IndisputableMonolith.CrossDomain.FibonacciPhiUniversality
This module establishes the universality of the golden ratio φ in linking Fibonacci sequences to powers of φ within Recognition Science. It supplies definitions for successive φ powers and Fibonacci values, plus lemmas proving their exact relations and bounds, all resting on the base identity φ² = φ + 1. Number theorists and physicists modeling discrete structures would cite it when constructing the phi-ladder for mass formulas. The module proceeds via direct algebraic definitions followed by inductive or computational lemmas that verify the φ–
claimThe golden ratio satisfies φ² = φ + 1. Higher powers obey φ^n = F_n φ + F_{n-1} where F_n denotes the nth Fibonacci number, with strict monotonicity of F_n and the bound φ^n bounded by F_n, certified universally across domains.
background
Recognition Science derives φ as the self-similar fixed point (T6) from the J-uniqueness relation in the forcing chain. The module imports the RS time quantum τ₀ = 1 tick from Constants and begins from the already-proved identity φ² = φ + 1. It introduces named objects phi_sq, phi_cubed, phi_fourth, phi_fifth, phi_eighth for explicit powers; fib_values for the Fibonacci sequence; phi_pow_fib for the linking identity; fib_strict_mono for monotonicity; phi_pow_bounded_by_fib for the bounding relation; and FibonacciPhiCert as the top-level certification object.
proof idea
This is a definition module. Successive powers are defined algebraically from the base relation φ² = φ + 1. Fibonacci values are introduced by standard recurrence. The core identity phi_pow_fib is established by direct computation or induction on n. Monotonicity and bounding lemmas follow by algebraic comparison or induction, using only the imported Constants relation and Mathlib arithmetic.
why it matters in Recognition Science
The module supplies the algebraic foundation for the phi-ladder appearing in mass formulas (yardstick · φ^(rung − 8 + gap(Z))). It supports the T6 forcing of φ as self-similar fixed point and enables cross-domain applications that later yield hbar = φ^{-5} and G = φ^5 / π. No parent theorems are listed in the used-by graph, yet the relations close the discrete universality step required by the overall Recognition framework.