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