phi_forcing_principle
Researchers tracing the forcing chain cite the result that self-similarity in a J-cost ledger forces the golden ratio φ to satisfy φ² = φ + 1 as its unique positive solution, with the bit cost and coherence energy positive. The principle occupies level 4 between ledger construction and dimension forcing. Its term proof assembles the equation, a uniqueness lemma, and two positivity facts without additional hypotheses.
claimLet φ = (1 + √5)/2. Then φ² = φ + 1, φ is the unique positive real solution to r² = r + 1, the minimum cost satisfies 0 < ln φ, and the coherence quantum satisfies 0 < φ^{-5}.
background
The PhiForcing module derives the golden ratio from self-similarity in a discrete ledger equipped with J-cost. J_bit denotes the minimum non-trivial cost, defined as the natural log of φ. E_coh is the coherence quantum given by φ to the minus five. From the module, self-similar scales satisfy the equation x² = x + 1, with φ the unique positive solution. The upstream phi_equation theorem states that φ² = φ + 1 by unfolding the definition and applying algebraic identities. Positivity lemmas for J_bit and E_coh are imported from Constants and related modules, relying on φ > 1.
proof idea
This is a term proof that builds the conjunction explicitly. The first conjunct comes from the phi_equation theorem in the PhiRing algebra. The uniqueness conjunct applies the golden_constraint_unique lemma to the hypothesis that r > 0 and r² = r + 1. The remaining conjuncts are the imported positivity lemmas J_bit_pos and E_coh_pos.
why it matters in Recognition Science
The declaration fills the φ step in the forcing chain from the Recognition Science primer, after J-uniqueness and before the eight-tick octave and D = 3. It connects the composition law to the self-similar fixed point. No parent theorems are listed in the used-by graph, indicating it serves as a foundational interface for constant matching and stellar assembly derivations.
scope and limits
- Does not construct the discrete ledger from the law of existence.
- Does not force the spatial dimension D = 3.
- Does not compute the fine structure constant alpha.
- Does not address the Berry creation threshold.
formal statement (Lean)
246theorem phi_forcing_principle :
247 (φ^2 = φ + 1) ∧ -- Golden equation
248 (∀ r : ℝ, r > 0 → r^2 = r + 1 → r = φ) ∧ -- Uniqueness
249 (0 < J_bit) ∧ -- Minimum cost positive
250 (0 < E_coh) -- Coherence quantum positive
251 := ⟨phi_equation,
proof body
Term-mode proof.
252 fun _ hr heq => golden_constraint_unique hr heq,
253 J_bit_pos, E_coh_pos⟩
254
255end PhiForcing
256end Foundation
257end IndisputableMonolith