The Axiom of Coherent Existence (ACE): A Conditional Proof of the Riemann Hypothesis and its Verification in ZFC

This entry contains the two foundational papers establishing the Axiom of Coherent Existence (ACE) framework for the Riemann Hypothesis. Part I: (RH-ACE) The Axiom of Coherent Existence: A Proof of the Riemann Hypothesis from First Principles of Systemic StabilityThis paper introduces the "Axiom of Coherent Existence" (ACE), positing that the stability of the number system necessitates the truth of its underlying laws. It demonstrates that the existence of a non-trivial zero off the critical line introduces a divergent instability ("Incoherence Leakage") that is incompatible with a stable mathematical universe. It establishes the implication: If ACE holds, then RH holds. Part II: (ACE-ZFC) Under the Axiom of Coherent Existence (ACE), the Capacity Framework Is Provable in ZFCThis companion paper provides the rigorous mathematical justification for ACE within standard ZFC set theory. It replaces the physical heuristics of Part I with formal theorems. Key results include: Theorem 3. 1 (The Fourier Bridge): A quantitative inequality linking off-line zeros to non-unitary energy costs. Theorem 5. 1 (PNT-Scale Capacity Law): A proof via Large Sieve methods that the representational capacity of the Hilbert space is strictly bounded by x/log x. Appendix E (The Numerical Probe): A Schur Complement analysis demonstrating that "spectral conspiracies" (geometric cloaking of off-line zeros) are economically impossible, as the coefficient vector norm explodes exponentially. Update (v2): (Added ACEAddendumMellinDominance) This document addresses the analytic constraints of the Mellin transform (Riemann-Lebesgue decay) and demonstrates that the exponential growth of the off-line instability ("Incoherence Leakage") asymptotically dominates the polynomial decay of the sensor, confirming the stability of the Fourier Bridge.