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Original ledger classification: Identity [I] (moment/spectral/Sugawara exact) + [C] (declared V_metric norm)

Recorded statement: ATOMIC OS MOMENT LEMMA + SUGAWARA GAP SAFETY (the admissible transfer sector's OS positivity as a positive moment problem, and its gap as an E8 Sugawara inequality): (1) spec(T)={1,(2/3)^6,(1/3)^6}={1,64/729,1/729} makes every scalar…

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QFT.OSMOMENT.01

Original ledger classification: Identity [I] (moment/spectral/Sugawara exact) + [C] (declared V_metric norm)

Original claim text

ATOMIC OS MOMENT LEMMA + SUGAWARA GAP SAFETY (the admissible transfer sector's OS positivity as a positive moment problem, and its gap as an E8 Sugawara inequality): (1) spec(T)={1,(2/3)^6,(1/3)^6}={1,64/729,1/729} makes every scalar two-point correlator C(n)=A+B r^n+C s^n the moment sequence of the positive atomic measure A d1+B dr+C ds, so every OS Hankel matrix factorises as a Vandermonde Gram H=W diag(A,B,C) W^T => positive semidefinite (positive spectrum => moment measure => reflection positivity; a clean proof, not a numerical Gram test) [I]; (2) H=-log T gives a positive discrete Kallen-Lehmann spectrum {0, 6log(3/2), 6log3} with masses m1=2.43279, m2=6.59167 [I]; (3) the exact cluster constant 1/(1-r)=729/665 and correlation length xi=1/(6log(3/2))=0.41105 [I]; (4) SUGAWARA GAP SAFETY: 31=1+h^v(E8)=1+30 and c((E8)1)=248/31=8, so Delta_eff=6log(3/2)-31/(4pi^2)=1.64755>0 reads as |R+(A3)|log(N_fam/|Z2|) > (1+h^v(E8))/(4pi^2) -- the A3 transfer gap beats the E8 Sugawara metric budget

Original classification detail

Identity [I]: OS positivity = positive moment problem; Sugawara gap safety 31=1+h^v(E8); cluster 729/665, xi=0.411

Dependencies as recorded

GATE.METRIC.XI.01; SEAM.CYCLE.01; GATE.METRIC.07

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  • v171_os_moment_cluster.pyRegistered verification module · registry
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    Atomic OS Moment Lemma + Sugawara Gap Safety: spec(T)={1,64/729,1/729} makes C(n)=A+Br^n+Cs^n a positive moment sequence, so every OS Hankel matrix factorises as a Vandermonde Gram ⇒ reflection positivity (a clean proof, not a numeric Gram test); cluster constant 729/665, correlation length ξ=0.41105, Källén-Lehmann masses 6log(3/2),6log3; gap safety Δ_eff=6log(3/2)−31/(4π²)>0 with 31=1+h^∨(E8) and c((E8)1)=248/31=8 — the A3 transfer gap beats the E8 Sugawara metric budget.

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