slug: dinov-2026-equipartition-duality-minimum-refutation bibkey: dinov2026kime doi: 10.48550/arxiv.2607.07851 url: https://arxiv.org/abs/2607.07851v1 triage: theorem motivation_gids:

• D5/S3/Quantum/Information/CovarianceSumBound

The within-DOF minimum clause of equipartition duality

Problem

Dinov, arXiv:2607.07851v1, Conjecture 3.22 (Equipartition duality):

Fix $n\ge 2$, an entropy value $s$, and an action marginal ${\rho }_{\mathbf{J}}$ on $(0,\infty {)}^n$. Among all states on ${\mathbb{T}}^n\times (0,\infty {)}^n$ with entropy $\ge s$ and action marginal ${\rho }_{\mathbf{J}}$, the phase-equipartitioned product state (unique when it exists) simultaneously (i) maximizes the entropy, (ii) minimizes every within-DOF uncertainty $u_j$, and (iii) is the unique state at which the per-DOF conjectured bound of Problem 3.17(b) is saturated for all $j$; moreover it is the unique fixed point, with the given marginal, of the multi-DOF kime-deformed semigroup ${\partial }_t\stackrel{=\sum j(-\omega j\partial \theta j+\epsilon \partial \theta j2)\rho }{\rho }$.

This dossier anchors the necessary minimum clause (ii). The other clauses are quoted to identify the original simultaneous assertion and receive no separate settlement.

Motivation

Covariance inequalities concern the actual coordinate law; uniformity of a different marginal does not determine that covariance. The existing D5/S3/Quantum/Information/CovarianceSumBound develops covariance constraints for density states. Here the question is whether fixed actions and an entropy lower bound force uniform phases to minimize each actual covariance determinant.

Gap

Preregistration #13139 specifies the original quantified assertion and this normalized-density route before the mathematical probes. The accepted independent source reviews found no resolution in their checked arXiv topic/history, OpenAlex citation and MathDB scopes. The source remains a v1 conjecture. Semantic Scholar returned HTTP 429; it contributes no negative literature evidence. These bounded readings are described in the associated Library note.

Route

Take two independent actions with density $e^{-J}$ on positive reals and two independent phases with density

$$\varphi (\theta )=\frac{1+\frac{1}{2}\cos (2\theta )}{2\pi }.$$

Use the actual source map $(q,p)=(\sqrt{2J}\sin \theta ,\sqrt{2J}\cos \theta )$. The comparator has uniform phases and the same joint action density $r(J_1,J_2)=e^{-(J_1+J_2)}$. Both entropies are finite and nonnegative, so both states satisfy the same threshold $s=0$.

For one degree of freedom the nonuniform state has mean zero and covariance $\mathrm{diag}(3/4,5/4)$; the comparator has covariance $I_2$. Independence gives full positive-definite covariances $\mathrm{diag}(3/4,5/4,3/4,5/4)$ and $I_4$. Both uncertainties are $\sqrt{15/16}<1$ in the nonuniform state. Both strict comparisons occur in the contradiction with the asserted minimum inequalities.

Falsifier

The original assertion for every $n\ge 2$ implies its minimum clause at $n=2$, which implies dinovTwoDofMinimumClause on the certified subclass. That predicate universally quantifies the entropy threshold, joint action density, Cartesian state and existing phase-uniform product comparator, then requires the comparator’s uncertainty to be no larger for both Bool degrees of freedom. result : ¬ dinovTwoDofMinimumClause refutes this necessary consequence and therefore the simultaneous assertion as written.

Evidence

D5/S3/Quantum/Information/DinovOriginalMinimumRefutation.lean retains the needed source definitions and one public result. All analytic and realization proofs are local to that result. The actual measures are AddCircle(2π) angular volume, positive-action Lebesgue measure and standard Cartesian volume. The proof certifies measurable nonnegative unit-mass densities, integrable entropy with the exact source equality, all four coordinates in L2, full positive-definite covariance, both angular fiber integrals, and both torus and Cartesian joint action pushforwards.

Triage

Tier 1 published conjecture; proof_shape: bind-only, escape_witness: none, and admission_basis: open-problem-resolution under #13139. The Scribe result carries OpenProblemResolutionClaim(Refuted) for this dossier, with the refutation attributed FromRepo. The original conjecture is literature. The utility is certified-instance with the typed negation of this module’s minimum predicate.

• Proved in result: all required state and source certificates, and failure of both minimum inequalities at the same realization and threshold.
• Failure mechanism: the second phase harmonic reallocates the q and p variances to $3/4$ and $5/4$ while preserving the joint action marginal. Their product is $15/16$, below the uniform-phase value. The entropy lower bound permits this anisotropy at $s=0$.
• Open: a corrected minimum principle with sufficient additional constraints. Fixed action marginals and a lower entropy bound alone do not provide those constraints.
• Boundary: failure of clause (ii) refutes the simultaneous conjecture; no separate theorem about entropy maximization, saturation uniqueness, Problem 3.17(b) on a fixed symplectic orbit, or the semigroup follows. Neighboring source inequalities and evolution results are not refuted.

ASSUMED-UNVERIFIED

The paper-to-predicate necessary-instance implication is source correspondence checked by the accepted independent reviews, not a generic-$n$ Lean theorem. The interpretation of latent phase as experimental kime is not an empirical conclusion. No global originality claim or Cartesian smoothness-at-zero certificate is made. Information-escape registration remains paused under CLAUDE.md §3.9; no registration-completion state is claimed.