Foundation Maximal Forcing Rsselection Example All Three Branches Realized

Three branches, one universe

In the Recognition Science framework, every claim about reality is sorted into one of three buckets. A claim is forced if it holds in every admissible realization of the framework's axioms; it is independent if it holds in some realizations and fails in others; and it is selected if it is not forced but a named principle picks it out. The declaration all_three_branches_realized proves, in the framework's machine-checked library of formal theorems, that all three buckets are genuinely occupied within a single small universe of discourse.

The universe in question is the class of real numbers satisfying the golden constraint r² = r + 1, with no positivity requirement. Over this class, the framework proves three things simultaneously. The trivial claim "True" is forced, since it holds for every number. The claim "r = phi" (the positive golden ratio, about 1.618) is selected: it is not forced, because the conjugate root psi = (1 - √5)/2 ≈ -0.618 also satisfies the same constraint and differs from phi, but a named selection principle, positivity, picks the expanding root. The claim "0 < r" is independent, since phi satisfies it and psi does not.

The theorem's real content is that the classification machinery is complete and non-degenerate. It rules out the failure mode where the classifier secretly labels everything forced or everything independent. The framework's library shows that a single closure can land a claim in any of the three buckets, with proofs, and that the selected bucket is not a place where claims go to die. The selection principle can be adopted as a tightening of the admissible class, which promotes the selected claim to forced. In this example, adding positivity to the golden constraint turns "r = phi" from selected into forced, so the selected tag always carries a proved resolution rather than a perpetual hold.

What the declaration does not claim is equally precise. It does not assert that the golden ratio is forced in general; over the golden-only class, it is explicitly not forced. It does not claim that the selection principle is itself derived from the framework's axioms; positivity is a named principle, not a proved consequence. And it does not claim that the three-branch classification applies to any particular physical system; the example is a formal demonstration within the framework's own language, not an empirical statement about the world.

/-- **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⟩

/-- **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⟩

/-- **"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 golden ratio is forced in general over the golden-only class. The positivity selection principle is derived from the framework's axioms. The three-branch classification applies to any particular physical system.

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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