IndisputableMonolith.CrossDomain.FibonacciPhiUniversality
This module collects algebraic identities and monotonicity results linking powers of the golden ratio φ to Fibonacci numbers, extending the base relation φ² = φ + 1 already established in Constants. Researchers modeling self-similar scaling across discrete and continuous domains in Recognition Science cite these lemmas when constructing the phi-ladder. The structure consists of direct computations for low powers, inductive arguments for general cases, and a final certification theorem.
claimThe module proves $\\phi^2 = \\phi + 1$ (reused from Constants), explicit values $\phi^3 = 2\\phi + 1$, $\phi^4 = 3\\phi + 2$, up to $\phi^8$, the relation $\phi^n = F_n \\phi + F_{n-1}$ for Fibonacci numbers $F_n$, strict monotonicity of the Fibonacci sequence, the bound $\phi^n$ bounded by Fibonacci terms, and the certification theorem FibonacciPhiCert asserting universality of the $\phi$-Fibonacci link.
background
Recognition Science derives φ as the unique positive fixed point of the self-similar map (T6) satisfying the quadratic equation already proved in Constants. The upstream Constants module supplies the RS-native time quantum τ₀ = 1 tick as the fundamental unit. This cross-domain module introduces no new primitives but assembles phi-power abbreviations and Fibonacci-sequence facts for later use in mass formulas and the phi-ladder.
proof idea
The module first defines the low-order phi powers and a fib_values table, then proves phi_pow_fib by direct expansion using the recurrence φ² = φ + 1. fib_strict_mono follows by induction on the Fibonacci recurrence; phi_pow_bounded_by_fib applies the same recurrence to obtain the inequality. The final FibonacciPhiCert and fibonacciPhiCert wrap the collection into a single certified statement.
why it matters in Recognition Science
These identities supply the discrete-to-continuous bridge required by the phi-ladder mass formula and the eight-tick octave (T7). Although no downstream declarations are listed, the results close the algebraic foundation for T6 self-similarity and prepare cross-domain applications that invoke the Recognition Composition Law. The module therefore sits between the Constants base and any later forcing-chain or mass-ladder constructions.
scope and limits
- Does not derive the Binet closed form or irrationality proofs for φ.
- Does not invoke the Recognition Composition Law or J-cost function.
- Does not address dimensional forcing (T8) or Berry threshold.
- Does not supply numerical approximations or alpha-band constraints.