Quantum Gravity from the Recognition Composition Law: Two Limits of a Single Operator, Zero Free Parameters

We show that quantum mechanics and general relativity emerge as complementary continuum limits of a single discrete operator R̂ derived from the Recognition Composition Law (RCL). The RCL uniquely determines a cost functional J(x) = ½(x + x⁻¹) − 1 on a discrete ledger whose self-similarity forces the golden ratio φ = (1 + √5)/2 as the fundamental scale. From φ alone: the Planck constant is ℏ = φ⁻⁵, Newton's constant is G = φ⁵/π, the Einstein gravitational coupling is κ = 8φ⁵, and their product satisfies the octave duality κℏ = 8. The high-frequency limit of R̂ yields the Schrödinger equation; the low-frequency limit yields the Einstein field equations with κ = 8πG. Black hole thermodynamics follows algebraically: S = πA/4, T_H = ℏ²/(8M), 2TS = M (Smarr), with πGℏ = 1 (Gauss–Bonnet unity). The 4D Lorentzian metric signature (−, +, +, +) is forced by the cost curvature J″(1) = 1 (spatial, positive) and the 8-tick recognition cycle (temporal, negative), giving spacetime dimension 1 + 3 = 4 with no alternative compatible with the forcing chain. The Yang–Mills mass gap Δ = J(φ) = (√5 − 2)/2 > 0 provides a natural UV completion. The framework has zero adjustable parameters: every constant, every structure, and every prediction derives from a single functional equation.