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A registered result in its source context

Original ledger classification: Identity [I] (cohomology basis + characters + residues + spectrum match) + Physical [P] (naturality residue)

Recorded statement: THE Q+ GRADING IS COHOMOLOGY (the R5/GATE.QGEO dictionary at the spectrum level; external review 2026-06-12, validated and sharpened): (1) BASIS + CHARACTERS [I]: b_1(P^1 minus mu_4) = 3 = N_fam with holomorphic basis omega_k =…

Statement, assumptions and sources

Paper context: Standard Model and flavor structure

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FLAV.QGEO.04

Original ledger classification: Identity [I] (cohomology basis + characters + residues + spectrum match) + Physical [P] (naturality residue)

Original claim text

THE Q+ GRADING IS COHOMOLOGY (the R5/GATE.QGEO dictionary at the spectrum level; external review 2026-06-12, validated and sharpened): (1) BASIS + CHARACTERS [I]: b_1(P^1 minus mu_4) = 3 = N_fam with holomorphic basis omega_k = z^(k-1)dz/(z^4-1), k = 1,2,3 (regular at infinity; the trivial character is absent - k = 4 would be the dlog form); the mu_4 character of omega_k is i^k, so H^1 = chi_1 + chi_2 + chi_3 with each nontrivial character exactly once; the residue vector of omega_k over the punctures is the character vector zeta^k/4 exactly. (2) THE GRADING IS Spec(Q+) [I]: the character degrees (1,2,3) are exactly Spec(Q+) (Q+ = 3*diag(cusp weights)+1, FLAV.QGEO.01); NEW: the cusp weights {0,1/3,2/3} are spec(A0*) (GATE.UWALL.07) - Mehta-Seshadri weights, cusp classes, cohomology characters and the flavor operator Q+ are ONE spectrum in four readouts. (3) A3 EXPONENTS [I]: (1,2,3) are the exponents of A3 (W(A3) = S4, GATE.UWALL.09) with Coxeter number h(A3) = 4 = |mu_4| - the Q+ generation grading IS the A3 exponent grading realised on the seam curve's cohomology: the R5 dictionary statement at spectrum level. (4) RESIDUE [P] (recorded): GATE.QGEO's remainder is the NATURALITY of the identification - the canonical map from the H^1 character basis to the V-axis generation basis (rowsum(V) = (1,2,3) verbatim, FLAV.SURF.01); spectrum equality is [I], the canonical map is the residue

Original classification detail

Identity: H^1 = chi_1+chi_2+chi_3 with degrees (1 2 3) = Spec(Q+) = A3 exponents; cusp weights = spec(A0*); residue = naturality of the canonical map

Dependencies as recorded

GATE.QGEO.01; FLAV.QGEO.01-03; GATE.UWALL.07; GATE.UWALL.09; FLAV.SURF.01

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Registered verification modules

  • v137_qplus_cohomology.pyRegistered verification module · registry
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    THE Q₊ GRADING IS COHOMOLOGY (the R5/GATE.QGEO dictionary at the spectrum level): BASIS + CHARACTERS [I]: b₁(ℙ¹∖μ₄) = 3 = N_fam with holomorphic basis ω_k = z^(k−1)dz/(z⁴−1), k = 1,2,3 (regular at infinity; the trivial character is absent); the μ₄ character of ω_k is i^k, so H¹ = χ₁⊕χ₂⊕χ₃ with each nontrivial character exactly once, and the residue vector over the punctures is the character vector ζ^k/4 exactly; THE GRADING IS Spec(Q₊) [I]: the character degrees (1,2,3) are exactly Spec(Q₊) (Q₊ = 3·cusp + 1, v69/v72), and NEW: the cusp weights {0, 1/3, 2/3} are spec(A₀*) (v115) — Mehta–Seshadri weights, cusp classes, cohomology characters and the flavor operator Q₊ are ONE spectrum in four readouts; A₃ EXPONENTS [I]: (1,2,3) are the exponents of A₃ (W(A₃) = S₄, v117) with Coxeter number h(A₃) = 4 = |μ₄| — the Q₊ generation grading IS the A₃ exponent grading, realised on the seam curve's cohomology; RESIDUE [P]: what remains of GATE.QGEO is the NATURALITY of the identification — the canonical map from the H¹ character basis to the V-axis generation basis (rowsum(V) = (1,2,3) verbatim, v135); spectrum equality is [I], the canonical map is the residue

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