possibility_is_not_singleton

formal source

  98  ⟨defect x, defect_nonneg h, le_refl _⟩
  99
 100/-- **Theorem**: There are infinitely many possible things. -/
 101theorem possibility_is_not_singleton :
 102    ∃ x y : ℝ, x ≠ y ∧ RSPossible x ∧ RSPossible y :=
 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. -/
 110theorem actuality_is_j_minimum :
 111    -- x = 1 is actual
 112    RSActual 1 ∧
 113    -- All other positive reals are merely possible (positive defect)
 114    (∀ x : ℝ, 0 < x → x ≠ 1 → ¬RSActual x) := by
 115  constructor
 116  · exact ⟨by norm_num, defect_at_one⟩
 117  · intro x hx hne ⟨_, hzero⟩
 118    exact hne ((defect_zero_iff_one hx).mp hzero)
 119
 120/-- **Theorem**: The actual world is unique — only one thing is actual. -/
 121theorem actuality_is_unique : ∃! x : ℝ, RSActual x :=
 122  necessity_is_unique_minimizer
 123
 124/-! ## IV. Impossibility = Non-Positive Ratio -/
 125
 126/-- **Theorem (Impossibility)**:
 127    Something is "impossible" iff it has non-positive ratio.
 128    Negative or zero ratios violate the positivity constraint of the recognition ledger. -/
 129theorem impossible_is_non_positive :
 130    ∀ x : ℝ, RSImpossible x ↔ ¬RSPossible x := by
 131  intro x