Foundation Maximal Forcing Rsselection Example Trivial Claim Forced

The trivially forced claim

A claim in the Recognition Science framework is a statement about a realization, which is a concrete way the world could be. A claim is forced when it holds in every realization that meets the governing admissibility conditions. The declaration trivialClaim_forced establishes that the claim "True" is forced over the golden-constraint class, meaning it holds in every realization without exception. This is a tautology, a statement that is true by definition, and the proof is immediate: the claim's truth condition is simply True, so no realization can falsify it.

The golden-constraint class, denoted Lgolden, admits all real numbers r satisfying r² = r + 1, without any positivity requirement. This class has two members: the golden ratio φ = (1 + √5)/2 and its conjugate ψ = (1 − √5)/2. The trivial claim holds for both, and indeed for any realization whatsoever, because its truth condition is unconditional. The declaration proves this by the trivial tactic, which closes the goal by reflexivity of the truth predicate.

This result is part of a larger demonstration that the framework's classification machinery is complete and non-degenerate. The triUniverse collects three claims over the golden-only class: the trivial claim (forced), the phi claim (selected by positivity), and the positivity claim (independent, witnessed by the two roots). The declaration all_three_branches_realized proves that all three branches of the trichotomy are realized in a single closure, ruling out the failure mode where the classifier is secretly always-forced or always-independent.

What trivialClaim_forced does not claim is substantive. It does not assert that the golden ratio itself is forced over the golden-only class; that claim is false, because the conjugate root ψ also satisfies the constraint. It does not assert that the trivial claim carries any physical content; it is a logical placeholder that holds everywhere. It does not assert that the framework's classification of the phi claim as selected is itself forced; that classification depends on the named selection principle of positivity, which is a definitional choice, not a theorem.

theorem trivialClaim_forced : Forced Lgolden.admissible trivialClaim := by
  intro _ _; trivial

/-- The golden-constraint class, without positivity. This is strictly looser than
`LphiGold`, which also requires `0 < r`. -/
def Lgolden : AdmissibilityClass ℝ where
  admissible := { r | satisfies_golden_constraint r }
  label := "golden-constraint ratios (no positivity)"

/-- **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 golden ratio itself is forced over the golden-only class, since the conjugate root also satisfies the constraint. The trivial claim carries any physical content beyond being a logical tautology. The classification of the phi claim as selected is itself forced, since it depends on a definitional selection 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)

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