cft-cmi

Weak link: numerical values of the tail constants A_1 <= 2.0 and A_3 <= 0.0374

Status: numerical — measured, with an error bar and a convergence order; not a theorem.

Area: Theorem A: exact recovery error of the free fermion · Card type: result · Knowledge-base id: tail-constants-numerical

Statement

The constants of the occupation-tail estimate of rigor/tail_bound.tex (Thm 4.2) satisfy A_1 = sup over kappa_c > 0 of |g1-hat(kappa_c/2 pi)| (1+kappa_c)^-1 e^{kappa_c} <= 2.0 and A_3 = sup over 0 < zeta <= 1 of zeta^-2 ||(g2)'''||_{L^1(0,inf)} <= 0.0374, hence C_1 <= 0.051 and C_2 = 4 pi A_3 <= 0.47. The supremum A_3 is the limit zeta -> 0, numerically about 0.0372 (tail_bound.tex l. 750-751, corrected 2026-09-24, Correction C10). Until 2026-09-24 the document, and this card, stated A_3 <= 0.036 and C_2 <= 0.46, read off finite-zeta samples that miss that limit.

How to verify

rigor/tail_bound.tex Thm 4.2 (thm:tau, l. 316-330) and Sec. 9 (3) (l. 740-751): A_1 from double-precision evaluations at kappa_c = 6..25.3 (numerics/pt_checks.py). A_3: the document now says (l. 750-751, corrected 2026-09-24, Correction C10) that zeta^-2 ||(g2)'''||_1 = 0.0355, 0.0326, 0.0275, 0.0214 at zeta = 0.05, 0.2, 0.5, 1 on the grid of rigor/pt_tailconst.py decrease in zeta, so the supremum A_3 is the limit zeta -> 0, which that script does not sample: 0.0364 on the same grid and 0.0372 on a refined grid, hence A_3 <= 0.0374 and C_2 = 4 pi A_3 <= 0.47. Until 2026-09-24 it read the four values as 'monotone decreasing in zeta, so A_3 <= 0.036 and C_2 <= 0.46'. Correction C10 (l. 1080-1112) gives the limit in closed form, g0(t) = -(1 + cosh(t)/2 - (3/2) sinh(t)/t)/(2 sinh^3(t/2)) = -t/30 + O(t^3), with ||g0'''||_1 = 0.03642 on the script's grid, 0.03718 on t = 0.002..80 and about 0.03721 including t near 0; on the refined grid the finite-zeta values stay below it (0.03698, 0.03617, 0.03331, 0.02175 at zeta = 0.01, 0.05, 0.2, 1). Check: from the cft_cmi root run python3 numerics/kb-checks-2026-09-24/a3_refined.py (needs mpmath, about 4 min) and compare with a3_refined.out; it also runs the original functions of rigor/pt_tailconst.py at zeta = 1e-4, 1e-3, 0.01 (0.036422, 0.036406, 0.0362456). The document's own verdict (l. 442-449) says these values are numerical, not a closed-form estimate, and l. 66 says they are not certified (Correction C4; 'certified numerically' until 2026-09-24).

Evidence

Related claims

None recorded.

Review

Review record: passed 2026-09-24 by REF-CARD-TAIL.

The referee's report and the card's event log stay in the private knowledge base; only the verdict, the date and the referee are published here.

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