Foundation Maximal Forcing Rsselection Example Positivity Promotes Selected To F

The promotion of a selected claim

The golden ratio φ is the positive solution of r² = r + 1. The equation also has a negative solution, ψ = (1 − √5)/2. If one only knows that a ratio satisfies r² = r + 1, the value is not determined: ψ is a perfectly good counterexample. The framework's ledger, a discrete record of recognition events, distinguishes three ways a claim can stand: forced, meaning it holds in every admissible case; independent, meaning it holds in some and fails in others; and selected, meaning it is not forced but is governed by a named principle that picks one outcome.

The declaration positivity_promotes_selected_to_forced establishes a precise relationship between these categories. It proves that the claim "r = φ" is selected over the golden-only class: it is not forced, because ψ satisfies the constraint, but a selection principle applies, namely positivity, which states that the physical scale ratio is the expanding root greater than 1. The declaration then proves that adopting positivity as a tightening of the admissible class promotes the claim to forced. Once the class is restricted to ratios that also satisfy 0 < r, the claim "r = φ" holds in every admissible case. The theorem states all three facts together: selected over the loose class, a tightening exists, and forced over the tightened class.

This is a theorem about the framework's classification machinery, not a physical claim. It does not assert that the golden ratio is the only possible scale ratio in nature, nor that positivity is the only selection principle one could name. It does not claim that the selection principle is derived from deeper axioms; the principle is adopted as a definitional choice. What the declaration shows is that the selected tag is not a dead end: a selected claim can be promoted to forced by making its governing principle explicit as a tightening of the admissible class. The proof is machine-checked, meaning the logical steps are verified by a computer, and it uses no framework-specific axioms beyond the standard ones of the ambient type theory.

The practical consequence is that the framework's three-way classification is not a permanent labeling. A claim sitting in the selected bucket is not stuck there; it carries within it the seed of its own resolution. The declaration demonstrates this by example, using the golden ratio as the concrete case. It also shows, in the same module, that the classification machinery is complete: a single universe can realize all three branches, with one claim forced, one selected, and one independent. This rules out the possibility that the classifier secretly always lands on one branch.

/-- **"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⟩

/-- **The maximal-forcing machinery is complete and non-degenerate.** A single
closure realizes all three branches of the trichotomy with proofs: one forced, one
selected, one independent. This rules out the failure mode where the classifier is
secretly always-forced or always-independent. -/
theorem all_three_branches_realized :
    Forced Lgolden.admissible trivialClaim ∧
    Selected Lgolden.admissible isPhiClaim ∧
    Independent Lgolden.admissible positiveClaim :=
  ⟨trivialClaim_forced, isPhi_selected_over_Lgolden, positiveClaim_independent⟩

What this page does not claim

The declaration does not claim that the golden ratio is the only possible scale ratio in nature. The declaration does not claim that positivity is derived from the framework's cost function. The declaration does not claim that every selected claim can be promoted to forced.

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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