All odd kinetic frequency moments

P. Tolias, T. Dornheim and J. Vorberger, Kinetic contribution to the arbitrary order odd frequency moments of the dynamic structure factor, arXiv:2508.17810v1, Sec. 2.3, Eq. (16), state: “Therefore, our conjecture states that the following result holds for the interacting uniform electron gas”, followed by

$$ M_{\mathrm{S,KIN}}^{(2k+1)}(q)= \left(\frac{\hbar q^2}{2m}\right)^{2k+1}\frac{1}{2k+2} \sum_{i=0}^{k}\binom{2k+2}{2i+1} \left(\frac{2}{\hbar q}\right)^{2i}\langle p^{2i}\rangle_0, $$

“where $k$ is an arbitrary non-negative integer.” The paper explicitly checks $k=0,1,2,3$. Its summary also asks whether similar expressions hold for other spin statistics. Issue #13703 fixes the operator representation and averaging conventions. The quotations and locators are in the literature note.

The formula expresses the kinetic contribution to every odd dynamic structure factor moment through finitely many per-particle radial momentum moments. D5/S3/Quantum/KineticMoments/OddMoment.result establishes it for every order and every split of the commutator nests under the stated momentum-measure hypotheses.

The preregistration records the arXiv version, citing-work, MathDB and repository searches. No all-order settlement was found in that searched scope. The published full text and the special-issue editorial are outside the verified literature scope. A bounded search does not establish exhaustive novelty.

A configuration is $P:\mathrm{Fin}(N)\to\mathbb R^3$. States are all complex-valued functions of $P$. The density operators are sums of pullbacks by $P_j\mapsto P_j\pm\hbar\boldsymbol q$, and the kinetic operator multiplies by $\sum_j|P_j|^2/(2m)$. This is the kinetic prescription $\hat H\equiv\hat K$ of Sec. 2.2 and Eq. (9).

Put $a=\hbar^2|\boldsymbol q|^2/(2m)$ and $b_j=(\hbar/m)\langle\boldsymbol q,P_j\rangle$. For $n=2k+1$ and $0\le\ell\le n$, the split nested commutator is multiplication by

$$ F_{k,\ell}(P)=(-1)^\ell\sum_j \bigl((b_j+a)^{2k+1}-(b_j-a)^{2k+1}\bigr). $$

Different-particle terms commute; same-particle opposite translations cancel. For an isotropic measure, the integrated directional moment satisfies

$$ \int\langle\boldsymbol q,P_j\rangle^{2i},d\nu =\frac{|\boldsymbol q|^{2i}}{2i+1} \int|P_j|^{2i},d\nu. $$

The proof derives this factor by comparing coefficients of integrated moment polynomials and solving a weighted recurrence. Radial integrability controls all directional integrals. The odd binomial expansion, the identity $\binom{2k+1}{2i}/(2i+1)=\binom{2k+2}{2i+1}/(2k+2)$ and the complex sum-rule prefactor yield Eq. (16).

The statement requires $N\ge1$, $\hbar>0$, $m>0$, $\boldsymbol q\ne0$, a simultaneous-$\mathrm{SO}(3)$-invariant probability measure and finite $2k$-th radial moments for each particle. Its algebraic operator carrier is the space of all configuration functions. A physical model with different operator or averaging conventions requires a separate bridge. Without isotropy the radial reduction is not asserted; without finite required moments the integral conclusion is not asserted.

The canonical sources are D5/S3/Quantum/KineticMoments/IsotropicAverage.lean and D5/S3/Quantum/KineticMoments/OddMoment.lean, with their Blueprint mirrors. OddMoment.claim includes both the exact operator identity and its normalized integral formula. OddMoment.result proves this complete conjunction. IsotropicAverage.isotropic_average proves the directional-to-radial identity for any isotropic measure with the specified integrable moment, without requiring a probability measure. The operator calculation uses the frozen ObserverCommutator.observer_read_update_commutator_formula and pinned Mathlib. The axiom closure of every public declaration is contained in ${\mathrm{propext},\mathrm{Classical.choice},\mathrm{Quot.sound}}$.

Tier 1 external conjecture, preregistered in #13703. Resolution: proved in the recorded momentum-space model. OddMoment.result has proof_shape: content and admission_basis: open-problem-resolution. IsotropicAverage has admission_basis: escape-witness, through isotropic_average and the private weighted_recurrence. integrable_directional is a consumed bind-only helper; the private operator-identity helpers are consumed by the settling proof. Utility is none: the delivery consists of universal operator and integral identities, with no finite certificate, enumeration or numerical reduction. Information-escape registration is paused under CLAUDE.md §3.9.

What the settlement shows

• Proved — every order and split. OddMoment.result establishes Eq. (16) for every $k\in\mathbb N$ and every $\ell\le2k+1$, including the Eq. (17) commutator form with its prescribed prefactor. The decisive mechanisms are translation cancellation and the isotropic weighted recurrence.
• Proved — other spin statistics, conditionally. OddMoment.result has no statistics, particle-independence, exchangeability or interaction hypothesis. It applies to any statistics whose operators and momentum distribution obey the fixed conventions and isotropic finite-moment assumptions. Open: construction of such physical equilibrium states, their thermodynamic limit and moment finiteness for a specified interaction or statistics.
• Open — dimension-$d$ analogue. The proposed angular factor $\Gamma(d/2)\Gamma(i+1/2)/(\sqrt\pi,\Gamma(i+d/2))$ is not proved here. The delivered theorem is for dimension three.
• Proved — direction-resolved algebra without isotropy. The private OddMoment.multiplier_expansion, used by multiplier_integral, gives $$F_{k,\ell}(P)=(-1)^\ell 2\sum_{i=0}^k \binom{2k+1}{2i}a^{2k+1-2i}(\hbar/m)^{2i} \sum_j\langle\boldsymbol q,P_j\rangle^{2i}.$$ It requires no isotropy. Open: a separately stated normalized anisotropic integral theorem; none is exported by this delivery.
• Proved — consequences for the source. Within the fixed model, Eq. (16) and Eq. (17) hold beyond the four explicitly evaluated orders. The identity uses the actual per-particle momentum distribution and requires no replacement by a Fermi distribution. Open: kinetic dominance of the full interacting moment, convergence of an infinite even-frequency series and the validity of numerical or QMC approximations that use these moments; the identity supplies no estimates for those questions.

The literature scope does not establish worldwide priority or the absence of an independent proof. The published full text and the special-issue editorial were not verified. Physical equilibrium-state existence, isotropy and finite moments for a particular interacting gas remain premises, not conclusions.