Prime counts and prime sums from multiplication tables: a self-similar formula system

Abstract

A multiplication table whose entries are all at least 2 manufactures exactly the composite numbers; the primes are what the table cannot make. Carrying that observation to completion yields a three-line inductive formula family S_j computing prime counts (j = 0), prime sums (j = 1), and prime power sums over arbitrary intervals, with no prime, primality test, or sieve at any depth — the de-duplication weights are Linnik's identity. The paper proves the interval structure (hyperbola shells, root-bounded folds, a finite quotient–root evaluation lattice), a duplicated-mass cancellation theorem, and a layer-birth law (dimension k is born at 2^k with a single newborn). A reference implementation passes forty exact interval tests and at the billion scale reproduces π(10^9) = 50,847,534 together with the primality and index of 999,999,937, touching 63,719 lattice points. The computational route through Linnik's identity was pioneered by McKenzie and the algorithms sit in the Legendre–Meissel family; the contribution is the geometric derivation, the structural theorems, the interval taxonomy, and the verification discipline. Developed in collaboration with Claude (Anthropic); a full AI-use disclosure appears in the paper. Verification code: https://github.com/carlgribble-caa/prime-moments v1.0: literature-review revision (expanded bibliography and related-work positioning, keywords and MSC 2020 added); mathematical content unchanged.