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Original ledger classification: Identity [I] (twisted average + resonance theorem + uniqueness) + Numerical [N] (sharpness control) + Physical [P] (scope)

Recorded statement: RESONANCE THEOREM (the GATE.UWALL.07 [N] findings promoted to [I] by a LINEAR computation - the anticipated Groebner step turned out unnecessary): (1) TWISTED-AVERAGE LEMMA [I, symbolic generic A0]: at infinity the equivariant Fuchsian…

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GATE.UWALL.08

Original ledger classification: Identity [I] (twisted average + resonance theorem + uniqueness) + Numerical [N] (sharpness control) + Physical [P] (scope)

Original claim text

RESONANCE THEOREM (the GATE.UWALL.07 [N] findings promoted to [I] by a LINEAR computation - the anticipated Groebner step turned out unnecessary): (1) TWISTED-AVERAGE LEMMA [I, symbolic generic A0]: at infinity the equivariant Fuchsian system reads dY/dw = -(1/w)(B0 + B1 w + ...)Y with B_m = sum_k i^{km} U^k A0 U^{-k}; the twisted mu_4 averages collapse to B0 = 4 diag(A0) and B1 = 4 a12 E_12 + 4 conj(a13) E_31 - only the eigenvalue-ratio -i cells survive. (2) RESONANCE THEOREM [I]: the anchor exponents diag(2,1,1) have resonance gap 1; the level-1 formal-gauge equation (1 - ad R0)H1 = R1 is singular exactly on the cells {(2,1),(3,1)}, where the obstruction entries of B1 are (0, 4 conj(a13)); all levels k >= 2 are non-resonant and the formal monodromy e^{-2 pi i diag(2,1,1)} = 1, hence M_inf = 1 <=> a13 = 0 - the transcendental Riemann-Hilbert condition collapses to ONE linear equation. (3) UNIQUENESS COROLLARY [I]: a13 = 0 + the (8,0,5)/144 moduli (GATE.UWALL.07) + phases = diagonal gauge (commutes with U, fixes the diagonal, rotates arg a12 and arg a23 freely) => the flat anchor locus is EXACTLY ONE GAUGE ORBIT = the exact A0*; within the mu_4-equivariant anchor class the U_wall bundle datum is ALGEBRAICALLY unique. (4) FALSIFICATION CONTROL [N]: ||M_inf - 1|| = 1.6e-10 at a13 = 0 vs 0.17 / 0.43 at a13 = 0.02 / 0.05 - the obstruction is sharp (~8.5 |a13|). (5) HONEST SCOPE [P] (recorded): the equivariant ansatz A_k = U^k A0 U^{-k} is the established D4 selector input (v19/v30); the residue side is now [I]; the holonomy values (harmonic diagonal (0, 1/2, 1/2)) remain [N] - transcendental monodromy of an exact system

Original classification detail

Theorem: M_inf = 1 <=> a13 = 0 (resonance obstruction); flat anchor locus = one gauge orbit = exact A0*; U_wall bundle datum algebraically unique

Dependencies as recorded

GATE.UWALL.07; GATE.UWALL.06; v19; v30; CONTRACT.U.01

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Registered verification modules

  • v116_resonance_uniqueness.pyRegistered verification module · registry
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    RESONANCE THEOREM (the v115 [N] findings promoted to [I] by a LINEAR computation — the anticipated Gröbner step was unnecessary): TWISTED-AVERAGE LEMMA [I] — at infinity the equivariant system reads dY/dw = −(1/w)(B₀ + B₁w + …)Y with B_m = Σₖ i^{km} Uᵏ A₀ U⁻ᵏ; the twisted μ₄ averages collapse to B₀ = 4 diag(A₀) and B₁ = 4a₁₂E₁₂ + 4ā₁₃E₃₁ (only the eigenvalue-ratio −i cells survive); RESONANCE THEOREM [I]: the anchor exponents diag(2,1,1) have resonance gap 1, the level-1 gauge equation is singular exactly on cells {(2,1),(3,1)} where B₁ = (0, 4ā₁₃), higher levels non-resonant, formal monodromy = 1 ⇒ M∞ = 1 ⟺ a₁₃ = 0 — the transcendental Riemann–Hilbert condition collapses to ONE linear equation; UNIQUENESS COROLLARY [I]: a₁₃ = 0 + the (8,0,5)/144 moduli + diagonal gauge ⇒ THE FLAT ANCHOR LOCUS IS EXACTLY ONE GAUGE ORBIT = the exact A₀* — within the μ₄-equivariant anchor class the U_wall bundle datum is ALGEBRAICALLY unique; FALSIFICATION CONTROL [N]: ‖M∞ − 1‖ ≈ 8.5|a₁₃| (sharp), 1.6e−10 at A₀*; honest scope: residue side [I], holonomy values (harmonic diagonal) remain [N]

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