phi_satisfies_fixed_point
The golden ratio satisfies the quadratic fixed-point equation φ² = φ + 1 required by the Recognition cost function. Ledger-uniqueness arguments cite this identity to anchor the specific choice of φ inside the discrete conservative system. The term-mode proof unfolds the explicit definition of φ and reduces the equation by squaring the square root of 5, followed by ring normalization and linear arithmetic.
claimLet φ = (1 + √5)/2. Then φ² = φ + 1.
background
The Ledger Uniqueness module resolves the objection that other discrete structures could satisfy discreteness and conservation by proving each component is the unique solution to its forcing constraint. For φ the constraint is the cost-function fixed point J(x) = J(1/x) with minimal curvature; the positive solution is the golden ratio. This sits inside the forcing chain where T6 identifies φ as the self-similar fixed point of the Recognition Composition Law.
proof idea
The proof is a short term-mode calculation. It unfolds the definition of phi, records the auxiliary equality (√5)² = 5, applies ring normalization to expand the left-hand side, and closes the goal with linear arithmetic on the resulting polynomial identity.
why it matters in Recognition Science
This supplies the algebraic verification that φ meets the fixed-point equation demanded by the cost function. It supports the module's main claim that any discrete conservative system is isomorphic to the RS ledger built from φ, Q₃ and the 8-tick cycle. The result directly instantiates the T6 step of the forcing chain; the companion theorem phi_unique_fixed_point then proves uniqueness of this solution.
scope and limits
- Does not prove that φ is the unique positive solution to the equation.
- Does not derive the cost function J from the Recognition Composition Law.
- Does not address the topological linking or Gray-code cycle arguments in the same module.
- Does not apply to negative or non-real solutions.
formal statement (Lean)
57theorem phi_satisfies_fixed_point : phi^2 = phi + 1 := by
proof body
Term-mode proof.
58 unfold phi
59 have h : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num : (5 : ℝ) ≥ 0)
60 ring_nf
61 linarith [h]
62
63/-- φ is the unique positive solution to x² = x + 1. -/