DefectCollapse
DefectCollapse(x) asserts that a positive real x has zero defect under the J-cost functional. Researchers formalizing the Law of Existence cite this predicate as the right-hand side of the biconditional linking existence to defect collapse. It is introduced as a direct abbreviation that conjoins the positivity guard with the zero-defect condition.
claimFor a real number $x$, the predicate DefectCollapse($x$) holds when $x > 0$ and defect($x$) = 0, where defect coincides with the J-cost functional.
background
The Law of Existence module supplies a literal formalization of the statement that x exists if and only if defect(x) = 0. The defect functional is defined in the same module as defect(x) := J(x) for positive reals, with J the cost function obeying the Recognition Composition Law. Upstream results include the toy Defect from CPM.LNALBridge, which returns zero when structural checks pass, and the self-reference structure from UniversalForcingSelfReference that records orbit and coherence axioms for meta-realization.
proof idea
This is a direct definition that conjoins the strict positivity condition 0 < x with the equality defect x = 0. No lemmas are applied; the declaration serves as an abbreviation for the conjunction that appears in the subsequent Law of Existence theorem.
why it matters in Recognition Science
DefectCollapse supplies the target predicate for the Law of Existence biconditional theorem, which states that x exists if and only if DefectCollapse x holds. It fills the right-hand side of the core equivalence x exists ⟺ defect(x) = 0 stated in the module documentation. Within the Recognition Science framework the predicate encodes collapse to unity under the J-cost, consistent with the forcing chain that derives the eight-tick octave and D = 3.
scope and limits
- Does not establish the equivalence between defect zero and x equaling one.
- Does not compute or approximate the numerical value of the defect functional.
- Does not address negative or zero arguments for x.
- Does not incorporate the full self-reference axioms from the imported structure.
Lean usage
theorem law_of_existence (x : ℝ) : Exists x ↔ DefectCollapse x := ⟨fun ⟨hpos, hdef⟩ => ⟨hpos, hdef⟩, fun ⟨hpos, hdef⟩ => ⟨hpos, hdef⟩⟩formal statement (Lean)
53def DefectCollapse (x : ℝ) : Prop := 0 < x ∧ defect x = 0
proof body
Definition body.
54
55/-! ## Core Equivalence Theorems -/
56
57/-- **Defect Zero Characterization**: defect(x) = 0 ⟺ x = 1 (for x > 0). -/