Foundation Maximal Forcing Rsselection Example Is Phi Selected Over Lgolden

The selected claim

The golden ratio φ is the positive solution to r² = r + 1, approximately 1.618. The equation also has a negative solution, ψ = (1 − √5)/2 ≈ −0.618, which satisfies the same algebraic constraint. Both numbers solve the same equation, yet only one is the famous expanding ratio. The Recognition Science framework formalizes this situation with a precise classification: over the class of all real numbers satisfying r² = r + 1, the claim "r = φ" is not forced. The conjugate root ψ is a counterexample, and the framework proves this in its machine-checked library of formal theorems.

What the framework's declaration isPhi_selected_over_Lgolden establishes is that the claim "r = φ" is Selected over this golden-constraint class. Selected is the framework's honest interim tag: the claim is not forced by the constraint alone, but a named selection principle resolves it. That principle is positivity. The physical scale ratio is the expanding root, greater than 1, and φ is the only positive solution. The framework states this as a theorem with a proof that combines the non-forcing result with the positivity principle.

The declaration also shows that Selected is not a dead end. The framework proves that adopting positivity as a tightening, moving from the golden-only class to the class that also requires r > 0, promotes the claim to Forced. In that tightened class, "r = φ" is forced, and the framework proves this promotion explicitly. The same declaration thus demonstrates both the honest intermediate tag and its resolution, so the Selected branch is never a place a claim goes to rest.

In plain terms, the framework's library shows that the golden ratio is not uniquely picked out by its defining equation alone. A second root hides in the same equation. The framework's machinery classifies the claim as Selected, governed by positivity, and then shows how adding that positivity as a constraint upgrades the claim to Forced. This is a worked example of the framework's three-branch classification: forced, selected, and independent. The same module also exhibits a trivially forced claim and an independent claim, so all three branches appear in one closure.

/-- "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 declaration does not claim that φ is forced by the golden constraint alone. The declaration does not claim that ψ is physically meaningful or admissible as a scale ratio. The declaration does not claim that the golden ratio's appearance in the framework's cost function is derived from this selection example.

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.

Derived articles

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