cooper_pairing_yields_persistent
Cooper pairing of any positive real x with its inverse produces a recognition event whose J-cost is zero, hence persistent. Researchers deriving observer emergence from coherent recognition structures cite this to obtain a reference frame without presupposing the identity tick. The proof constructs the event state explicitly and invokes the zero-cost property of the paired state.
claimFor any real number $x > 0$, there exists a recognition event $e$ (a positive real state) such that the J-cost of $e$ equals zero.
background
In the Observer-Forcing module a recognition event is a positive real state whose J-cost is well-defined and non-negative. IsPersistent holds exactly when this cost vanishes, justified because any positive cost would shift under context changes while zero is the global minimum. The upstream cooper_pair_cost_zero theorem shows that the state $x · x^{-1}$ collapses to J-cost zero for any $x > 0$, supplying persistence even when no event sits directly at the identity tick 1. The module's seven-step argument uses this to promote coherent recognition to an observer by attaching a persistent reference.
proof idea
The tactic proof constructs a RecognitionEvent record whose state field is $x · x^{-1}$, proves positivity by rewriting with mul_inv_cancel₀ and norm_num, then applies the cooper_pair_cost_zero theorem to obtain J-cost zero and therefore IsPersistent.
why it matters in Recognition Science
This theorem supplies the structural source of persistence used by cooper_paired_reference_yields_observer and the master observer_forcing_certificate. It completes step 6 of the module argument: Cooper pairing yields a zero-cost reference from any positive starting state. In the Recognition Science chain it supports the forcing of the observer from non-trivial coherent recognition, consistent with J-uniqueness and the identity-tick minimum.
scope and limits
- Does not claim that every persistent reference must arise via Cooper pairing.
- Does not address persistence under dynamical evolution or perturbations.
- Does not extend the construction beyond real-valued states.
- Does not prove that the resulting observer is unique.
Lean usage
obtain ⟨ref, hpers⟩ := cooper_pairing_yields_persistent x hxformal statement (Lean)
124theorem cooper_pairing_yields_persistent
125 (x : ℝ) (hx : 0 < x) :
126 ∃ e : RecognitionEvent, IsPersistent e := by
proof body
Tactic-mode proof.
127 refine ⟨{
128 state := x * x⁻¹,
129 state_pos := by
130 rw [mul_inv_cancel₀ (ne_of_gt hx)]; norm_num
131 }, ?_⟩
132 show Cost.Jcost (x * x⁻¹) = 0
133 exact cooper_pair_cost_zero x hx
134
135/-! ## §5. The Observer -/
136
137/-- An observer is a coherent recognition structure equipped with a
138 persistent reference event. The observer integrates multiple
139 distinguishable recognition events against a single fixed
140 identity-tick reference. -/