persistent_state_unique
Any persistent reference event has state exactly 1. Researchers tracing the observer-forcing chain cite this to fix the reference frame at the J-cost minimum. The proof is a one-line application of the Jcost zero equivalence lemma to the hypothesis that cost vanishes.
claimLet $ref$ be a recognition event with state $x > 0$. If $J(x) = 0$, then $x = 1$.
background
The ObserverForcing module shows that coherent recognition forces an observer by requiring a persistent reference frame. A recognition event is a positive real state under recognition whose J-cost is defined via the Recognition Composition Law. IsPersistent holds exactly when this cost is zero, because any positive cost would shift under context changes while zero is the unique global minimum. Upstream, Jcost_eq_zero_iff states that J(x) = 0 if and only if x = 1 for positive x.
proof idea
The term proof applies the forward direction of Jcost_eq_zero_iff to the hypothesis that the reference cost equals zero, directly yielding the state equality.
why it matters in Recognition Science
This supplies step 5 of the observer-forcing argument: the unique state with Jcost x = 0 is x = 1. It is invoked by observer_forcing_certificate to bundle the structural facts and by reference_unit_state to confirm an observer's reference sits at the identity tick. In the framework it anchors the persistent reference required for T5 J-uniqueness and the construction of the eight-tick octave.
scope and limits
- Does not establish existence of any persistent reference.
- Does not apply to states with non-positive values.
- Does not address coherence among multiple events beyond the reference frame.
Lean usage
theorem reference_unit_state (obs : Observer) : obs.recognition.reference.state = 1 := persistent_state_unique obs.recognition.reference obs.persistentformal statement (Lean)
97theorem persistent_state_unique
98 (ref : RecognitionEvent) (h : IsPersistent ref) :
99 ref.state = 1 :=
proof body
Term-mode proof.
100 (Cost.Jcost_eq_zero_iff ref.state ref.state_pos).mp h
101
102/-- Persistence is preserved under definitional substitution: any
103 persistent reference event has the same state as the canonical
104 identity event. -/