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Original ledger classification: Formal

Recorded statement: LEAN: FLAT-AWAY HEAT POSITIVE-DEFINITENESS (2026-06-18). Lean 4 formalisation (TfptCarrier/SeamDeckClosure.lean Part 5) of the algebraic core of the v292/v295 heat route: the heat-trace deviation as a quadratic form heatDeviation s W g =…

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FORM.FLATAWAY.01

Original ledger classification: Formal

Original claim text

LEAN: FLAT-AWAY HEAT POSITIVE-DEFINITENESS (2026-06-18). Lean 4 formalisation (TfptCarrier/SeamDeckClosure.lean Part 5) of the algebraic core of the v292/v295 heat route: the heat-trace deviation as a quadratic form heatDeviation s W g = sum_{k in s} W k * (g k)^2 in the smooth Z4 curvature modes. THEOREMS: heatDeviation_nonneg (non-negative for non-negative weights) and heatDeviation_eq_zero_iff (with STRICTLY POSITIVE weights W_k=c_k(t), the deviation vanishes IFF every curvature mode vanishes) -- i.e. matching the flat (E8)_1 heat data forces the off-mark curvature to zero (flat away from the four mu4 marks). The positivity of the weights is the v295 analytic input (convexity); this is the formal 'Delta=0 <=> flat' step. Built (lake build TfptCarrier.SeamDeckClosure, 10s) and axiom-checked: depends ONLY on propext, Classical.choice, Quot.sound (no sorry, no native_decide).

Original classification detail

[E] Lean-proven: positive-weighted sum of squares is positive-definite (heatDeviation_eq_zero_iff), the formal Delta=0 <=> flat core of the Flat-Away heat route. Built + axiom-clean. Does NOT close the open input (positivity of the weights is consumed from v295).

Dependencies as recorded

FLATAWAY.A2.01; FLATAWAY.HEAT.01; FORM.QGEO.04

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  • v295_flataway_a2_exact.pyRegistered verification module · frontier
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    FLATAWAY.A2.01 — the EXACT analytic a₂ coefficient, upgrading the v292 positive-definiteness from numerical to a proof. [E] first variation vanishes: d/dε Tr e^{−tΛ(ε)}|₀ = −t f̂(0) Σe^{−t|n|} = 0 (smooth off-mark curvature has zero mean, Gauss-Bonnet); [E] convexity ⇒ global minimum: φ(x)=e^{−tx} strictly convex ⇒ Λ↦Tr e^{−tΛ} convex (Klein/Peierls), so ε=0 is the GLOBAL minimum and ΔTr≥0 for ALL ε (verified even in ε over [−1,2]), =0 iff f=0; [E] exact a₂ kernel ΔTr⁽²⁾=(1/4)Σ_n[Φ(|n|,|n+4k|)+Φ(|n|,|n−4k|)]g² with the divided-difference kernel of e^{−tx} (Duhamel), matched to numerics to rel.err<1e−6; [E] degenerate-split closed form 2e^{−2kt}(cosh(tg/2)−1)≥0; [O] reduction now analytic: Flat-Away ⟸ 'the seam fixes a₂'. Lean-mirrored (FORM.FLATAWAY.01).

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