SelfRefQuery
A real configuration c qualifies as a self-referential stabilization query precisely when its defect satisfies defect(c) = 0 if and only if defect(c) ≠ 0. Researchers formalizing cost-theoretic resolutions of Gödel incompleteness in Recognition Science cite this structure to encode self-reference as an assertion about non-stabilization. The definition is a direct structure that packages the real number and the contradictory biconditional via the defect functional.
claimA real number $c$ is a self-referential stabilization query when its defect satisfies $defect(c) = 0$ if and only if $defect(c) ≠ 0$.
background
The Gödel Dissolution module translates Gödel sentences into Recognition Science by modeling them as self-referential stabilization queries under cost minimization. The defect functional is defined as defect(x) := J(x), where J is the J-cost from the Law of Existence; Stabilizes(c) holds exactly when defect(c) = 0, i.e., when iterated dynamics converge to zero defect. Diverges(c) is the complementary predicate that defect(c) exceeds every bound.
proof idea
This is a structure definition with no proof body. It directly asserts the configuration field and the self_ref field carrying the biconditional (defect config = 0) ↔ ¬(defect config = 0). No lemmas or tactics are invoked.
why it matters in Recognition Science
SelfRefQuery supplies the contradictory object used by GodelDissolutionTheorem and complete_godel_dissolution to conclude that Gödel sentences become non-configurations outside the RS ontology. It supports the paper claim that RS closure requires only a unique cost minimizer, not arithmetic completeness, and aligns with the forcing chain by treating stabilization as convergence under the phi-ladder and eight-tick dynamics.
scope and limits
- Does not claim that every self-referential sentence is contradictory.
- Does not address provability or arithmetic truth directly.
- Does not model non-stabilization queries outside the defect functional.
- Does not resolve halting or other undecidability results.
formal statement (Lean)
73structure SelfRefQuery where
74 /-- The configuration -/
75 config : ℝ
76 /-- The self-referential property: c encodes "I don't stabilize" -/
77 self_ref : (defect config = 0) ↔ ¬(defect config = 0)
78
79/-- **THEOREM 1**: Self-referential stabilization queries are contradictory.
80
81 If c encodes "c ⟺ ¬Stab(c)", then assuming c has any definite
82 stabilization status leads to contradiction.
83
84 This is the heart of the Gödel dissolution: such c cannot exist
85 as consistent configurations. -/