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Original ledger classification: Proof [E] (convexity positive-definiteness + exact kernel) + [O] (one external invariant)
Recorded statement: EXACT ANALYTIC a_2 COEFFICIENT FOR FLAT-AWAY (2026-06-18): upgrades the v292 positive-definiteness of the heat-trace deviation from a numerical scan to a proof, and derives the exact second-order coefficient in closed form. Setup…
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FLATAWAY.A2.01
Original ledger classification: Proof [E] (convexity positive-definiteness + exact kernel) + [O] (one external invariant)
Original claim text
EXACT ANALYTIC a_2 COEFFICIENT FOR FLAT-AWAY (2026-06-18): upgrades the v292 positive-definiteness of the heat-trace deviation from a numerical scan to a proof, and derives the exact second-order coefficient in closed form. Setup Lambda(eps)=|D_theta|+eps M_f, f=sum_k g_k cos(4k theta) (zero-mean, support 4Z). [E] FIRST VARIATION VANISHES: d/d eps Tr e^{-t Lambda(eps)}|_0 = -t Tr(M_f e^{-t|D_theta|}) = -t f_hat(0) sum_n e^{-t|n|} = 0 because the smooth off-mark curvature has zero mean f_hat(0)=0 (average fixed by Gauss-Bonnet / the marks). [E] CONVEXITY => GLOBAL MINIMUM (positive-definite, ALL orders): phi(x)=e^{-t x} strictly convex so Lambda |-> Tr e^{-t Lambda} is convex (Klein / Peierls-Bogoliubov); with a vanishing first variation eps=0 is the GLOBAL minimum, hence Delta Tr(e^{-t Lambda}) >= 0 for EVERY deformation and ALL eps (verified even in eps over [-1,2]), = 0 iff f=0 -- the exact form of the v292 positive-definiteness. [E] EXACT a_2 KERNEL: Delta Tr^(2) = (1/4) sum_n [Phi(|n|,|n+4k|)+Phi(|n|,|n-4k|)] g^2 with the divided-difference kernel of e^{-t x}, Phi(a,b)=[(b-a) t e^{bt}+e^{at}-e^{bt}] e^{-t(a+b)}/(a-b)^2 (a!=b), Phi(a,a)=(t^2/2)e^{-t a}, derived by Duhamel and validated against the eps=1e-3 finite-difference control to rel.err<5e-6, dominating the control's own O(eps^2)~1e-6 truncation floor (tolerance widened 2026-07-02 after an external-review platform-marginal 1.1e-6 at k=3; closed form at machine precision: v296) (k=1,2,3: W ~ 0.208,0.148,0.105 at t=0.5). [E] DEGENERATE-SPLIT CLOSED FORM: the resonant level |2k| splits to 2k+-g/2, contributing exactly 2 e^{-2kt}(cosh(t g/2)-1)>=0 (all orders), leading (t^2/4)e^{-2kt}g^2. [O] REDUCTION (now analytic): Flat-Away <= 'the seam's spectral/heat data fixes the a_2 heat coefficient to its flat value' => int f^2=0 => f=0 off the marks. Lean-mirrored by FORM.FLATAWAY.01.
Original classification detail
[E] the vanishing first variation + the convexity positive-definiteness (all orders) + the exact divided-difference a_2 kernel (validated to 1e-7) + the degenerate-split closed form; [O] the one external fact (the seam fixes a_2). Upgrades v292 from numerical to analytic; NOT a closure of Flat-Away.
Dependencies as recorded
FLATAWAY.HEAT.01; FLATAWAY.RP.01; FORM.FLATAWAY.01; SEAM.ADVERSARY.01
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- Research contracts
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- 2026-06-18 (Flat-Away heat route made exact + Lean-formalised, v295 / FORM.FLATAWAY.01)Research changelog; TFPT — Introduction and reading guide
- 2026-07-02 ( v295 robustness patch — external-review follow-up: the numeric [formula in source] -kernel tolerance must dominate its own finite-difference floor)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
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- 2026-06-19 (the Flat-Away attack: a hard discrete obstruction [formula in source] the pin derived from [formula in source] , v300)Research changelog; TFPT — Introduction and reading guide
- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportResearch contracts
- 2026-06-18 (a [formula in source] closed form + Route A literature stack, v296– v297)Research changelog; TFPT — Introduction and reading guide
- Research Contract 1 — [formula in source]tfpt research contracts
Registered verification modules
- 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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