Proved
Airy factorization and curvature identity ↗
The normalized one-sided factor and fixed Brownian/drift convention.
Chernoff’s Density Is Strongly Log-Concave · Equations (2)–(6); Sections 2–4 →
Source dated Jul 21, 2026 · Observed Oct 14, 2026
Theorem · version 1
Airy factorization of the density and curvature
Statement
For a positive normalized factor g with bilateral Laplace transform M(s) = Ai(0)/Ai(2^(−1/3)s), the standard Chernoff density has f(x) = C_f g(x)g(−x), where C_f = 2^(−1/3)/Ai(0)², and κ(x) = v(x)+v(−x) for v = −(log g)″.
Assumptions
- The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero. Factor normalization is fixed; multiplication by a positive constant does not change logarithmic derivatives.
Evidence
Source reports
Connections
In this problem · publication v8 · Selected context
Strong log-concavity of the Chernoff density →
Supporting result · Proved
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Chernoff’s Density Is Strongly Log-Concave →
- Equations (2)–(6); Sections 2–4 → · Original passage ↗
Recorded wording
The Chernoff density factors as f(x) = C_f g(x)g(−x), where the normalized one-sided factor has reciprocal-Airy bilateral Laplace transform. The curvature is v(x)+v(−x).
paraphrase · Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.
Xianyang Zhang · Quan Zhou · Preprint · Publication date Jul 21, 2026
Source-reported results and external checks are separate from independent OpenMath assessment.
Revision & attribution
Source revision: v1. Observed Oct 14, 2026.
The authors attribute the proof to GPT-5.6 Sol, describe an initial algebraic failure and later corrected approaches, and report carefully checking the argument and revising the exposition.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Airy factorization and curvature identity ↗
The normalized one-sided factor and fixed Brownian/drift convention.
Source dated Jul 21, 2026 · Observed Oct 14, 2026 · Source passage →
Record details
- Claim ID
- claim-chernoff-airy-factor
- Statement version
- 1
- SHA-256
- 2bae29f81fe38915f4a3f071735e7baed518ac97c9cdcf124050a92ff4d9a9c8