Certified exact sums of primes over dyadic intervals via a ladder of generalized factorials

Abstract

For S(m), the sum of primes in the dyadic window (m, 2m], the paper constructs a ladder of generalized factorials whose integer exponent sequences are Chebyshev polynomials rescaled to take integer values. The p-adic valuation at a window prime p equals the exponent polynomial Q_j(p), so each rung yields one exact linear functional (moment) of the window primes, computable from the primes up to m together with logarithms of the integers up to 2m, without identifying any prime inside the window — rung 0 is Chebyshev's 1852 identity. A near-minimax Chebyshev combination of K+1 moments approximates t/log t; a rigorous sandwich encloses S(m) in an interval of certified width, and the integrality of S(m) converts certified width below 1 into the exact value. A 260-bit interval-arithmetic implementation certifies S(10^3) = 200,923, S(10^4) = 15,434,795, and S(10^5) = 1,255,204,276 using 6, 8, and 10 moments, with empirical window decay of roughly one order of magnitude per additional moment above the proven geometric floor 3 + 2√2. The method is a certified exact procedure of logarithmic description length — not an asymptotically fast algorithm, and the paper says so plainly. Developed in collaboration with Claude (Anthropic); a full AI-use disclosure appears in the paper. Verification code: https://github.com/carlgribble-caa/prime-moments