cooper_pairing_yields_persistent
plain-language theorem explainer
Cooper pairing of any positive real x with its inverse yields a recognition event whose J-cost is exactly zero, hence persistent by definition. Researchers tracing the emergence of observers in Recognition Science cite this to establish that persistence is forced by the algebra of the J-function without presupposing a unit state. The proof is a direct construction that builds the event from the paired state and invokes the zero-cost lemma for that state.
Claim. For every positive real number $x$, there exists a recognition event $e$ (a positive real state) such that the J-cost of $e$ equals zero.
background
A recognition event is a positive real state equipped with a positivity witness. IsPersistent holds precisely when that event's J-cost vanishes; the justification is that only the global minimum cost remains invariant under arbitrary comparison contexts. The module sets out seven steps showing how coherent recognition forces an observer, with Cooper pairing supplying a structural source of zero-cost references even when no event sits exactly at the identity tick. The upstream cooper_pair_cost_zero theorem states that Jcost(x · x^{-1}) = 0 for any positive x, obtained by reduction to the unit case Jcost(1) = 0.
proof idea
The tactic proof constructs a RecognitionEvent whose state is x · x^{-1}, discharges the positivity obligation by mul_inv_cancel₀ followed by norm_num, then applies cooper_pair_cost_zero directly to obtain the required IsPersistent witness.
why it matters
This result supplies one of the six facts bundled in the observer_forcing_certificate and is invoked inside cooper_paired_reference_yields_observer to build an observer from an arbitrary positive state rather than the canonical identity. It closes step 6 of the module's forcing chain: Cooper pairing provides the persistent reference needed to promote coherent recognition to an observer. The construction aligns with the Recognition Composition Law and the uniqueness of the J-cost minimum at the identity tick.
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