A 2-adic obstruction for Bessel quadrature formulas with nodes in ℚ(i) ∩ S¹
Abstract
Let w(z)=-e^(-2/z)/(4π i) be the weight on the unit circle for which the Bessel polynomials are orthogonal, and let K=ℚ(i) ∩ S¹ be the set of rational points of the circle. Matsumura asks, in part (2) of his Problem 1.2, whether for a given r there is a quadrature formula of degree r+1 for w whose r+1 nodes all lie in K. He answers no for r=1 and for r=2, the latter by exhibiting a genus-one curve of Mordell–Weil rank 0, and writes: "It seems hard to extend the proof of Theorem 1.4 to r ≥ 3." We settle r=3, in the negative, and the nodes are not assumed distinct. The proof is a count at the prime 1+i of ℤ[i]. Writing each node as ζ/barζ with ζ=n+mi primitive and clearing denominators turns the degree-4 moment criterion into the single equation F(ζ)=0 in ℤ[i], where F=128textstyle∏ nⱼ-32∏ mⱼ+13∏ζⱼ +(8-10i)∏(nⱼ-mⱼ)+(8+10i)∏(nⱼ+mⱼ), and the valuation of F at 1+i equals the number of j with nⱼ,mⱼ both odd, hence is at most 4; so F ≠ 0. The same count re-proves the two published cases r=1 and r=2 with no elliptic curve and no computer algebra. We also record an explicit degree-4 formula with four distinct rational nodes, so that the obstruction is caused by the circle condition and not by the degree, and we measure, outside the formal development, exactly how far the obstruction reaches: it is confined to r ≤ 3. Every theorem and proposition below has been formally verified in Lean 4 against Mathlib, with the four exceptions that the closing section lists; no statement rests on an exhaustive search.
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Stated results
BQ1routine2026-08-30
The degree-4 moment criterion: any quadrature formula of degree 4 for the Bessel weight with four nodes forces 2 + 5e1 + 10e2 + 15e3 + 15e4 = 0 on the elementary symmetric functions of the nodes
BQ2routine2026-08-30
Non-vacuity controls: an explicit degree-4 formula with four distinct RATIONAL nodes 0, 1, -1, -4/5 (the source's Theorem 1.5 at r = 3), and the affirmative r = 0 case of Problem 1.2 (2) (one node -1, weight 1)
BQ3candidate2026-08-30
The Diophantine core: for four primitive Gaussian integers the system bqRe = bqIm = 0 has no solution, because v₁₊ᵢ(F) equals the number of ζⱼ with both coordinates odd and is therefore at most 4
BQ4candidate2026-08-30
Matsumura's Problem 1.2 (2) at r = 3, answered: there is NO quadrature formula of degree 4 for Bessel polynomials with 4 nodes on Q(sqrt-1) cap S¹ (nodes not assumed distinct)
BQ5known2026-08-30
The source's Theorem 1.3 (1) (no degree-2 formula with 2 nodes on Q(sqrt-1) cap S¹) re-proved by the same 2-adic count: the criterion forces prod zetaⱼ = 12 prod nⱼ, so 16 divides a product of two primitive norms
BQ6known2026-08-30
The source's Theorem 1.4 (no degree-3 formula with 3 nodes on Q(sqrt-1) cap S¹) re-proved without an elliptic curve: the criterion forces 4 to divide both coordinates of prod zetaⱼ, so 16 divides a product of three primitive norms
BQ7measurement2026-08-30
Frontier measurement: the 2-adic obstruction is confined to r <= 3, and a bounded exhaustive search finds no r = 4 witness
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- The Bessel polynomials yₙ(x) = ∑_(k≤n) ((n+k)! / ((n-k)! k!)) (x/2)ᵏ are orthogonal on the unit circle S¹ ⊂ ℂ with respect to the weight
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- 2026-09-07 03:53 UTC
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