phi_unique_self_similar
Any positive real r obeying the golden constraint r² = r + 1 equals the golden ratio φ. Modelers of self-similar discrete ledgers with J-cost cite the result to fix scale ratios. The proof is a direct one-line application of the uniqueness theorem for the golden constraint.
claimLet r be a positive real number. If r satisfies the equation r² = r + 1, then r equals the golden ratio φ = (1 + √5)/2.
background
The Phi Forcing module derives the golden ratio from self-similarity of a discrete ledger equipped with J-cost. The predicate satisfies_golden_constraint r is defined to hold exactly when r² = r + 1; the module documentation states that this algebraic relation arises because scaling by r must be composable in the ledger, yielding the Fibonacci-like identity r² = r + 1. The upstream theorem golden_constraint_unique proves that the only positive solution is φ by solving the quadratic and discarding the negative root.
proof idea
The declaration is a one-line term-mode wrapper that applies the theorem golden_constraint_unique to the hypotheses 0 < r and satisfies_golden_constraint r.
why it matters in Recognition Science
The result supplies the unique positive solution required by the module's forcing chain, completing the step from self-similar ledger to φ as stated in the module documentation. It is invoked by phi_harmonic_forced and the corresponding uniqueness theorem in FrequencyLadder. In the Recognition Science framework it realizes the self-similar fixed point (T6) that later forces the eight-tick octave and D = 3.
scope and limits
- Does not prove existence of any solution to r² = r + 1.
- Does not address negative or complex scale ratios.
- Does not derive the golden constraint from ledger axioms.
- Does not quantify stability under small perturbations of r.
formal statement (Lean)
186theorem phi_unique_self_similar {r : ℝ} (hr_pos : 0 < r) (hr_eq : satisfies_golden_constraint r) :
187 r = φ :=
proof body
Term-mode proof.
188 golden_constraint_unique hr_pos hr_eq
189
190/-! ## Discrete Ledger with Self-Similarity -/
191
192/-- A discrete ledger structure (from LedgerForcing). -/