Exact radix kernels for truncated Neumann series: no four-product radix-15 kernel exists, and μ₁₅=5

Abstract

Let Tm(m)(B) = I + B + … + Bᵐ⁻¹ be the radix-m kernel of the truncated Neumann series and let μₘ be the least number of matrix products with which a straight-line program (products of linear combinations, starting from I and B; additions and scalings free) evaluates Tm(m). Sao (arXiv:2602.11843) gives an exact three-product radix-9 kernel, an approximate four-product radix-15 kernel whose output leaks 0.185 B¹⁵ + 0.458 B¹⁶, and asks whether exact radix-15 kernels exist. We prove that they do not, over every field: a four-product program outputs a polynomial of degree exactly 16 or of degree at most 12; degrees 13, 14 and 15 are impossible. The proof is a degree-parity argument — both factors of the last product are forced to have the same degree, so deg P₄ is even, and deg P₄ = 14 would force deg P₃ = 7, itself twice a degree. Consequently μ₁₄,μ₁₅,μ₁₆ ≥ 5, the degree-16 spillover that Sao observes numerically is forced, and with the five-product construction Tm(15)(B) = Tm(3)(B) Tm(5)(B³) of Dimitrov and Coelho one gets μ₁₅ = 5 exactly. The exact radix-15 update therefore costs 7/log₂ 15 ≈ 1.79 products per doubling of the series length, worse than radix 9's 5/log₂ 9 ≈ 1.58; more precisely, radix 9 strictly beats every exact radix-m kernel with 2 ≤ m ≤ 16, and radix 17 is the first radix the argument leaves open. Two attributions in the source are corrected: the quinary kernel μ₅ = 2 is not in the paper it is credited to, and the "degree 15 in 4 products" scheme it cites is a degree-16 polynomial. Every statement is machine-checked in Lean 4 with no computational reflection and no axiom beyond the three standard ones.

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