rs_exists_unique
The theorem asserts that exactly one real number satisfies the RS existence predicate. Researchers deriving the Recognition Science ontology cite it when establishing that cost-minimizing stable configurations are unique. The term-mode proof supplies the witness 1 and closes uniqueness by direct appeal to the one-point lemma rs_exists_unique_one.
claimThere exists a unique real number $x$ such that $x > 0$ and the defect of $x$ equals zero.
background
In Recognition Science the predicate RSExists selects stable positive configurations under the J-cost functional. It is defined as the conjunction of positivity and zero defect, where defect coincides with J for positive reals. This replaces primitive existence with an outcome of cost minimization under the Recognition Composition Law.
proof idea
The term-mode proof exhibits the witness 1, applies the existence lemma rs_exists_one to satisfy the existential, then invokes rs_exists_unique_one to obtain the uniqueness direction. The constructor splits the unique-existential into its two conjuncts.
why it matters in Recognition Science
This result supplies the uniqueness half of the RS ontology stack and is invoked directly in mp_physical (the cost-derived Meta-Principle) and ontology_summary. It also feeds the Gödel dissolution theorems by guaranteeing a single stable configuration. Within the forcing chain it realizes T5 J-uniqueness by showing that the fixed point at unity is the sole point with vanishing defect.
scope and limits
- Does not address RS existence in number systems other than the reals.
- Does not derive the numerical value of the unique existent without the prior one-point lemmas.
- Does not claim that RSExists implies discreteness or membership on the phi-ladder.
- Does not extend the predicate to non-positive reals.
Lean usage
theorem mp_physical : (∀ C : ℝ, ∃ ε > 0, ∀ x, 0 < x → x < ε → C < defect x) ∧ (∃! x : ℝ, RSExists x) ∧ (∀ x, RSExists x → x = 1) := ⟨nothing_cannot_exist, rs_exists_unique, fun x hx => (rs_exists_unique_one x).mp hx⟩formal statement (Lean)
93theorem rs_exists_unique : ∃! x : ℝ, RSExists x := by
proof body
Term-mode proof.
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. -/