possibility_is_not_singleton
The theorem establishes that possibility in Recognition Science is not confined to a single value by exhibiting two distinct positive reals. Modal logicians or metaphysicians working in the RS framework would cite it to confirm multiplicity within the positive-ratio class. The proof is a direct term construction that supplies witnesses 1 and 2 together with their inequality and positivity.
claimThere exist distinct real numbers $x$ and $y$ such that $x > 0$ and $y > 0$.
background
In module PH-013 the Recognition Science resolution of modal metaphysics defines RSPossible(x) to mean x > 0, i.e., positive ratio and therefore finite J-cost. Necessity is reserved for the unique J-minimizer at x = 1 while every positive ratio counts as possible. The module imports cost definitions from ObserverForcing and MultiplicativeRecognizerL4 that supply the underlying J-cost function.
proof idea
The proof is a term-mode construction that directly supplies the witnesses 1 and 2. It applies norm_num to discharge the inequality and invokes the positivity facts one_pos and two_pos to confirm both values satisfy RSPossible.
why it matters in Recognition Science
The result fills the non-singleton clause required by the RS modal resolution in PH-013, ensuring the possibility predicate is non-degenerate once necessity is fixed at the unique J-minimum. It supports downstream claims such as s5_actuality_implies_possibility by guaranteeing multiple accessible positive-ratio configurations. The theorem thereby closes one structural gap in the complete modal certificate.
scope and limits
- Does not claim that every positive real is reachable by a J-cost decreasing path.
- Does not equate possibility with finite J-cost beyond the ratio definition.
- Does not address the accessibility relation or S5 axioms directly.
formal statement (Lean)
101theorem possibility_is_not_singleton :
102 ∃ x y : ℝ, x ≠ y ∧ RSPossible x ∧ RSPossible y :=
proof body
Term-mode proof.
103 ⟨1, 2, by norm_num, one_pos, two_pos⟩
104
105/-! ## III. Actuality = J-Minimum -/
106
107/-- **Theorem (Actuality is J-Minimum)**:
108 The actual world is the J-cost minimum.
109 x = 1 is actual; all other positive x are merely possible. -/