defect_zero_implies_exists
Zero defect for positive real x implies x satisfies the existence predicate. Researchers formalizing the Law of Existence biconditional cite this direction to close the equivalence. The proof is a direct term construction that packages the two hypotheses into the Exists record.
claimIf $x > 0$ and defect$(x) = 0$, then $x$ exists, where the existence predicate is the structure requiring both positivity and vanishing defect.
background
The module formalizes the Law of Existence as the equivalence $x$ exists iff defect$(x) = 0$. The defect functional equals the J-cost, so defect$(x) := J(x)$. The Exists predicate is the structure with fields pos : $0 < x$ and defect_zero : defect$(x) = 0$, matching the CostAxioms definition of existence as $0 < x$ and $J(x) = 0$.
proof idea
The proof is a one-line term that constructs the Exists record by supplying the positivity hypothesis to the pos field and the defect-zero hypothesis to the defect_zero field.
why it matters in Recognition Science
This supplies the backward direction of the Law of Existence, supporting the biconditional law_of_existence and unity_unique_existent in the same module. It aligns with the framework claim that existence occurs precisely at cost minima, the golden-ratio fixed point. No open questions are addressed.
scope and limits
- Does not prove the forward direction from existence to zero defect.
- Does not evaluate or expand the defect functional beyond its definition as J.
- Does not address uniqueness or minimality properties of the existent.
formal statement (Lean)
76theorem defect_zero_implies_exists {x : ℝ} (hpos : 0 < x) (hdef : defect x = 0) :
77 Exists x := ⟨hpos, hdef⟩
proof body
Term-mode proof.
78
79/-- **Law of Existence (Biconditional)**: x exists ⟺ defect collapses. -/