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A registered result in its source context

Original ledger classification: Derived spectral gap [E] + OS/clustering [C]

Recorded statement: THE LAST LEVER -- THE SEAM GAP IS THE DERIVED RECOVERY GAP (2026-06-19): after v300 (Route B = Route A's rationality) and v301 (Route A's invertibility discharged), the WHOLE open content of SEAM.EQUIV.01 was the single spectral statement…

Statement, assumptions and sources

Paper context: tfpt research contracts

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SEAM.EQUIV.GAP.01

Original ledger classification: Derived spectral gap [E] + OS/clustering [C]

Original claim text

THE LAST LEVER -- THE SEAM GAP IS THE DERIVED RECOVERY GAP (2026-06-19): after v300 (Route B = Route A's rationality) and v301 (Route A's invertibility discharged), the WHOLE open content of SEAM.EQUIV.01 was the single spectral statement 'the quasi-free seam bulk is gapped'. This shows that gap is NOT a free input but the already-established RECOVERY GAP of the seam transfer operator, derived from the carrier {c3,g_car}. [E] TRANSFER SPECTRUM: spec(T) = {1, (2/3)^6, (1/3)^6} = {1, 64/729, 1/729} (v160/v162) -- a simple leading Perron eigenvalue 1 (the stationary/vacuum mode), the rest strictly <1. [E] THE GAP IS DERIVED: Delta = -ln((2/3)^6) = 6 ln(3/2) = 2 N_fam ln(3/2) ~ 2.4328 > 0 (multiplicative gap 1-(2/3)^6 = 665/729); 6 = 2 N_fam, (2/3)^6 the mu4-deck transfer eigenvalue -- a number FORCED by the carrier, not an assumption. [E] POSITIVITY MARGIN: Delta - 31/(4 pi^2) = 6 ln(3/2) - 31/(4 pi^2) ~ 1.648 > 0 (the v76 decoupling margin) -- the gap strictly beats the threshold, robust not marginal; Perron-Frobenius primitivity makes the leading mode simple. [C] OS/QUASI-FREE CLUSTERING: for a reflection-positive quasi-free system a transfer-operator gap Delta>0 <=> exponential clustering (correlation length 1/Delta) <=> a mass gap of the OS-reconstructed Hamiltonian (Osterwalder-Schrader transfer-matrix gap = mass gap; quasi-free decay) -- so the bulk mass gap = the transfer gap, no extra input. [O] VERDICT: 'the quasi-free seam bulk is gapped' is realised by the DERIVED Recovery gap Delta = 6 ln(3/2) > 0, so SEAM.EQUIV.01's residual is now a composition of STANDARD cited theorems (OS/quasi-free clustering + Kitaev free-fermion invertibility v301 + the v297 AQFT theorem stack) over ESTABLISHED TFPT facts (16 quasi-free Majoranas v148/v160; transfer gap 6 ln(3/2) v160/v162): NO undischarged TFPT-internal assumption remains. SEAM.EQUIV.01 stays [O] (not machine-proved end-to-end), but the open front is reduced to citable machinery over a spectral gap that is provably positive.

Original classification detail

[E] the frozen transfer spectrum + the derived Recovery gap 6 ln(3/2) + the positivity margin; [C] the OS/quasi-free 'transfer gap = mass gap' identification; [O] the verdict. Identifies the last input with an established carrier-forced quantity; reduces SEAM.EQUIV.01 to a composition of cited theorems over established TFPT facts; does NOT claim SEAM.EQUIV.01 is machine-proved.

Dependencies as recorded

SEAM.EQUIV.A.INV.01; GATE.UWALL.07; SYNTH.INVENTORY.02; SEAM.EQUIV.01

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Registered verification modules

  • v302_seam_gap.pyRegistered verification module · frontier
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    SEAM.EQUIV.GAP.01 — the LAST lever: after v300/v301 the whole open content of SEAM.EQUIV.01 was 'the quasi-free seam bulk is gapped'; this shows that gap is NOT a free input but the established RECOVERY GAP of the seam transfer operator. [E] TRANSFER SPECTRUM spec(T)={1,(2/3)^6,(1/3)^6}={1,64/729,1/729} (v160/v162), simple leading Perron eigenvalue 1, rest <1. [E] THE GAP IS DERIVED: Δ=−ln((2/3)^6)=6 ln(3/2)=2 N_fam ln(3/2)≈2.4328>0 (mult. gap 1−(2/3)^6=665/729); 6=2 N_fam, (2/3)^6 the μ4-deck transfer eigenvalue — forced by the carrier, not assumed. [E] POSITIVITY MARGIN Δ−31/(4π²)=1.6476>0 (the v76 decoupling margin), robust not marginal. [C] OS/QUASI-FREE CLUSTERING: for an RP quasi-free system a transfer gap Δ>0 ⟺ exponential clustering ⟺ a mass gap of the OS-reconstructed Hamiltonian (bulk mass gap = transfer gap). [O] VERDICT: the last TFPT-internal input is realised by the derived Recovery gap 6 ln(3/2)>0, so SEAM.EQUIV.01's residual is a composition of cited theorems (OS/clustering + Kitaev v301 + the v297 AQFT stack) over established TFPT facts — no undischarged TFPT-internal assumption remains; stays [O] (not machine-proved end-to-end).

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