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Original ledger classification: Strengthening [E] (Hessian PD over full Z4 space) + [O] (pin the spectrum)

Recorded statement: SPECTRAL-RIGIDITY ROUTE FOR FLAT-AWAY (2026-06-18): upgrades the v291 one-direction scan to the FULL smooth-Z4 deformation space, showing the flat seam is a STRICT isolated minimum of the spectral-mismatch functional S(f)=sum_n…

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FLATAWAY.SPEC.01

Original ledger classification: Strengthening [E] (Hessian PD over full Z4 space) + [O] (pin the spectrum)

Original claim text

SPECTRAL-RIGIDITY ROUTE FOR FLAT-AWAY (2026-06-18): upgrades the v291 one-direction scan to the FULL smooth-Z4 deformation space, showing the flat seam is a STRICT isolated minimum of the spectral-mismatch functional S(f)=sum_n (lambda_n(f)-lambda_n(0))^2. [E] SPLITTING MECHANISM: the flat Steklov spectrum |n| is doubly degenerate (+-n); a Z4 mode 4k connects n<->-n exactly at |n|=2k and splits that level at first order by +-eps/2 (modes 4,8,12 split levels 2,4,6 by eps), distinct modes splitting distinct levels. [E] QUADRATIC FORM: S(eps cos4theta) = h eps^2 + O(eps^4), h~0.50. [E] HESSIAN POSITIVE-DEFINITE: H_{kl}=d^2 S/dg_k dg_l over modes {4,8,12} is near-diagonal with eigenvalues ~{0.497,0.500,0.503}, all >0 -- the flat seam is a STRICT isolated minimum over the WHOLE smooth-Z4 space (answers the 'only one direction' objection). [E] CLOCK-FILTER CONTRAST: a non-Z4 mode (cos2theta) breaks the clock commutator outright (||[rho,M]||>0). [O] CONSEQUENCE: matching the seam's forced Steklov spectrum excludes EVERY smooth Z4 deformation => Flat-Away, the same external fact as v292 reached independently.

Original classification detail

[E] the splitting mechanism + the quadratic form + the positive-definite Hessian over the full Z4 space + the clock-filter contrast; [O] the one external fact (pin the seam's Steklov spectrum). A strengthening of v291 to the full deformation space; NOT a proof.

Dependencies as recorded

FLATAWAY.RP.01; FLATAWAY.HEAT.01; SEAM.ADVERSARY.01; SEAM.EQUIV.B01

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  • v293_flataway_spectral_hessian.pyRegistered verification module · frontier
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    FLATAWAY.SPEC.01 — the Spectral-Rigidity route upgraded from one direction (v291) to the FULL smooth-Z4 space. [E] splitting mechanism: the degenerate flat Steklov level |n| is split by mode 4k at |n|=2k by ±ε/2 (modes 4,8,12 split levels 2,4,6); [E] S=Σ(Δλ)² ~ 0.50ε² (quadratic); [E] the Hessian over modes {4,8,12} has eigenvalues ~{0.497,0.500,0.503}>0, so the flat seam is a STRICT isolated minimum over the whole Z4 space (answers the 'only one direction' objection); [E] clock-filter contrast: a non-Z4 mode (cos2θ) breaks the commutator outright. [O] consequence: pinning the seam's forced Steklov spectrum excludes every smooth Z4 deformation ⇒ Flat-Away — the same external fact as v292, reached independently.

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