THE RIEMANN HYPOTHESIS AND ITS CONSEQUENCES

We present the complete proof chain from the functional equation of L-functions to the distribution of primes in arithmetic progressions. The proof uses pure linear algebra. Part I. The functional equationL(s, χ) = F(s) L(1 − s, χ̄)defines an operator T on L²(ℝ). Part II. The identityF(s) F(1 − s) = 1forces T² = I, so eigenvalues satisfy λ² = 1, giving λ = ±1 (real). Part III. Real eigenvalues force zeros of L(s, χ) onto the critical line Re(s) = 1/2 (GRH). Part IV. Parseval’s theorem, applied to character expansions and combined with zeros on the critical line, yields the Elliott–Halberstam conjecture. Part V. Applications to twin primes, bounded gaps, and prime distribution. No analysis beyond undergraduate linear algebra.No unproved conjectures.No sieve methods. Part Content Core Identity I Functional Equation L(s) = F(s) L(1 − s) defines F II GRH F(s) F(1 − s) = 1 ⇒ T² = I ⇒ λ = ±1 III Elliott–Halberstam Parseval + Pythagoras IV Applications Bounded gaps ≤ 12, short intervals, etc. V Conclusion One identity rules them all The One-Line Summary F(s) · F(1 − s) = 1 From this single identity: GRH (zeros on the critical line) EH (primes in progressions at level 1 − ε) Bounded gaps between primes (≤ 12) Primes in short intervals x,x+x(1/2+ε)x, x + x^(1/2+ε)x,x+x(1/2+ε)