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Original ledger classification: Identity [I] (gap forced by mu4-equivariance + spec(A0*); fixed = rank-8 Calderon) + [P]/[A] (A) factors into the already-open A2 (net existence) + R1 (seam clock)

Recorded statement: THE FINAL IDENTIFICATION, FORCED AS FAR AS IT GOES (the v161/GATE.METRIC.17 one-particle statement carries NO new open content; premise (A) factors into the two already-open items A2 + R1): the residual was 'the seam transport one-particle…

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GATE.METRIC.18

Original ledger classification: Identity [I] (gap forced by mu4-equivariance + spec(A0*); fixed = rank-8 Calderon) + [P]/[A] (A) factors into the already-open A2 (net existence) + R1 (seam clock)

Original claim text

THE FINAL IDENTIFICATION, FORCED AS FAR AS IT GOES (the v161/GATE.METRIC.17 one-particle statement carries NO new open content; premise (A) factors into the two already-open items A2 + R1): the residual was 'the seam transport one-particle symbol is the gapped Calderon Bogoliubov map - (alpha) fixed = rank-8 K_Sigma, (beta) gap = (2/3)^6'. (1) THE GAP VALUE IS FORCED [I] - the cusp weights = spec(A0*) = {0,1/3,2/3} (the parabolic residue, forced by anchor+torsion, v114-v117), the transfer eigenvalue is (1-alpha)^{p2} with p2 = 6 = |R+(A3)| (v124/v126), giving {1,(2/3)^6,(1/3)^6} and gap 6 log(3/2) - every entry carrier data, no free knob; (2) MU4-EQUIVARIANCE FORCES DIAGONAL-IN-CUSP [I] - sum_{k=0}^{3} U^k X U^{-k} = |mu4| diag(X) for U = diag(1,i,-i) and the commutant of U is exactly the diagonal algebra, so a mu4-equivariant gapped generator is diagonal with spectrum = the cusp weights (the gapped equivariant generator is unique, v116/v140); with (1) its gap is (2/3)^6; (3) CUSP GRADING = A3 GRADING [I] - degrees (1,2,3) = Spec(Q+) = A3 exponents, h(A3)=4=|mu4|, three levels = N_fam (v137), so the lift from the 3-dim generation space to the 16 Majoranas is the established A3 subnet grading; (4) FIXED = rank-8 CALDERON POLARIZATION [I] - DtN symbol |k|>=0 (v156/v157), polarization P=(1+iA)/2 rank 8 = rank E8 = c (v113), the IR fixed point of the gapped free-boundary RG is the conformal free-fermion vacuum with this covariance. CLOSURE [P]/[A]: with v161 (cone gap = one-particle gap) and v160 (unique quasi-free fixed point => kappa_{2n}=0), (A) CLOSES given only mu4-equivariance + admissibility of the seam RG generator, whose two non-derived inputs are (i) 'the seam RG generator is mu4-equivariant + admissible' = the R1 seam-clock identification (HOR.CLOCK.*; v105 relocation) and (ii) the operator-algebraic net existence = the A2 residual (Kawahigashi-Longo). BOTH already open => premise (A) carries NO independent open content; it factors exactly into A2 + R1 with all numbers forced, the count of independent open items does NOT increase, and (A) is closed CONDITIONALLY on A2 + R1 (a unification, NOT an unconditional proof)

Original classification detail

Identity [I]: gap (2/3)^6 forced + fixed = rank-8 K_Sigma; (A) factors into the already-open A2 + R1 (no new open content)

Dependencies as recorded

GATE.METRIC.16; GATE.METRIC.17; v56; v113; v115; v137; v156; v157

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Registered verification modules

  • v162_seam_transport_identification.pyRegistered verification module · registry
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    the final identification, forced as far as it goes — the v161 one-particle statement carries NO new open content: (1) the gap (2/3)^6 is FORCED — cusp weights = spec(A0*) = {0,1/3,2/3} (parabolic residue, v115), transfer eigenvalue (1-α)^{p2}, p2 = 6 = |R+(A3)|, all carrier data [E]; (2) μ4-equivariance forces the generator diagonal in the cusp basis — Σ_k U^k X U^{-k} = |μ4| diag(X), commutant of U = diagonal algebra — so the gapped μ4-equivariant generator has spectrum = cusp weights, gap (2/3)^6 [E]; (3) the cusp grading IS the A3 grading on the 16 Majoranas (v137) [E]; (4) the fixed space = rank-8 Calderón polarization (v113/v156/v157) [E]. CLOSURE [C/O]: with v160+v161, (A) closes given only μ4-equivariance + admissibility, whose two non-derived inputs are the ALREADY-open A2 (net existence) and R1 (seam clock, HOR.CLOCK) — so (A) factors exactly into A2 + R1, adds NO new open item, and is closed CONDITIONALLY on them (a unification, not an unconditional proof).

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