cost_to_recognition_bridge
plain-language theorem explainer
The theorem assembles J-symmetry of the cost function, existence of a zero-cost self-recognition event, extraction of recognition structures from any observable mechanism, and carrier-preserving recognition-like structures from J-stable carriers. A researcher deriving observer emergence from ledger costs would cite it to close the cost-to-recognition link. The proof is a single term that packages four independent facts: the LedgerForcing symmetry lemma, the global-minimum self-recognition result, the extraction functor, and the stability-forces
Claim. Let $J$ be the cost function on nonzero reals. Then $J(x)=J(x^{-1})$ for all $x≠0$, there exists a recognition event of ratio 1 with zero cost, every observable extraction mechanism on a type $S$ yields a recognition structure on $S$, and every $J$-stable structure (a carrier equipped with a bounded-below cost) admits a recognition-like structure on the same carrier.
background
The module establishes that recognition structures are forced by the underlying cost foundation. Key local definitions: ObservableExtractionMechanism(S) consists of a map extract:S→ℝ that is non-constant; RecognitionStructure(S) equips S with a reflexive symmetric relation recognizes; JStableStructure is a carrier type together with a cost function bounded below; RecognitionLikeStructure is the corresponding reflexive-symmetric relation on a carrier. RecognitionEvent is a triple (source,target,ratio) with ratio>0. Upstream, LedgerForcing.J_symmetric states J(x)=J(x^{-1}) for x≠0, while the global-minimum lemma (global_minimum_is_self_recognition) supplies the zero-cost event at ratio 1.
proof idea
Term-mode construction of a four-component conjunction. The first component is the function that applies LedgerForcing.J_symmetric; the second is the global_minimum_is_self_recognition lemma; the third is the lambda that sends any extraction mechanism M to the pair (recognition_from_extraction M, trivial); the fourth is the stability_forces_recognition lemma. No further tactics are used.
why it matters
This bridge theorem sits inside the Recognition Forcing module and supplies the cost-to-structure direction of the master theorem recognition_forcing_complete. It therefore participates in the chain that derives recognition events from the J-cost axioms (T5 J-uniqueness and T6 phi fixed point). Because the module proves recognition is forced by cost, the result supports the larger claim that observer structures emerge necessarily from the single functional equation without additional postulates.
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