stable_to_recognition
This definition converts a J-stable structure (carrier with bounded-below cost) into a recognition-like structure whose relation equates elements of identical cost. Researchers tracing the cost-to-recognition forcing step cite it as the explicit bridge construction. The definition is a direct structure builder that inherits the carrier, sets the relation to cost equality, and verifies reflexivity and symmetry from equality axioms.
claimLet $S$ be a J-stable structure consisting of a carrier set $C$ and a real-valued cost function bounded below. The associated recognition-like structure has the same carrier $C$ and the relation $x R y$ if and only if the cost of $x$ equals the cost of $y$; this relation is reflexive and symmetric.
background
The Recognition Forcing module establishes that recognition structures are forced by cost foundations. A JStableStructure comprises a carrier type together with a cost map to the reals that admits a lower bound. RecognitionLikeStructure is the precursor object: a carrier equipped only with a reflexive and symmetric relation, without yet requiring transitivity or a recognition interpretation.
proof idea
The definition constructs the output structure by direct assignment: carrier is copied from the input, the relation is defined pointwise as equality of the input cost values, reflexivity is supplied by rfl on equality, and symmetry is supplied by the symmetry of equality. No external lemmas beyond core equality tactics are required.
why it matters in Recognition Science
This definition supplies the concrete map invoked by the downstream theorem stability_forces_recognition, which concludes that every J-stable structure yields a recognition-like structure on the same carrier. It occupies the explicit-construction slot in the module's progression from bounded cost to recognition, supporting the Recognition Science claim that recognition is forced by cost minima. The construction aligns with the broader forcing chain in which cost equality identifies recognition events.
scope and limits
- Does not establish transitivity of the induced relation.
- Does not require the cost function to attain its lower bound.
- Does not identify the cost with the J-cost or any specific functional form.
- Does not prove that the resulting relation corresponds to an observable extraction mechanism.
Lean usage
theorem stability_forces_recognition (S : JStableStructure) : ∃ (R : RecognitionLikeStructure), R.carrier = S.carrier := ⟨stable_to_recognition S, rfl⟩formal statement (Lean)
141def stable_to_recognition (S : JStableStructure) : RecognitionLikeStructure := {
proof body
Definition body.
142 carrier := S.carrier
143 rel := fun x y => S.cost x = S.cost y
144 refl := fun _ => rfl
145 symm := fun _ _ h => h.symm
146}
147