config_classification
Every real configuration satisfies either stabilization (defect exactly zero) or lies outside the ontology (neither stabilizes nor diverges). Logicians and physicists studying cost-theoretic treatments of incompleteness theorems cite this trichotomy. The argument proceeds by case split on the stabilization predicate, then invokes the separate impossibility of unbounded real defects.
claimFor every real number $c$, either the defect of $c$ equals zero or $c$ is outside the ontology, meaning its defect is neither zero nor greater than every real bound.
background
The module recasts Gödel incompleteness inside Recognition Science by translating self-referential sentences into stabilization queries under cost minimization. Stabilization holds precisely when defect vanishes; divergence requires the defect to exceed every real number; outside the ontology means failure of both predicates. The local setting treats RS closure as the existence of a unique cost minimizer rather than arithmetic completeness, so Gödel sentences become non-configurations that oscillate without fixed point.
proof idea
Case analysis on whether stabilization holds for the input real. The affirmative case yields the left disjunct directly. The negative case combines the negation of stabilization with the upstream result that divergence is impossible for any real defect, since an unbounded defect would exceed its own value.
why it matters in Recognition Science
The classification supplies the exhaustive trichotomy required by the Gödel dissolution theorem, ruling out divergence for real configurations and forcing self-referential queries into the outside category. It thereby supports the paper claim that RS selection by cost is orthogonal to provability gaps. No downstream uses are recorded, but the result closes the logical space for diverging configurations in the foundation layer.
scope and limits
- Does not classify concrete configurations as stabilizing or outside.
- Does not extend the trichotomy beyond real numbers.
- Does not supply a decision procedure for the stabilization predicate.
- Does not address self-reference in non-RS formal systems.
formal statement (Lean)
193theorem config_classification (c : ℝ) : RSStab c ∨ RSOutside c := by
proof body
Tactic-mode proof.
194 by_cases hs : RSStab c
195 · left; exact hs
196 · right
197 exact ⟨hs, diverge_impossible c⟩
198
199/-! ## The Gödel Dissolution -/
200
201/-- **THE GÖDEL DISSOLUTION THEOREM**
202
203 The Gödel phenomenon is dissolved, not denied:
204 1. Self-referential stabilization queries are contradictory (can't exist)
205 2. Gödel sentences translate to such queries
206 3. Therefore Gödel sentences are "non-configurations" - outside the ontology
207
208 RS closure means "unique cost minimizer exists", not "all arithmetic true."
209 These are different claims about different targets.
210 Gödel constrains provability; RS constrains selection.
211 Orthogonal. -/