Black Hole Thermodynamics from the Fibonacci Cascade: Hawking Radiation as the Bilateral Born Rule

Abstract

We apply the Fibonacci Cascade Conformal Layer (FCCL) to black hole thermodynamics. The Hawking temperature maps exactly to a cascade level: 𝑛_𝐻(𝑀) = ln(8𝜋𝑀/𝑚𝑃) / ln(𝜑). A universal offset 𝑛_𝐻 −𝑛_𝑆 = ln(4𝜋)/ ln 𝜑 = 5.26 separates the Hawking cascade level from the Schwarzschild cascade level for every black hole, regardless of mass. The Hawking spectrum is identified as the bilateral Born rule 𝑃(𝑛|𝑛𝐻) = 𝜑^(−2|𝑛−𝑛_𝐻|)/ √5 centered on 𝑛𝐻 — quantum probabilities and black hole radiation share the same Fibonacci structure. The black hole information paradox is resolved by the bilateral cascade structure: information enters in the 𝜉+ (matter) direction and is returned in the 𝜉− (Hawking radiation) direction, with the normalization 𝑍 = 𝑍₊ + 𝑍₋ = 𝜑 + 1/𝜑 = √5 exactly conserved. The Page time corresponds to 𝑛_𝐻 → 𝑛_𝐻/2, giving 𝑀_Page = √[𝑀₀𝑚_𝑃 /(8𝜋)] — the geometric mean of initial mass and Planck mass. The Bekenstein-Hawking entropy is recovered as 𝑆𝐵𝐻 ≈ 𝜑^(2𝑛_𝑆) × 𝑆_Born where 𝑆_Born = 1.666 nats is the Shannon entropy of the Born rule. A black hole of mass 𝑀 and an elementary particle of the same energy are FCCL solitons at cascade levels 𝑛 ≈ 0 and 𝑛 ≈ 𝑛_𝐻 respectively — dual objects in the Fibonacci cascade hierarchy.