What was already known
Hillion and Johnson’s 2019 Conjecture 4.4 asked for this parameter-monotonicity range. The cases q=0, 1 and 2 were known, with counterexamples above 2. Joint concavity, including a 2026 preprint about its threshold, is a different statement.
AI involvement
JD Jones’s disclosure attributes the mathematical arguments and extensions to GPT-6 Astra through Codex. Separate agents developed and audited the Bernoulli foundation, association argument and proof exposition; Jones directed and integrated the project. The disclosure supports AI-led work, not task-only autonomy.
The project disclosure attributes the mathematical arguments and extensions to GPT-6 Astra, with collaborating agents developing the proof components. JD Jones directed and integrated the work. A task-only, human-free run is not documented.
Importance
Reports the full monotonicity range for two standard entropy families, including boundary probabilities.
Limitations
Monotonicity concerns the coin probabilities while the entropy order stays fixed. It is not joint concavity or a theorem for orders above 2. The reported full formal contract covers the two entropy statements, a supporting finite inverse-score inequality and their concrete definitions. Quantitative gains, strict equality and continuous-law extensions lie outside that contract. No named human account explaining the central proof or independent human specialist review is documented.
Technical verification
Palomar reports full-contract checking on 17 October 2026 with Lean, NanoDa and con-ron. The contract covers both all-orders entropy theorems and their concrete probability/entropy definitions, not just the historical association-only result.
Formal checking
Reported full-contract checking, distinct from earlier association-only checks.
- Checker
- Palomar mechanical verifier: Lean, NanoDa and con-ron
- Scope
- RenyiMonotonicity.renyi_monotone and tsallis_monotone for every finite n, q in [0,2] and 0≤pᵢ≤rᵢ≤1/2, plus finite inverse_score_association and eight concrete definition equations; only propext, Classical.choice and Quot.sound.
- Basis
- Reported by the linked checker
- Check date
- 2026-10-17
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. The available proof exposition is AI-generated. No named human account explaining the central proof is documented. This is an evidence gap, not a finding that people cannot understand it.
Correction and publication history
Exact primary theorem, original conjecture, adjacent concavity paper, explicit central-AI disclosure and full-contract mechanical checking reviewed under CURATION.md. Assessments: AI-led, Formal (Lean; source-reported), Few by default for undocumented human understanding, Notable. This is a scoped new contribution; no registry-only eligibility or unverified extension promoted.
Read this published revision