CoherentRecognition
A structure for coherent recognition consists of a sequence of recognition events over the naturals, a fixed reference event, and a witness that at least two events have distinct states. Researchers deriving observer emergence from non-trivial recognition streams cite this minimal data package as the starting point for forcing a persistent reference frame. The definition is a direct structure declaration that assembles the sequence, reference, and nontriviality condition from the underlying event type.
claimA coherent recognition structure consists of a function $e : ℕ → ℛ$, a reference event $r ∈ ℛ$, and a witness that there exist $n, m ∈ ℕ$ with the state of $e(n)$ differing from the state of $e(m)$, where $ℛ$ denotes the type of recognition events (positive real states).
background
RecognitionEvent is the structure carrying a positive real state together with its positivity proof; its cost is the J-cost drawn from the Cost module. The module sets out a seven-step chain in which multiple distinguishable events require an invariant reference whose cost cannot shift with context, forcing that cost to zero by the uniqueness lemma Jcost_eq_zero_iff (J(x) = 0 iff x = 1 for positive x). Upstream results include the nontrivial kinship systems and the active-edge count A, but the immediate dependencies are the cost interpretation and the zero-cost uniqueness statement.
proof idea
The declaration is a structure definition that directly assembles the three fields: the event sequence, the reference event, and the existential nontriviality condition on distinct states. No tactics or lemmas are applied; the construction is purely packaging of the RecognitionEvent type and the state inequality.
why it matters in Recognition Science
This structure is the direct input to the Observer definition, which attaches an IsPersistent reference to it. It occupies the second step in the module's argument that non-trivial recognition forces an observer, as described in the module documentation. Within the Recognition Science framework it marks the passage from J-uniqueness to the requirement of a fixed reference frame, prior to cooper pairing and the eight-tick octave. It leaves open the concrete construction of the reference via cooper pairing when no event sits exactly at the identity tick.
scope and limits
- Does not assert existence of any coherent recognition structure.
- Does not define or compute J-cost values.
- Does not invoke the Recognition Composition Law.
- Does not extend to continuous or higher-dimensional event spaces.
formal statement (Lean)
74structure CoherentRecognition where
75 events : ℕ → RecognitionEvent
76 reference : RecognitionEvent
77 nontrivial : ∃ n m : ℕ, (events n).state ≠ (events m).state
78
79/-! ## §3. The Persistent Reference -/
80
81/-- A reference is *persistent* if its J-cost is zero.
82
83 Justification: if the reference cost were `c > 0`, then the
84 comparison `J(eᵢ) − c` against this reference would shift if `c`
85 itself depended on the comparison context. The only invariant
86 cost across arbitrary recognition events is `c = 0`, since
87 `Jcost = 0` is the unique global minimum (`Cost.Jcost_eq_zero_iff`).
88 A reference at any other cost is not a fixed frame; it is itself
89 in motion. -/