The Forced Measure Golden-Ratio Gibbs Weights are the Unique Instance Weighting of a Self-Similar Ledger

The recognition forcing chain determines the shape of physical law: a unique convex comparison cost, a unique self-similar scale (the golden ratio φ), a minimal period of eight ticks, and three spatial dimensions. It does not say how much of reality occupies each allowed state. This paper closes that gap. On the discrete grading of recognition steps I impose two requirements: the weight of an independent composite is the product of the weights, and the single-step weight is a fixed point of the ledger's own reciprocal balance, ρ = 1/(1 + ρ). The unique solution is w(n) = φ⁻ⁿ. In the continuum, any factorizing antitone weight calibrated at f(1) = φ⁻¹ equals φ⁻ᵗ for every nonnegative real t; monotonicity alone excludes the pathological solutions of the underlying Cauchy equation, with no continuity or measurability assumption. Equivalently the weight is a Gibbs law e^(−t ln φ) whose temperature is forced, (ln φ)⁻¹ in step units, where maximum-entropy derivations leave the temperature as a multiplier fixed by data. The normalized measure is geometric with partition function exactly φ², ground share φ⁻², mean recognition depth exactly φ, variance φ³, and Shannon entropy (2 + φ) ln φ. Constants derived separately in the framework, including ℏ = φ⁻⁵ in recognition-native units, the dark-energy occupancy φ⁻⁴, the rung-44 baryon scale φ⁻⁴⁴, and the cosmological redshift kernel 1/(1 + z), are exhibited as values of this single measure. Two consequences follow at once. The measure is cost-blind, so chirality selection cannot come from it. And the dark-energy deviation amplitude collapses from a free real parameter to one integer, giving the dated equilibrium prediction w₀ ∈ [−0.884, −0.882) with w_a = −(1 + w₀), testable against DESI, Roman, and Euclid. I state exactly what is theorem, what is physical premise, and what remains open: the dynamical H-theorem and the Born bridge.