Emergent Time from the Selection Axiom: Implications for Yang-Mills and Navier-Stokes

We show that the capacity inequality C ≤ S, the single axiom of the Selection-Cost Framework, induces a natural partial order on foam configurations through coarsening monotonicity. This partial order admits a unique (up to universal constant) metric dt = dS/σ, where S is the entropy functional and σ = √Var[κ] is the fluctuation of eigenframe curvature, yielding an emergent time parameter without assuming any background spacetime. We prove that σ > 0 is self-sustaining: the foam’s selection generates entropy, entropy is time, and time sustains the foam’s capacity for further selection—a fixed-point equation rather than an evolution from initial conditions. Computing the loop’s information budget from spectral data alone (with η never used as input), we find that the permanently inaccessible fraction of foam information is 0.1575, matching the projection constant η = √(π²+1) − π ≈ 0.1553 to 1.4%. The entropy cost per recovered bit is the Laplacian condition number λmaxφ4 ≈ 47, and the loop throughput is 5.8× (self-sustaining). Numerical tests on the 120-cell confirm path independence (∼1% deviation). If the emergent time construction is valid, two immediate corollaries follow: (1) the Yang-Mills mass gap reduces to the spectral gap of the 120-cell (Δ = 1/φ4 > 0), and (2) Navier-Stokes regularity follows from the boundedness of κ on a finite foam. We identify the uniqueness of the S/σ metric on S² geometry as the single remaining theorem required to complete both arguments.