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Original ledger classification: Identity/Lattice [I]/[L] (duality, projector, orbits, fusion) + [P] (the one conformal-net step)

Recorded statement: GRADED FROBENIUS (R2 identification at the finite Lie level): the Z4-graded hull (E8.GRADE.01/v128) carries (1) EXACT COSET DUALITY coset(-r) = -coset(r) mod 4 for all 240 roots (-C1=C3 bijectively, C0/C2 self-dual; sizes 52,64,60,64) =>…

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GATE.METRIC.09

Original ledger classification: Identity/Lattice [I]/[L] (duality, projector, orbits, fusion) + [P] (the one conformal-net step)

Original claim text

GRADED FROBENIUS (R2 identification at the finite Lie level): the Z4-graded hull (E8.GRADE.01/v128) carries (1) EXACT COSET DUALITY coset(-r) = -coset(r) mod 4 for all 240 roots (-C1=C3 bijectively, C0/C2 self-dual; sizes 52,64,60,64) => the Killing form restricts to a NONDEGENERATE pairing g_k x g_{-k} - a Frobenius grading; (2) the glue character average (1/4)sum_k i^{k deg} is EXACTLY the carrier projector (1 on C0, 0 on C1,C2,C3) - the conditional-expectation shadow of the v125 Q-system with quasi-basis index |Z4|=4; (3) graded bracket additivity re-verified self-contained (6720 pairs); (4) each glue sector is ONE W(D5)xW(A3) orbit with multiplicity-one dimensions 64=16*4=(16,4b), 60=10*6=(10,6) - SIMPLE CURRENTS, sector ring = Z4 = the C[Z4] fusion, halfway subgroup {0,2} = SO(16)_1 (v92). CONSEQUENCE: the C[Z4] Q-system (v125) is realised by the 248-dim hull as its graded Frobenius module - R2's identification HOLDS AT THE FINITE LIE LEVEL. RESIDUE [P]: exactly one conformal-net statement remains ('the seam-Calderon inclusion of the 16-Majorana net carries this Q-system'), with no remaining finite freedom

Original classification detail

Identity/Lattice [I]/[L]: Q-system realised at the finite Lie level; residual = the net statement

Dependencies as recorded

E8.GRADE.01; GATE.METRIC.08; GATE.METRIC.07; CAR.QFREE.01; v125; v128

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Registered verification modules

  • v143_graded_frobenius.pyRegistered verification module · registry
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    graded Frobenius (R2): the ℤ₄-graded hull carries exact coset duality (−C₁=C₃), the glue average is the carrier projector (index 4), each sector is one Weyl orbit (simple current) — the ℂ[ℤ₄] Q-system realised at the finite Lie level; residual = the conformal-net step

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