Theorem · version 1
For the standard Chernoff density f, κ(x) = −(log f)″(x) satisfies for every real x, and .
Compare the quantity and hypotheses within each group. A finite check, conditional theorem and proposed improvement cover different scopes.
| Result | Value | Conditions |
|---|---|---|
| Pointwise and tail curvature bounds → Proved Selected source date: | > 4|x| | Standard Brownian variance, drift t² and every real x; the inequality is strict. |
Proved
Standard two-sided Brownian motion with unit quadratic drift; no explicit optimal uniform constant is asserted.
Chernoff’s Density Is Strongly Log-Concave · Theorem 1 and Section 5 →
Source dated Jul 21, 2026 · Observed Oct 14, 2026
In this problem · publication v8 · Selected context
Bound · Proved
Relationship not independently reviewed. This concerns its use in this problem.
The paper establishes v(x) > 0 and v(x) ≥ 4x for the normalized one-sided factor, hence κ(x) > 4|x| and inf κ > 0 for the Chernoff density.
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Xianyang Zhang · Quan Zhou · Preprint · Publication date Jul 21, 2026
Source-reported results and external checks are separate from independent OpenMath assessment.
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.
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Proved
Standard two-sided Brownian motion with unit quadratic drift; no explicit optimal uniform constant is asserted.
Source dated Jul 21, 2026 · Observed Oct 14, 2026 · Source passage →