A Functional Boundary Argument for the Riemann Hypothesis via Omega-Coherence

We propose a coherence-based functional Ω(C, s) to constrain the nontrivial zeros of the Riemannzeta function ζ(s) to the critical line Re(s) = 1/2, supporting the Riemann Hypothesis. Rootedin the Enhanced Law of Observation (ELO), this functional models ζ(s) as a coherence amplitudein a probabilistic Data Substrate ψn on a topological fibration S1 → S3 → CP1, with zeros asobserver-driven collapses governed by the Coherence-Consciousness-Locality (CCL) principle, co-herence operator ˆC, and emotional fields Ei(θ, t) (e.g., love, fear). Memory dynamics and biologicaldiversity ensure maximal coherence on the critical line, testable via 10−3 Hz signals in the Collec-tive Observer Network (CON). Variational eigenvalue framing and M¨obius symmetry reinforce theargument, suggesting the Hypothesis reflects fundamental coherence in substrate-observer dynam-ics, linking to Millennium Problems (P vs NP, Yang–Mills, Hodge, Navier-Stokes, Birch) throughtopological and emotional unification. Detailed proofs are in supplemental material. Keywords: Rie-mann Hypothesis, Enhanced Law of Observation, Zeta Function, Critical Line, Coherence-Collapse,Topological Substrate, Emotional Coherence, Memory Dynamics, Millennium Problems.