A registered result in its source context
Original ledger classification: Red-team [E] (adversary passes commutator, shifts spectrum/heat) + [O] (consequence)
Recorded statement: ROUTE-B RED-TEAM: Z4 BLOCK-DIAGONALITY != MARK-LOCALITY (2026-06-18). Constructs the strongest adversary against Route B -- a SMOOTH Z4-symmetric off-mark curvature f(theta)=eps*cos(4 theta) -- and shows HONESTLY (no manufactured pass):…
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Paper context: Research contracts
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SEAM.ADVERSARY.01
Original ledger classification: Red-team [E] (adversary passes commutator, shifts spectrum/heat) + [O] (consequence)
Original claim text
ROUTE-B RED-TEAM: Z4 BLOCK-DIAGONALITY != MARK-LOCALITY (2026-06-18). Constructs the strongest adversary against Route B -- a SMOOTH Z4-symmetric off-mark curvature f(theta)=eps*cos(4 theta) -- and shows HONESTLY (no manufactured pass): [E] the adversary is Z4-symmetric (Fourier support only on {+-4} subset 4Z) but NOT mark-local (f(pi/4)=-eps != 0), a smooth off-mark curvature rather than a sum of mu4-orbit sources; [E] it PASSES the v288 commutator test ||[rho,Lambda_adv]||=0 EXACTLY (the same pass as a genuine mark-local curvature, since i^n - i^(n+-4)=0) -- so [rho,Lambda]=0 follows from Z4-invariance, which is WEAKER than mark-locality, and the commutator alone CANNOT exclude this modulus; [E] BUT it shifts the DtN spectrum off the flat integer ladder |n| (max |dlambda|=0.155 at eps=0.3) and the heat trace Tr(e^{-t Lambda}) (delta=0.019 at t=0.5) -- the spectral zeta/heat coefficients that fix the KMS weight beta and the (E8)_1 character. [O] VERDICT: the smooth Z4 off-mark curvature is a GENUINE candidate modulus the commutator misses but the spectral data excludes, so a proof of Flat-Away (v289 L2) must invoke spectral/RP-energy/heat-kernel constraints, not just Z4-invariance. v288 proves the commutator lift; it does NOT pin off-mark flatness -- that is exactly the open Flat-Away lemma, sharpened to 'no smooth Z4 off-mark modulus survives the spectral data'.
Original classification detail
[E] the adversary construction + the commutator pass + the spectrum/heat shift (the discriminators); [O] the consequence that Flat-Away must use spectral input. An honest red-team test; it does NOT manufacture a pass and sharpens (not closes) the open lemma.
Dependencies as recorded
SEAM.EQUIV.B01; MARKLOCAL.RAW.01; FLATAWAY.RP.01; QGEO.SUBPRIN.01
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- Research contracts
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- 2026-06-18 (Route-B red-team: the smooth [formula in source] adversary + Flat-Away as a mini-theorem, v290– v291)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
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- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportResearch contracts
- Research Contract 1 — [formula in source]tfpt research contracts
Registered verification modules
- v290_z4_smooth_curvature_adversary.pyRegistered verification module · frontier
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SEAM.ADVERSARY.01 — the Route-B red-team that Z4 block-diagonality is NOT mark-locality. Constructs the strongest adversary, a SMOOTH Z4-symmetric off-mark curvature f=ε·cos(4θ), and shows honestly: [E] it is Z4-symmetric (Fourier support {±4}⊂4ℤ) but NOT mark-local (f(π/4)=−ε≠0); [E] it PASSES the v288 commutator ‖[ρ,Λ]‖=0 (same as mark-local; i^n−i^(n±4)=0) — so [ρ,Λ]=0 is necessary but NOT sufficient; [E] BUT it shifts the DtN spectrum off the flat ladder (max|Δλ|=0.155 at ε=0.3) and the heat trace Tr(e^{−tΛ}) (Δ=0.019) — the KMS weight + (E8)₁ character shift; [O] verdict: a genuine candidate modulus the commutator misses but the spectral data excludes — Flat-Away must use spectral/energy/heat input. An honest red-team, no manufactured pass.
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