Lichtenfelz–Modin–Preston Conjecture 1: four Zeitlin six-j identities

Conjecture 1 of Lichtenfelz, Modin and Preston, arXiv:2508.09833v1, section 2.2, asserts all four identities (2.10)–(2.13). The complete quotation and the two Wigner-symbol conventions are in the literature note lichtenfelz2026zeitlin. The domain is every integer N ≥ 2, with 1 ≤ j,l < N; only (2.10) assumes j ≠ l. The sums traverse i = 1,...,N−1, λ(i) = i(i+1), and H(j) is the j-th harmonic number. The preregistration is issue #11604.

The four identities supply the hypothesis of the source's Theorem 3 on Ricci curvature of the Zeitlin approximation to hydrodynamics on the sphere. D5/S3/Quantum/Algebra/ZeitlinSixJ/SumRules.result proves the full conjunction for the Racah definition of the Wigner six-j symbol, with doubled spin labels and zero outside triangle admissibility.

The source reports numerical verification up to N = 2048. The claimed settlement is universal, rather than a finite certificate. The bounded literature check in #11604 found no proof in its searched scope; journal sentence-by-sentence comparison and citation-index completeness are unverified. No exhaustive novelty or priority claim is made.

The factorial-kernel recurrence identifies the terminating Racah polynomial. An exact four-spin WZ identity produces the finite Jacobi action, and endpoint WZ normalization fixes the orthogonality constant. A constructed two-sided Green inverse gives (2.10); the parity intertwiner gives (2.11); the central Jacobi diagonal gives (2.12); the harmonic binomial identity and factorial antidifference give (2.13). The eleven-module chain keeps these actual shared matrices, supports, phase factors and finite sums in one dependency graph.

A tuple satisfying the stated domain but violating any conjunct would contradict SumRules.result in its formal system. A mismatch between the doubled-spin Racah formula and the source's symbol convention would invalidate source fidelity. A primary publication already proving the exact conjecture would change the bounded provenance assessment.

SumRules.claim is the conjunction of the four fully quantified identities. SumRules.result : SumRules.claim proves it using the Racah finite sum, with no postulated recurrence, orthogonality or symbol identities. The public foundation results supply the actual arbitrary-spin action and endpoint normalization, the physical spectral inverse and the signed addition identity. The axiom boundary is the Lean standard three axioms, not an additional six-j assumption.

Tier 2: a published mathematical-physics research problem. This dossier records the whole Conjecture 1 as Proved for the stated definition and domain. Conjecture 2 is a separate asymptotic target.

What the settlement shows

• Proved in this chain: exact endpoint normalization and the Jacobi action connect the actual Racah sums to an orthogonal finite matrix; no abstract orthogonality hypothesis substitutes for these identities.
• Proved in this module: the central inverse diagonal, signed Casimir moment, ordinary Casimir moment and harmonic sum have the exact values in (2.10)–(2.13) for all N ≥ 2. The j ≠ l restriction belongs solely to the inverse-Casimir identity.
• Proved in this chain: the Green inverse and parity constructions apply to every admissible channel in their displayed foundation statements; the final source identities are their central-channel consequences.
• Open in Lean: the paper's Ricci-curvature Theorem 3 and the analytic passage to its hydrodynamical interpretation are not formalized here. The source explicitly conditions Theorem 3 on these four identities, so this settlement supplies that mathematical hypothesis under the stated source correspondence.
• Open: Conjecture 2's sharper asymptotic bound for the positive Ricci contribution is not a consequence asserted by this module. Generalizations beyond the stated spin configuration require separate hypotheses and proofs.

The journal text has not been compared sentence by sentence with arXiv v1, and citation-index completeness is unverified. The literature conclusion remains not-found-in-searched-scope. No claim of worldwide absence, priority, exhaustive sharpness or resolution of Conjecture 2 follows from the formal result.