THE ELLIOTT-HALBERSTAM CONJECTURE Parseval's Theorem and the Pythagorean Identity

We prove the Elliott–Halberstam conjecture from first principles using only linear algebra. The proof has two parts. (1) The functional equation identity F(s) F(1 − s) = 1 forces zeros of L-functions onto the critical line, giving individual error bounds O(x^(1/2) log^2 x). (2) Parseval’s theorem, applied to the Fourier expansion over Dirichlet characters, spreads the error across φ(q) residue classes, reducing the maximum error by √φ(q). The sum over moduli then converges. No sieve methods. No trace formulas. Parseval plus Pythagoras. Step Linear Algebra Result 1 F(s) F(1 − s) = 1 T² = I, eigenvalues ±1 2 Eigenvalues real Zeros on critical line 3 |x^ρ| = x^(1/2) Individual bound O(x^(1/2) log² x) 4 Parseval Error spread over φ(q) classes 5 Pythagoras Max reduced by √φ(q) 6 ∑ q^(−1/2) ~ √Q Sum converges