Foundation Maximal Forcing Rsselection Example Positive Claim Independent

A claim that stays undecided

The golden ratio φ, about 1.618, is the positive solution to the equation r² = r + 1. That equation has a second solution, ψ = (1 − √5)/2, about −0.618. Both numbers satisfy the same algebraic constraint; they differ only in sign. The Recognition Science framework classifies claims about such ratios into three buckets: forced, selected, and independent. A claim is independent when the available constraints neither prove it nor disprove it, so the framework's machinery leaves it undecided.

The declaration positiveClaim_independent establishes exactly this for the claim "0 < r" (that the ratio is positive) over the class of numbers satisfying r² = r + 1 without any positivity requirement. The theorem proves this claim is independent: the positive root φ makes it true, the negative root ψ makes it false, and both roots are admissible under the constraint. So the framework cannot force positivity from the golden equation alone. This is a proved result in the machine-checked library of formal theorems, not a guess or a convention.

The same library also shows why this independence is not a dead end. A separate selection principle, named positivity, chooses the expanding root φ because a physical scale ratio must be greater than 1. Adding that principle as a tightening of the constraint promotes the claim from independent to forced. The framework thus demonstrates all three classification branches in one example: a trivial claim is forced, the phi claim is selected, and the positivity claim is independent. This completeness matters because it rules out a classifier that secretly always lands on one answer.

What the declaration does not claim is broader. It does not say positivity is unprovable in general, only that it is unprovable from the golden constraint alone. It does not assign any physical meaning to the numbers; the framework's own physical claims about scale ratios live elsewhere. And it does not say the independent status is permanent: the same library proves that adding the positivity principle resolves the claim. The independence is a precise statement about one constraint class, not a verdict on the golden ratio itself.

theorem positiveClaim_independent : Independent Lgolden.admissible positiveClaim := by
  refine ⟨φ, psi, ?_, ?_, ?_, ?_⟩
  · show satisfies_golden_constraint φ
    exact phi_equation
  · show satisfies_golden_constraint psi
    exact psi_golden
  · show (0 : ℝ) < φ
    exact phi_pos
  · intro h
    have hpos : (0 : ℝ) < psi := h
    have h5 : (1 : ℝ) < Real.sqrt 5 := by
      have hlt : Real.sqrt 1 < Real.sqrt 5 := by
        apply Real.sqrt_lt_sqrt <;> norm_num
      simpa using hlt
    have hneg : psi < 0 := by simp only [psi]; linarith
    linarith

/-- 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

/-- The conjugate root differs from phi (it is the contracting root). -/
theorem psi_ne_phi : psi ≠ φ := by
  have h5 : 0 < Real.sqrt 5 := Real.sqrt_pos.mpr (by norm_num)
  have hlt : psi < φ := by
    simp only [psi, φ]
    linarith [h5]
  exact ne_of_lt hlt

/-- **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 positivity is unprovable in general, only from the golden constraint alone. It does not assign physical meaning to the golden ratio or its conjugate root. It does not claim the independent status is permanent, since the library proves a resolution via the positivity principle.

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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