Foundation Maximal Forcing Rsselection Example Is Phi Not Forced Over Lgolden

The golden constraint alone

The golden ratio φ is the positive solution to r² = r + 1, the equation of extreme and mean ratio that Euclid described around 300 BC. The same quadratic has a second solution, ψ = (1 - √5)/2 ≈ -0.618. Both roots satisfy the equation; they differ in sign and in size. The positive root expands when squared, the negative root contracts. Classical accounts of the golden ratio usually mention this conjugate root only to discard it, because geometry and aesthetics work with lengths, and lengths are positive.

The Recognition Science framework formalizes this classical fact as a theorem about its own classification machinery. The framework's machine-checked library of formal theorems defines a class Lgolden of ratios that satisfy r² = r + 1 with no positivity requirement. Over that class, the claim "r = φ" is not forced: the conjugate root ψ is an admissible member of the class, it satisfies the same constraint, and it differs from φ. The theorem isPhi_not_forced_over_Lgolden states exactly this, that the golden ratio is not forced by the golden constraint alone.

In Recognition Science, this places the claim in the third branch of a trichotomy. A claim can be forced (true in every admissible realization), independent (true in some and false in others), or selected (not forced, but governed by a named principle). The phi claim over Lgolden is selected. The named principle is positivity: the physical scale ratio is the expanding root, greater than 1. That principle is not a theorem; it is a definitional choice the framework makes explicit. The framework then proves that adding positivity as a tightening promotes the claim from selected to forced, so the selected tag is a waystation, not a resting place.

The theorem's value is precision about what the golden constraint alone can and cannot do. It cannot single out φ; the equation admits two roots. The framework's own derivation of φ as the unique self-similar scaling therefore depends on the positivity selection, not on the algebraic constraint alone. The machine-checked theorem makes that dependence legible, and the same module shows all three classification branches realized in one universe, so the machinery is not secretly biased toward forcing.

/-- The conjugate root satisfies the golden constraint `r^2 = r + 1`. -/
theorem psi_golden : satisfies_golden_constraint psi := by
  unfold satisfies_golden_constraint psi
  have hs : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num)
  linear_combination (1 / 4 : ℝ) * hs

/-- "r = phi" is **not forced** over the golden-only class: the conjugate root is
an admissible counterexample. -/
theorem isPhi_not_forced_over_Lgolden : ¬ Forced Lgolden.admissible isPhiClaim := by
  intro hF
  have hpsi : psi = φ := hF psi psi_golden
  exact psi_ne_phi hpsi

/-- **"r = phi" is Selected over the golden-only class.** Not forced, but governed
by the positivity selection principle. This is the third branch of the trichotomy,
reached honestly. -/
theorem isPhi_selected_over_Lgolden : Selected Lgolden.admissible isPhiClaim :=
  ⟨isPhi_not_forced_over_Lgolden, ⟨positivitySelection⟩⟩

/-- **Drainage of the Selected tag.** Selected is not an endpoint: adopting the
positivity principle as a tightening (`Lgolden → LphiGold`) promotes the claim to
`Forced`. The promotion is exactly `forced_isPhi`. So this Selected entry has a
proved resolution, not a perpetual hold. -/
theorem positivity_promotes_selected_to_forced :
    Selected Lgolden.admissible isPhiClaim ∧
    Nonempty (Tightening Lgolden LphiGold) ∧
    Forced LphiGold.admissible isPhiClaim :=
  ⟨isPhi_selected_over_Lgolden, ⟨tighten_Lgolden_LphiGold⟩, forced_isPhi⟩

What this page does not claim

The theorem does not claim that φ is the only solution to r² = r + 1. The theorem does not claim that positivity is itself forced or derived. The theorem does not claim that the golden constraint alone determines any physical scale.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/MaximalForcing/RSSelectionExample.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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