Entropy-Gated Cognitive Field Collapse: A Rigorous Proof of the τ -α-ν(x) Trifecta for Belief Evolution

I present a rigorous mathematical framework for entropy-gated cognitive field col-lapse, proving that belief evolution dynamics are completely characterized by a tri-fecta of parameters: memory persistence α ∈ (0, 2), spatial entropy variation ν(x) :M → (0, 1], and temperature gating τ (x) : M → (0, ∞). Using fractional calculusand stochastic differential geometry, I establish existence and uniqueness of solutions,demonstrate monotonic entropy descent, and prove that belief collapse converges toDirac measures as τ → 0. The framework yields a natural metric structure on beliefspace recoverable from field observations alone. I show these three parameters areboth necessary and sufficient—no subset suffices, and no additional parameters areneeded. The theory makes specific testable predictions: memory decay follows t−α/2power laws, different cognitive domains exhibit distinct entropy scaling exponents,and decision probabilities follow entropy-gated softmax distributions. Applicationsinclude cognitive decision optimization, adaptive learning dynamics, and multi-agentsynchronization. This work provides the first complete mathematical foundation forunderstanding how cognitive systems transition from distributed uncertainty to crys-tallized belief states, with direct implications for artificial intelligence, neuroscience,and collective intelligence systems.