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Original ledger classification: Reduction [E] (positive-definite heat form) + [O] (one external invariant a_2)

Recorded statement: HEAT-KERNEL ROUTE FOR FLAT-AWAY (2026-06-18): reduces the open Flat-Away lemma to ONE named spectral invariant by showing the heat-trace deviation is a positive-definite quadratic form in the off-mark curvature. Setup Lambda=|D_theta|+M_f,…

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FLATAWAY.HEAT.01

Original ledger classification: Reduction [E] (positive-definite heat form) + [O] (one external invariant a_2)

Original claim text

HEAT-KERNEL ROUTE FOR FLAT-AWAY (2026-06-18): reduces the open Flat-Away lemma to ONE named spectral invariant by showing the heat-trace deviation is a positive-definite quadratic form in the off-mark curvature. Setup Lambda=|D_theta|+M_f, f=sum_k g_{4k} cos(4k theta). [E] QUADRATIC LEADING ORDER: Delta Tr(e^{-t Lambda}) = c(t) eps^2 + O(eps^4) (Delta Tr/eps^2 constant ~0.208 at t=0.5), no first-order term (average curvature fixed by Gauss-Bonnet/marks). [E] POSITIVE-DEFINITE PER MODE: c_k(t)>0 for every Z4 mode 4k (k=1,2,3: ~0.066,0.047,0.033). [E] POSITIVE-DEFINITE ON COMBINATIONS: random Z4 combinations give Delta Tr>0, no cancellation, so Delta Tr=0 <=> f=0. [E] LEADING INVARIANT = ||f||_L^2 (Parseval). [O] REDUCTION: Flat-Away <= 'the seam's spectral/heat data fixes the a_2 heat coefficient (=Tr e^{-t Lambda}) to its flat value', then int f^2=0 => f=0 off the marks -- ONE external invariant, far smaller than 'prove Flat-Away'.

Original classification detail

[E] the quadratic, positive-definite structure of the heat-trace deviation over the full smooth-Z4 space + the L^2 identification; [O] the one external fact (the seam fixes a_2). A genuine reduction of Flat-Away to one spectral invariant; NOT a proof.

Dependencies as recorded

FLATAWAY.RP.01; SEAM.ADVERSARY.01; SEAM.EQUIV.B01; MARKLOCAL.RAW.01

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  • v292_flataway_heat_reduction.pyRegistered verification module · frontier
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    FLATAWAY.HEAT.01 — the Heat-Kernel route made precise: reduces Flat-Away to ONE named spectral invariant. [E] the heat-trace deviation ΔTr(e^{−tΛ}) = c(t)ε²+O(ε⁴) is a positive-definite quadratic form in the off-mark curvature (c_k(t)>0 for every Z4 mode 4k: 0.066/0.047/0.033; positive on all combinations, no cancellation), so ΔTr=0 ⟺ f=0; leading invariant ‖f‖_L² (Parseval). [O] reduction: Flat-Away ⟸ 'the seam fixes the a₂ heat coefficient (=Tr e^{−tΛ}) to its flat value' ⇒ ∫f²=0 ⇒ f=0 off the marks. One external invariant, far smaller than 'prove Flat-Away'.

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