-fixed kappa_{2n} lies in the gapped complement and decays geometrically -> 0; (4) GAP>0 SUFFICES [I] - kappa_{2n}->0 for any sub-leading rate in (0,1); the value (2/3)^6 only sets the decay rate. RESIDUAL/THE LOAD [P]: v56's gapped PF uniqueness is established on the 3-dim scalar transfer spectrum {1,(2/3)^6,(1/3)^6} (flavor gap / Page recovery), NOT on the Schwinger cone (the 16-Majorana seam already has C(16,4)=1820, C(16,6)=8008 connected-cumulant directions, dim >> 3); non-circular only if uniqueness holds on the FULL cone (incl. non-Gaussian functionals). So (A) reduces from 'prove Gaussianity' to the single internal, exact-testable statement 'the admissible TFPT seam transport is a primitive, spectrally gapped, RP-cone-preserving map on the full Schwinger cone, not only the readout spectrum of v56' - strictly smaller, but unproven; v56 as it stands does not supply it

Original classification detail

Identity [I]: quasi-free => kappa_{2n}=0 + PF dichotomy; (A) reduced to ONE scope premise (v56 on the full Schwinger cone)

Dependencies as recorded

GATE.METRIC.12; GATE.METRIC.15; v56; v113; v155; v158

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  • v160_seam_gaussianity_from_pf.pyRegistered verification module · registry
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    seam Gaussianity as a CONDITIONAL fixed-point theorem (the proposed closure of premise A, honestly typed): (1) quasi-free ⇒ all higher connected cumulants κ_{2n}=0 (signed set-partition/Pfaffian inversion, exact on a pure Majorana covariance); (2) Wick functoriality ⇒ a kernel-preserving transport fixes the whole quasi-free state; (3) the Perron-Frobenius dichotomy (a fixed κ_{2n} = a SECOND fixed ray, kills uniqueness; a non-fixed κ_{2n} decays by the gap) forces Gaussianity; (4) the qualitative claim needs only gap>0, the value (2/3)^6 only sets the rate. THE LOAD [P]: v56's gapped PF uniqueness lives on the 3-dim scalar transfer spectrum {1,(2/3)^6,(1/3)^6}, while the seam correlation cone has C(16,4)=1820, C(16,6)=8008 cumulant directions ⇒ (A) reduces to the single internal statement 'the admissible TFPT transport is primitive + spectrally gapped on the FULL Schwinger cone, not only the readout spectrum'.

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