rs_exists_unique

formal source

  90theorem rs_exists_one : RSExists 1 := ⟨by norm_num, defect_at_one⟩
  91
  92/-- There exists exactly one RSExistent value. -/
  93theorem rs_exists_unique : ∃! x : ℝ, RSExists x := by
  94  use 1
  95  constructor
  96  · exact rs_exists_one
  97  · intro y hy
  98    exact (rs_exists_unique_one y).mp hy
  99
 100/-! ## Nothing Cannot RSExist -/
 101
 102/-- For any threshold, sufficiently small positive values have defect exceeding it.
 103    This means "approaching nothing" has unbounded cost. -/
 104theorem nothing_unbounded_defect :
 105    ∀ C : ℝ, ∃ ε > 0, ∀ x, 0 < x → x < ε → C < defect x :=
 106  nothing_cannot_exist
 107
 108/-- No value near zero is RSExistent.
 109    This is the operational content of "Nothing cannot recognize itself". -/
 110theorem nothing_not_rs_exists :
 111    ∃ ε > 0, ∀ x, 0 < x → x < ε → ¬RSExists x := by
 112  obtain ⟨ε, hε_pos, hε⟩ := nothing_unbounded_defect 1
 113  use ε, hε_pos
 114  intro x hx_pos hx_small ⟨_, hdef⟩
 115  have hC : 1 < defect x := hε x hx_pos hx_small
 116  rw [hdef] at hC
 117  linarith
 118
 119/-! ## RSTrue: Truth as Stabilized Recognition -/
 120
 121/-- A configuration-to-cost bridge: maps a configuration to the scalar
 122    cost-input via observable and scale maps relative to a reference. -/
 123structure CostBridge (C : Type*) where