The Holographic Ledger: Black Hole Thermodynamics from the Recognition Composition Law

We derive the complete thermodynamics of Schwarzschild black holes—the Bekenstein–Hawking entropy, Hawking temperature, and the Smarr mass formula—from the Recognition Composition Law (RCL), the single functional equation that generates Recognition Science. The derivation uses zero free parameters. The key results are: 1. The Bekenstein–Hawking entropy is S = πA/4 in RS-native units, forced by Gℏ = 1/π. 2. The Octave-Entropy Identity: κℏ · S(A) = 2πA, equivalently 8S = 2πA, locks black hole information to the 8-tick recognition cycle. 3. Gauss–Bonnet Unity: πGℏ = 1, the identity that makes the Smarr formula work. 4. The Smarr formula: 2TS = M for Schwarzschild black holes, derived from (πGℏ)² = 1. 5. The "¼" in S = A/(4Gℏ) equals 1/2^(D−1) for D = 3, connecting the entropy formula to the dimension-forcing theorem. Every constant entering the derivation—ℏ, G, κ, and D—is derived from the golden ratio φ = (1 + √5)/2, which is itself forced by the RCL. No postulates from general relativity or quantum field theory are assumed.