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A registered result in its source context

Original ledger classification: Identity [I] (mu4-average + (8 0 5)/144 + exact normal form) + Numerical [N] (RH verification + uniqueness scan) + Physical [P] (residue)

Recorded statement: THE ANCHOR PINS THE RESIDUE (exact normal form for the U_wall flat-bundle residue; answers the GATE.UWALL.06 branch question): (1) MU4-AVERAGE LEMMA [I, symbolic generic X] - sum_k U^k X U^{-k} = 4 diag(X): mu_4 conjugation-averaging is…

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Paper context: Research contracts

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GATE.UWALL.07

Original ledger classification: Identity [I] (mu4-average + (8 0 5)/144 + exact normal form) + Numerical [N] (RH verification + uniqueness scan) + Physical [P] (residue)

Original claim text

THE ANCHOR PINS THE RESIDUE (exact normal form for the U_wall flat-bundle residue; answers the GATE.UWALL.06 branch question): (1) MU4-AVERAGE LEMMA [I, symbolic generic X] - sum_k U^k X U^{-k} = 4 diag(X): mu_4 conjugation-averaging is the diagonal projection, so the exponents at infinity are 4 diag(A0) and the anchor splitting O(-2)+O(-1)^2 holds IFF diag A0 = (2,1,1)/4 = a/|mu_4| - the residue diagonal IS the anchor over mu_4 (the v33 diagonal was forced, not chosen). (2) THE (8,0,5)/144 LEMMA [I]: anchor diagonal + cusp spectrum {0,1,2}/N_fam pin the off-diagonal weight sum to 13/144; with a13 = 0 the det = 0 condition solves UNIQUELY to (|a12|^2, |a23|^2) = (8, 5)/144 - numerators (rank E8, g_car), denominator (|mu_4| N_fam)^2 = 144, total 13 = Delta_Q = 2p_2(a)+1 (the quark denominator 117 = 9x13). (3) EXACT RESIDUE NORMAL FORM [I]: A0* = [[1/2, sqrt2/6, 0],[sqrt2/6, 1/4, sqrt5/12],[0, sqrt5/12, 1/4]] has char poly lam(lam-1/3)(lam-2/3) exactly. (4) A0* IS THE RIEMANN-HILBERT SOLUTION [N]: big-circle monodromy trivial to 1.6e-10, prod M_k = 1, unitarised |diag M~_0| = (0, 1/2, 1/2), tr(M~_0 U) = 1 - the v33/v40 numerical point is gauge-equivalent to this exact matrix (a12/a23 phases = diagonal gauge). (5) THE ANCHOR FORCES THE d1 = 0 SLICE [N]: multi-seed RH scan over the anchor-pinned spectral manifold lands at every flat solution on the SAME gauge orbit with |diag M~_0| = (0, 1/2, 1/2) - combined with GATE.UWALL.06 the whole harmonic diagonal is anchor + torsion forced; delta = 1/2 needs no selection. (6) RESIDUE [P]: a13 = 0 + gauge-orbit uniqueness numerical (one component across seeds); promotion to [I]/[L] via Groebner on the RH constraints = the remaining U_wall point-selection step

Original classification detail

Exact residue normal form A0* (diag = anchor/mu4 spectrum = cusp weights off-weights = (8 0 5)/144); anchor forces d1 = 0; residue = Groebner promotion

Dependencies as recorded

GATE.UWALL.06; GATE.UWALL.04; v33; v40; CONTRACT.U.01

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Registered verification modules

  • v115_anchor_residue.pyRegistered verification module · registry
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    THE ANCHOR PINS THE RESIDUE (exact normal form for the U_wall flat-bundle residue; answers the v114 branch question): μ₄-AVERAGE LEMMA [I] — Σₖ Uᵏ X U⁻ᵏ = 4 diag(X): μ₄ conjugation-averaging IS the diagonal projection ⇒ exponents at infinity = 4 diag(A₀) ⇒ anchor splitting 𝒪(−2)⊕𝒪(−1)² ⟺ diag A₀ = (2,1,1)/4 = ANCHOR/|μ₄| (v33's diagonal was forced, not chosen); THE (8,0,5)/144 LEMMA [I]: anchor diagonal + cusp spectrum pin Σ|off|² = 13/144, and a₁₃ = 0 + det = 0 solve UNIQUELY to (|a₁₂|², |a₂₃|²) = (8,5)/144 — numerators (rank E8, g_car), denominator (|μ₄|N_fam)², total 13 = Δ_Q (the quark denominator 117 = 9·13); EXACT NORMAL FORM [I]: A₀* = [[1/2, √2/6, 0],[√2/6, 1/4, √5/12],[0, √5/12, 1/4]] with char poly λ(λ−⅓)(λ−⅔) exactly; A₀* IS THE RIEMANN–HILBERT SOLUTION [N]: big-circle monodromy trivial to 1e−10, unitarised |diag M̃₀| = (0,½,½) — the v33/v40 numerical point is gauge-equivalent to this exact matrix; ANCHOR FORCES d₁ = 0 [N]: every flat solution in a multi-seed RH scan lands on the same gauge orbit ⇒ the harmonic diagonal is anchor+torsion forced, δ = ½ needs no selection; residue [P]: Gröbner promotion of a₁₃ = 0 + uniqueness

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