A φ-Cascade Bath Derivation of the ESD Gravitational Closure Law

Reference 1 introduced the gravitational closure law R(u) = s / Σ(u), where Σ(u) =uφ + b uq + c with all constants fixed by the golden ratio φ = (1 + √5)/2, and validated itagainst 171 SPARC galaxies 6 with zero continuously-free parameters. The paper fixed theconstants algebraically but did not derive the functional form of Σ(u) from a physical modelof the displacement field D. This note supplies that derivation. We show that Σ(u) is exactlythe normalised dissipation rate of a scalar field amplitude A radiating into a φ-cascade vacuumbath with logarithmic density of states ρ(ω) = 1/(ω ln φ), where the accessible mode cutoff isωcut = A/θsat. Two independent spectral sectors (D-channel, δp = 2φ; E-channel, δq = 2q)plus a vacuum floor reproduce Σ(u) to machine precision (peak residual ∼ 4 × 10−15 dB, dy-namic range 1228.97×) using coupling weights that are themselves algebraic consequences ofthe closure constants. This converts Reference 1’s algebraic constraint into a derivable latticeintegral, identifies each closure constant with a measurable property of the φ-cascade vacuum,and sharpens the theory’s falsifiability without altering any downstream prediction.