recognition_cost
recognition_cost defines the cost of a recognition event as the J function applied to its ratio. Researchers tracing the forcing chain from ledger costs to recognition structures cite it when proving that unit-ratio events are cost-free and that deviations carry positive cost. It is a direct one-line definition that delegates to the LedgerForcing J implementation.
claimLet $e$ be a recognition event with ratio $r > 0$. Its recognition cost equals $J(r)$, where $J(x) = (x + x^{-1})/2 - 1$.
background
The RecognitionForcing module shows that recognition structures are forced by the underlying cost foundation. RecognitionEvent, imported from LedgerForcing, is the structure carrying source, target, and a positive real ratio; positivity ensures the J-cost is defined. Upstream, ObserverForcing.cost likewise sets the cost of an event to its J-cost on the state, while MultiplicativeRecognizerL4.cost derives an analogous function from a comparator on positive ratios. The module doc states that the section proves recognition is forced by the cost foundation.
proof idea
This is a one-line definition that applies LedgerForcing.J directly to the ratio field of the supplied recognition event.
why it matters in Recognition Science
The definition supplies the cost map used by every subsequent result in the module. It is invoked by recognition_is_cost_structure to equate zero cost with unit ratio, by global_minimum_is_self_recognition to exhibit the self-recognition zero-cost case, and by recognition_forcing_complete as part of the master bridge from cost to recognition structures. It instantiates the J-uniqueness step (T5) of the unified forcing chain and supplies the cost side of the Recognition Composition Law.
scope and limits
- Does not prove non-negativity of the cost.
- Does not characterize the zero-cost locus.
- Does not incorporate source or target labels into the numerical value.
- Does not extend to ratios that fail positivity.
formal statement (Lean)
21noncomputable def recognition_cost (e : LedgerForcing.RecognitionEvent) : ℝ :=
proof body
Definition body.
22 LedgerForcing.J e.ratio
23
24/-- Recognition events with ratio = 1 are cost-free. -/