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Original ledger classification: Reduction [E] (full-L^2 lift, lifts v201/v284) + [O] (mark-locality residual)

Recorded statement: ROUTE B (DtN) FULL-L^2 SUB-PRINCIPAL Z4 BLOCK-DIAGONALITY (2026-06-18): proves (numerically, on the full Toeplitz operator) the advisor's 'probably provable' step, lifting v201/v284 from the finite H^1 block to the full L^2 operator.…

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SEAM.EQUIV.B01

Original ledger classification: Reduction [E] (full-L^2 lift, lifts v201/v284) + [O] (mark-locality residual)

Original claim text

ROUTE B (DtN) FULL-L^2 SUB-PRINCIPAL Z4 BLOCK-DIAGONALITY (2026-06-18): proves (numerically, on the full Toeplitz operator) the advisor's 'probably provable' step, lifting v201/v284 from the finite H^1 block to the full L^2 operator. Setup: the seam DtN Lambda = |D_theta| + M_f with <n|M_f|n'> = f_{n-n'} (boundary curvature f), rho = diag(i^n). [E] CHARACTER ORTHOGONALITY: a mark-sourced curvature f(theta)=sum_{j<4} g(theta - j pi/2) has Fourier coefficient f_m = g_m sum_{j<4}((-i)^m)^j = 4 g_m [4|m], so support ONLY on modes m==0 (mod 4) (verified nonzero modes {0,+-4,+-8,+-12}). [E] FULL-L^2 BLOCK-DIAGONALITY: on the full mode range -50..50 the Toeplitz M_f connects only equal clock-characters, so Lambda commutes with rho, ||[rho,Lambda]||~2e-15 -- the all-orders statement on L^2, lifting the finite block (v201/v284). [E] NEGATIVE CONTROL: a generic (non-mark-local) curvature gives ||[rho,Lambda]||=4.44>0. [E] LEAN CONSISTENCY: the operator form of SeamDeckClosure.geom_sum_fourth_root + markLocal_blockDiagonal. [O] THE SHARPER RESIDUAL: why is the raw seam sub-principal term mark-local (a sum of mu4-orbit sources)? -- the curvature-locality half of SEAM.EQUIV.01.

Original classification detail

[E] character orthogonality + the full-L^2 Z4 block-diagonality (lifts v201/v284 to the full operator) + the negative control + Lean consistency; [O] the one sharper residual (raw sub-principal mark-locality). Real progress on Route B; does NOT close the premise.

Dependencies as recorded

SEAM.EQUIV.01; QGEO.SUBPRIN.01; QGEO.ROUTEI.01; FORM.QGEO.04; QGEO.MODULAR.01

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  • v288_full_l2_subprincipal_z4.pyRegistered verification module · frontier
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    SEAM.EQUIV.B01 — Route B (DtN): the full-L² lift of the sub-principal Z4 block-diagonality, proving (numerically, on the full Toeplitz operator) the 'probably provable' step and lifting v201/v284 from the finite block to L². [E] character orthogonality: a mark-sourced curvature f=Σ_{j<4} g(θ−jπ/2) has Fourier support only on m≡0 mod 4 (nonzero modes {0,±4,±8,±12}; the 4th-root mark sum Σ_{j<4}((−i)^m)^j=4[4|m]); [E] full-L² block-diagonality: Λ=|n|+M_f commutes with ρ=diag(i^n), ‖[ρ,Λ]‖~2e-15 on −50..50; [E] negative control: a generic curvature gives ‖[ρ,Λ]‖=4.44>0; [E] Lean consistency (geom_sum_fourth_root + markLocal_blockDiagonal); [O] the sharper residual — why is the raw seam sub-principal term mark-local? (the curvature-locality half of SEAM.EQUIV.01).

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