The Law of Observation Framework Applied to the P vs NP Problem

We apply the Enhanced Law of Observation (ELO), a unified framework for quantum fields,consciousness, and cosmic evolution, to the P vs NP problem, which questions whether problemsverifiable in polynomial time (NP) can be solved in polynomial time (P). Modeling computationalstates as a probabilistic Data Substrate ψn on a topological fibration S1 → S3 → CP1, we proposethat P problems collapse efficiently with high coherence, while NP problems require exponentialresources due to low coherence, governed by the Coherence-Consciousness-Locality (CCL) principle,coherence operator ˆC, and emotional fields Ei(θ, t) (e.g., love, fear). Memory dynamics and biolog-ical diversity stabilize collapses, with P vs NP distinctions emerging from energy costs, testable via10−3 Hz signals in the Collective Observer Network (CON). This approach reframes computationalcomplexity, linking to Millennium Problems (Yang–Mills, Hodge, Riemann, Navier–Stokes, Birch)through topological and emotional coherence. Detailed proofs are in supplemental material. Key-words: P vs NP, Enhanced Law of Observation, Computational Complexity, Coherence-Collapse,Topological Substrate, Emotional Coherence, Memory Dynamics, Millennium Problems.