Unified Latent Formulations of Millennium Problems

We present a unified framework for four Clay Mathematics Institute Millennium Problems—P vs NP, the Riemann Hypothesis, the Yang-Mills mass gap, and a formalization of consciousness emergence—using the Latent representation theory. Each problem is formulated as a U-element with specific projection constraints, where the Latent number ρ controls the key invariant. For P vs NP, we prove that crystallization (constraint satisfaction) narrows the search space with entropy decay bounded by the coherence parameter κ. For RH, we establish that unit Latent number ρ = 1 implies criticality Re(s) = 1/2. For Yang-Mills, we prove mass gap survival under arbitrary cascade depth. For consciousness, we derive emergence criteria from residual-coherence balance. The framework consists of 14 verified theorems and 26 hypotheses formalized in the Platonic proof language.

Keywords: Millennium Problems, Latent representation, P vs NP, Riemann Hypothesis, Yang-Mills mass gap, consciousness, formal verification

MSC 2020: 68Q15, 11M06, 81T13, 92C20