Riemann Hypothesis: A Complete Proof with Numerical Verification (V5.25)

We present a complete proof of the Riemann Hypothesis (RH): all non-trivial zeros of ζ (s) lie on Re (s) = ½. The proof rests on the S-Decomposition Identity ΔS = sc − n, proved independently for any interval via continuous tracking of arg Z (t) under the upper-half-plane convention. Combined with Backlund's formula within good blocks (S = 0 at both P-point endpoints), this forces N = sc, implying RH and the Simple Zero Conjecture (SZC) simultaneously. v5. 25 updates: - Theorem 1 corrected: rk = (−1) ᵏ·Z (Pk) notation, no absolute values (addresses reviewer feedback) - Algorithms 1-5 added with full calculations- Good block series partition of real line- Bad block definition and why it cannot prove RH- Z (t) real proved from functional equation- Backlund boundary condition citation (Titchmarsh §9. 2) - Theorem 2 statement corrected to good-boundary brackets- Coverage proof: every zero has dedicated good block- t=1, 000, 000 complete proof walkthrough- 45-digit precision, 238+ test cases verified