The maximal gap in an elementary lower bound for the Gaussian isoperimetric function: a certified bracket and the location of the maximiser
Abstract
Let φ and Φ be the density and the distribution function of the standard Gaussian law, and let I(p)=φ(Φ⁻¹(p)) be the Gaussian isoperimetric function. In a recent study of one-dimensional sub-Gaussian comparison in convex order, Zhang proves the elementary lower bound I(p) ≥ L(p):=(2)/(sqrtπ) p(1-p)√(log(1-p)/(p)/(1-2p)) and remarks, on the strength of a numerical evaluation only, that L is never below I by more than 0.00720 and that the maximal gap occurs at p₀ ≈ 0.10125. Nothing about the size or the location of that gap is proved there. We prove I(p)-L(p) ≤ 0.0072 for every p ∈ (0,1) with p ≠ 1/2; we exhibit an explicit point, p_w=Φ(-1.2745), at which the gap lies in [0.00719656,0.0071965623], so that Δ^*:=supₚ(I(p)-L(p)) lies in [0.00719656,0.0072]; and we prove that every p at which the gap attains the certified level 0.00719656 satisfies p ∈ [0.1006,0.1019] or 1-p ∈ [0.1006,0.1019], which brackets the maximiser to three digits and confirms the value 0.10125 quoted in the source. We also exhibit a point at which the leading constant 2/sqrtπ=1.128379… already fails when raised to 1.12839, and identify p=1/2 as a removable singularity at which the bound is an equality. The upper bounds rest on exhaustive interval sweeps in exact rational arithmetic — 8070 intervals for the global bound, 19 245+17 917 more for the maximiser — over a self-contained rational enclosure of Φ and exp, built here because the library we verify in has no error function and nothing that evaluates or bounds the normal distribution function, and reusable elsewhere. All of this is machine-checked in Lean 4; Section [sec:verif] records exactly what the kernel checks and what it does not.
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Source snapshot 2026-09-07 03:53 UTC
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Claim ledger
Stated results
GG1candidate2026-09-03
supₚ (I(p) - LB(p)) <= 0.0072 for every p in (0,1) with p!= 1/2, by an 8070-interval certified sweep of x = Phi⁻¹(p) in [-3.0310, 0] plus the elementary tail |x| >= 2.85
GG2candidate2026-09-03
The gap is at least 0.00719656 at p = Phi(-1.2745) and at most 0.0071965623 there, so supₚ (I(p) - LB(p)) lies in [0.00719656, 0.0072] and the value at the witness is pinned to eight digits
GG3routine2026-09-03
p = 1/2 is a removable singularity of the source's bracket: log((1-p)/p)/(1-2p) is 0/0 there, its limit is 2, and with that limit (eqn:I) is the equality (2/sqrt pi)(1/2)(1/2)sqrt 2 = phi(0) = I(1/2); under Lean's y/0 = 0 convention the literal formula gives 0 instead
GG4routine2026-09-03
The constant 2/sqrt(pi) 1.1283792 of (eqn:I) cannot be raised to 1.12839: at p = Phi(-0.005) the modified right-hand side strictly exceeds I(p)
GG5routine2026-09-03
Controls: 0.00719 is not an upper bound for the gap; 0.0072 is not attained at the witness; and dropping p!= 1/2 makes GG1 false, because gap(0) = phi(0) = 0.3989 under the junk-value convention
GG6routine2026-09-03
Kernel-clean rational two-sided enclosures for exp on [0,8] and for the standard normal c.d.f. Phi at any rational x <= 0, from the alternating Taylor sandwich for exp on the negative axis integrated against the Gaussian density
GG7measurement2026-09-03
Cost of the certificate: 8070 intervals in 30 uniform-step blocks, 42.4 s wall / 15.5 s user in one native_decide at 6.77 GB peak RSS (dominated by the import-Mathlib baseline) (marginal cost 0.72 ms per interval); the tangent bound Phi(x) >= Phi(x1) - phi(x1)(x1-x) cuts the ideal interval count from 8200 to 6060 and halves the number of Phi evaluations
This ledger entry is reported in prose and is not bound to a Lean theorem.GG8candidate2026-09-03
Every point at which the gap reaches the certified value 0.00719656 lies in Phi⁻¹([0.1006, 0.1019]) (or its reflection), so the maximiser p₀ is bracketed to three digits, confirming the source's p₀ 0.10125
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Yihan Zhang, *Sharp One-Dimensional Sub-Gaussian Comparison in Convex Order*, arXiv:2604.26819. Primary category math.PR; cross-listed cs.IT, math.ST, stat.ML (read off the live abs page 2026-09-03 — the scout report journal/2026-09-03-scout-ml-rl-back.md says "cs.LG, cs.IT cross" and "live v1 = corpus copy"; both are wrong: there is no cs.LG cross-list, and the live version is v2, submitted 2026-06-14). All quotations in this family are taken from the v2 e-print (cxₘgf.tex).
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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