persistent_event_state_eq_identity
plain-language theorem explainer
Any persistent recognition event must occupy the same state as the canonical identity event. Researchers deriving observer emergence from coherent recognition structures cite this when fixing the reference frame at zero J-cost. The proof is a one-line rewrite via the persistent-state uniqueness lemma followed by reflexivity.
Claim. Let $ref$ be a recognition event. If $ref$ is persistent (i.e., its J-cost equals zero), then the state of $ref$ equals the state of the identity event, which is $1$.
background
A RecognitionEvent is a structure carrying a positive real state. IsPersistent is the predicate that the event's J-cost is zero; the module justifies this by noting that only a zero-cost reference remains invariant under arbitrary comparison contexts, since J-cost zero is the unique global minimum. The module's seven-step argument shows that any non-trivial coherent recognition structure forces an observer by attaching such a persistent reference frame. Upstream, the identity event is defined with state exactly 1, and Cost.Jcost_eq_zero_iff supplies the uniqueness of the zero-cost state.
proof idea
This is a term-mode proof. It rewrites the goal by applying the sibling lemma persistent_state_unique to the hypotheses ref and h, then closes with reflexivity.
why it matters
The declaration completes step 5 of the observer-forcing argument: the unique zero J-cost state is the identity tick. It supports the master theorem nontrivial_recognition_forces_observer that promotes any non-trivial recognition stream to an observer. Within the Recognition Science framework it instantiates the uniqueness property of the J-cost minimum, aligning with T5 J-uniqueness and the construction of persistent references via cooper pairing.
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