Recognition: 2 theorem links
· Lean TheoremA Counterexample to the Gaussian Completely Monotone Conjecture
Pith reviewed 2026-05-13 00:55 UTC · model grok-4.3
The pith
An explicit probability measure on the real line makes the fifth time derivative of entropy positive along the heat flow, disproving the Gaussian completely monotone conjecture.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We provide an explicit probability measure on R for which the fifth time derivative of the entropy along the heat flow is positive at some time. This disproves the Gaussian completely monotone (GCM) conjecture and therefore also the Gaussian optimality conjecture and the entropy power conjecture. Our proof also implies the existence of a log-concave probability measure on R for which the GCM conjecture fails at some order.
What carries the argument
The explicit probability measure on the real line, together with direct computation showing that the fifth-order time derivative of its entropy under the heat flow is positive at some time.
If this is right
- The Gaussian completely monotone conjecture is false.
- The Gaussian optimality conjecture is false.
- The entropy power conjecture is false.
- There exists a log-concave probability measure on R for which the GCM conjecture fails at some order.
Where Pith is reading between the lines
- The existence of an explicit counterexample means other researchers can now test related monotonicity questions by direct calculation on the same measure.
- The result shows that sign changes in entropy derivatives can occur at finite order even for measures that are otherwise well-behaved.
- Computational search methods can locate counterexamples to conjectures about functional inequalities when purely analytic approaches have not succeeded.
Load-bearing premise
The constructed measure must be a valid probability distribution and the fifth-order time derivative of its entropy under the heat flow must have been correctly computed to be positive at the claimed time.
What would settle it
Direct numerical or symbolic computation of the fifth time derivative of entropy for the given explicit measure at the specified time, which the paper claims is positive.
read the original abstract
We provide an explicit probability measure on $\mathbb{R}$ for which the fifth time derivative of the entropy along the heat flow is positive at some time. This disproves the Gaussian completely monotone (GCM) conjecture (Cheng-Geng '15) and therefore also the Gaussian optimality conjecture (McKean '66) and the entropy power conjecture (Toscani '15). Our proof also implies the existence of a log-concave probability measure on $\mathbb{R}$ for which the GCM conjecture fails at some order. The explicit counterexample was found by GPT-5.5 Pro.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper constructs an explicit probability measure on ℝ for which the fifth time derivative of the entropy along the heat flow is positive at some t > 0. This is presented as a counterexample to the Gaussian completely monotone conjecture of Cheng-Geng (2015), with consequent disproofs of the Gaussian optimality conjecture (McKean 1966) and the entropy power conjecture (Toscani 2015). The construction is also shown to yield a log-concave counterexample at some order.
Significance. If the explicit measure is valid and the sign of the fifth derivative is correctly established, the result resolves multiple long-standing conjectures in information theory and functional inequalities by exhibiting a concrete non-Gaussian example where higher-order monotonicity fails. The explicit, searchable nature of the counterexample is a methodological strength that permits independent verification.
major comments (2)
- [§3] §3 (Construction of μ): The explicit density must be confirmed to be non-negative and to integrate exactly to 1. The normalization constant is load-bearing for all subsequent derivative signs; any scaling error would invalidate the claimed positivity.
- [§4] §4 (Fifth-derivative computation): The sign of d⁵/dt⁵ Ent(μ_t) at the stated t > 0 rests on five successive differentiations under the integral (or via the Fokker-Planck equation) together with integration-by-parts identities. The paper must exhibit the full expanded expression or a reproducible numerical evaluation at that time point; an algebraic slip in any coefficient would reverse the sign and leave the GCM conjecture intact.
minor comments (2)
- [Abstract] The abstract states that the example was found by GPT-5.5 Pro; move this remark to an acknowledgment or footnote so that the mathematical content remains the focus of the abstract.
- [§2] Notation for the heat flow μ_t and the entropy functional should be introduced once in §2 and used consistently thereafter.
Simulated Author's Rebuttal
We thank the referee for the careful reading and for identifying points where additional explicit verification will strengthen the manuscript. We address each major comment below and have prepared revisions that directly respond to the concerns raised.
read point-by-point responses
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Referee: [§3] §3 (Construction of μ): The explicit density must be confirmed to be non-negative and to integrate exactly to 1. The normalization constant is load-bearing for all subsequent derivative signs; any scaling error would invalidate the claimed positivity.
Authors: The density in §3 is given by an explicit, closed-form expression (a compactly supported piecewise polynomial) whose non-negativity is immediate by inspection of the coefficients on each interval. The normalization constant is the exact reciprocal of the integral of the unnormalized function, which evaluates to a simple rational number that we now state explicitly. In the revised version we add a short subsection that computes this integral in closed form and confirms the resulting measure is a probability measure. This removes any ambiguity about scaling for the subsequent derivative calculations. revision: yes
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Referee: [§4] §4 (Fifth-derivative computation): The sign of d⁵/dt⁵ Ent(μ_t) at the stated t > 0 rests on five successive differentiations under the integral (or via the Fokker-Planck equation) together with integration-by-parts identities. The paper must exhibit the full expanded expression or a reproducible numerical evaluation at that time point; an algebraic slip in any coefficient would reverse the sign and leave the GCM conjecture intact.
Authors: We agree that reproducibility of the sign is essential. The original computation was performed symbolically; the revised manuscript now includes the fully expanded algebraic expression for the fifth time derivative evaluated at the indicated t, together with a high-precision numerical check (to 20 decimal places) that confirms the value is positive. Both the symbolic form and the numerical verification are placed in an appendix so that any reader can reproduce the result independently. This directly addresses the possibility of an algebraic error. revision: yes
Circularity Check
No circularity: explicit counterexample construction is self-contained
full rationale
The paper's central claim is the existence of an explicit probability measure on R for which the fifth time derivative of entropy under the heat flow is positive at some t>0. This is established by direct construction of the measure (found via GPT-5.5 Pro) followed by explicit computation of the derivatives via the Fokker-Planck equation or repeated differentiation under the integral. No step reduces a claimed prediction to a fitted parameter, self-citation chain, or definitional equivalence; the disproof is external to the conjectures being refuted. The argument is therefore self-contained against the stated assumptions of measure validity and derivative sign.
Axiom & Free-Parameter Ledger
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclearWe provide an explicit probability measure on R for which the fifth time derivative of the entropy along the heat flow is positive at some time.
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IndisputableMonolith/Foundation/RealityFromDistinction.leanreality_from_one_distinction unclearH^{(5)}_μ(1/3) ∈ (0.36, 0.37) via Hermite recursion and ball-arithmetic certificate
Reference graph
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discussion (0)
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