Rosenzweig–Stanfill even residue vanishing

Bart Rosenzweig and Jonathan Stanfill, On the fundamental solutions of two nonlocal parabolic equations related to logarithmic Laplacians, arXiv:2606.04225v1, Open Problem 1.6(iv), printed page 4:

Consider the notation of Theorem 1.5. Then the following are conjectured to be true:

(iv) For every α ∈ (0, π), (1.13) is equal to zero whenever m ≥ 0 is even.

The Lean proposition is ∀ m : ℕ, Even m → ∀ α : ℝ, 0 < α → α < Real.pi → paperBracket m α = 0. It includes m = 0 and α = π/2. The bracket uses the literal finite sums in (1.13), the primary recursive c-array in (1.14), its separate Bernoulli formula at π/2, and the partial ordinary Bell profile sum in (1.20)–(1.21). The convention is B₁ = −1/2. Natural subtraction and integer division are used only in indices; coefficient arithmetic takes place in ℂ. The external factor 2 exp(γ_E m)/π is nonzero, so the zero bracket gives the zero value of (1.13).

The even-index cancellation is part of the paper's proposed description of the zero and nonzero residues of K₂(t,x,x). The preregistered question is issue #11546; the admission basis is open-problem-resolution. The mathematical target is the full universal clause (iv), not a finite collection of even indices or angles.

The arXiv version history contains only v1, and its HTML states clause (iv) as conjectured. OpenAlex work W7163512061 reports zero citations. Google Scholar reports two citing works: R. A. Chandru, Arithmetic identification from interval Weil spectra, and Finite noisy determination of cuspidal representations from interval Weil spectra. Their complete Google-cached HTML texts cite the source's heat-kernel continuation; neither treats this Bell/Bernoulli residue identity. Both texts contain zero occurrences of “Bernoulli” and “Bell polynomial”. These external readings were collected at 2026-10-02 09:49–09:52 UTC. Semantic Scholar returned HTTP 429, so that index supplies no current reading. Historical openness beyond the checked sources is unverified; no exhaustive search or priority claim is asserted.

Let C(s) be the ordinary series of the primary c-array. The module derives its square inverse-denominator identity from the Euler-derivative polylogarithm recursion, using an arbitrary-order Stirling coefficient identity obtained by normalizing the frozen Stirling inclusion–exclusion theorem. At α = π/2, the separate Bernoulli formula gives a logistic series and the same square identity. Thus exp(−s/2)C(s) is even.

The b-convolution gives A(s) = C(s)E(s), with E even, so H(s) = exp(−s/2)A(s) is even and H′ is odd. The residue bracket is the negative of the Bernoulli residue functional applied to exp(s/2)H′(s). Bernoulli-polynomial translation at 1/2 identifies the functional on each monomial. Reflection Bₙ(1−t) = (−1)ⁿBₙ(t) kills odd coefficients at 1/2; for every even m the functional therefore annihilates H′. All coefficient sums are finite. Formal power-series substitution requires zero constant coefficient, not analytic convergence.

The public result has proof_shape content. Its live intermediate content includes linear_ODE_unique, F_translation, primary_square_inverse and bernoulli_annihilates. Existing Bernoulli and geometric-series definitions are used directly. Information-escape registration is paused under CLAUDE.md §3.9.

An even natural m and a real α strictly between 0 and π with a nonzero paperBracket would refute the statement. Finitely many zero values cannot establish it. The theorem imposes no restriction excluding π/2 and no positive lower bound on m beyond natural-number membership.

The kernel-checked theorem is D5.S3.ArithSums.LogLaplacianEvenResidueVanishing.result. The statement uses the original recursion and Bell polynomials, rather than an assumed generating-function characterization of the c-array. The axiom closure is propext, Classical.choice and Quot.sound.

Resolution: proved for Open Problem 1.6(iv). The utility classification is none: the module proves an unbounded symbolic identity and contains no finite enumeration, certificate checker, numerical reduction or certified numerical instance.

What the settlement shows

• Proved in this module: the decisive mechanism is the combination of reflection-evenness after multiplication by exp(−s/2) and Bernoulli translation at 1/2. The cancellation covers every even m, including zero, and the entire open interval of angles, including π/2.
• Proved in this module: the convolution argument preserves the reflection-evenness for arbitrary complex d-arrays and weights. The specific finite-difference coefficients are needed to identify the paper's expression; their detailed values do not drive the parity cancellation.
• Open: clauses 1.6(i)–(iii), nonvanishing at odd indices, endpoint angles, and analytic convergence are outside this settlement. The proof gives no nonvanishing or sharpness statement for odd m.
• Proved in this module: clause (iv) supplies the even-index vanishing statement for the source's residue formula. Conclusions that use this cancellation can invoke the proved clause. Open: a complete pole classification still needs the source's remaining nonvanishing assertions; no additional analytic theorem is claimed here.

Exhaustive historical openness and priority beyond the searched sources are unverified. Semantic Scholar was unavailable. The Google-cached citing texts were read; their original ResearchGate downloads returned HTTP 403. No claim about model-family independence follows from the carrier labels.