stability_forces_recognition
Any J-stable structure (a type with a real-valued cost function bounded below) forces existence of a recognition-like structure on the identical carrier. Researchers assembling the recognition forcing chain cite this lemma when building the master theorem. The proof is a one-line wrapper that invokes the stable_to_recognition construction and closes by reflexivity on carriers.
claimLet $S$ be a J-stable structure: a type equipped with a cost function $c$ to the reals that is bounded from below. Then there exists a recognition-like structure $R$ (a type with a reflexive symmetric relation) such that the carrier of $R$ equals the carrier of $S$.
background
The RecognitionForcing module proves that recognition structures arise necessarily from cost foundations. JStableStructure packages a carrier type with a cost map to reals together with the assertion that the cost admits a finite lower bound. RecognitionLikeStructure packages the same carrier with a reflexive and symmetric binary relation. The upstream construction stable_to_recognition turns any JStableStructure into a RecognitionLikeStructure by declaring two elements related precisely when their costs coincide.
proof idea
One-line wrapper that applies stable_to_recognition to the input JStableStructure and uses reflexivity to witness the carrier equality.
why it matters in Recognition Science
The theorem is invoked by cost_to_recognition_bridge and by the master theorem recognition_forcing_complete. It supplies the direct step from bounded-cost structures to recognition-like structures, consistent with the Recognition Science claim that cost minima coincide with recognition events. No open scaffolding questions are closed by this declaration.
scope and limits
- Does not establish transitivity of the induced relation.
- Does not prove uniqueness of the recognition-like structure.
- Does not derive numerical constants or dimensional constraints.
- Does not address extraction mechanisms or observability beyond cost equality.
formal statement (Lean)
148theorem stability_forces_recognition (S : JStableStructure) :
149 ∃ (R : RecognitionLikeStructure), R.carrier = S.carrier :=
proof body
Term-mode proof.
150 ⟨stable_to_recognition S, rfl⟩
151
152/-! ## Part 4: Master Theorem -/
153