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Original ledger classification: Closed form [E] (geometric tail + finite middle)

Recorded statement: EXACT a_2 COEFFICIENT IN CLOSED FORM (2026-06-18): evaluates the v295 infinite mode sum analytically. W_k(t)=(1/2) sum_{n in Z} Phi(|n|,|n+4k|) with the divided-difference kernel of e^{-tx}. [E] TAILS TELESCOPE: for n>=0,…

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FLATAWAY.A2.CLOSED.01

Original ledger classification: Closed form [E] (geometric tail + finite middle)

Original claim text

EXACT a_2 COEFFICIENT IN CLOSED FORM (2026-06-18): evaluates the v295 infinite mode sum analytically. W_k(t)=(1/2) sum_{n in Z} Phi(|n|,|n+4k|) with the divided-difference kernel of e^{-tx}. [E] TAILS TELESCOPE: for n>=0, Phi(n,n+4k)=e^{-nt} C_k/(16k^2) with FIXED C_k=4kt+e^{-4kt}-1 (geometric in n); likewise n<=-4k with D_k=e^{4kt}-4kt-1; using C_k + D_k e^{-4kt}=4kt(1-e^{-4kt}) the two tails close to W_k^tail=t(1-e^{-4kt})/(8k(1-e^{-t})). [E] MIDDLE FINITE: the modes -4k<n<0 give (1/2) sum_{m=1}^{4k-1} Phi(m,4k-m), containing the degenerate diagonal Phi(2k,2k)=(t^2/2)e^{-2kt} (the splitting term). [E] CLOSED FORM: W_k(t)=t(1-e^{-4kt})/(8k(1-e^{-t})) + (1/2) sum_{m=1}^{4k-1} Phi(m,4k-m); W_1(t)=t(2t e^t+e^{3t}+3e^{2t}+e^t-1)e^{-3t}/8; reproduces v295 to machine precision (k=1,2,3). [E] POSITIVITY MANIFEST term by term (tail>0, each middle Phi>0). [E] SMALL-t: W_k(t)=t/2+O(t^3) -- leading 1/2 mode-independent, the t^2 term cancels (tail's (1/4-k) vs middle's (4k-1)/4), k enters at O(t^3). An exact analytic upgrade of v295.

Original classification detail

[E] the closed-form evaluation of the exact a_2 coefficient (geometric tail + finite middle) + manifest positivity + the small-t law with the t^2 cancellation; reproduces v295 to machine precision. An exact analytic upgrade; does NOT close Flat-Away.

Dependencies as recorded

FLATAWAY.A2.01; FLATAWAY.HEAT.01; FORM.FLATAWAY.01

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  • v296_flataway_a2_closed.pyRegistered verification module · frontier
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    FLATAWAY.A2.CLOSED.01 — the exact a₂ coefficient in CLOSED FORM (replacing v295's validated-numeric sum). [E] the operator tails telescope to a geometric series (Φ(n,n+4k)=e^{−nt}C_k/(16k²), C_k+D_k e^{−4kt}=4kt(1−e^{−4kt})), giving W_k^tail=t(1−e^{−4kt})/(8k(1−e^{−t})); [E] the middle is finite (4k−1 terms incl. the degenerate diagonal Φ(2k,2k)=(t²/2)e^{−2kt}); [E] closed form W_k(t)=t(1−e^{−4kt})/(8k(1−e^{−t}))+½Σ_{m=1}^{4k−1}Φ(m,4k−m), with W_1(t)=t(2t eᵗ+e³ᵗ+3e²ᵗ+eᵗ−1)e^{−3t}/8, reproducing v295 to machine precision; [E] positivity manifest term by term; [E] small-t W_k=t/2+O(t³) (mode-independent leading 1/2, the t² term cancels).

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