prop_cost
prop_cost maps a PropConfig to the defect of its ratio field. Researchers modeling logic as cost minimization cite it to define stability thresholds. The definition is a direct one-line application of the defect functional imported from LawOfExistence.
claimFor a PropConfig $c$ with positive ratio $r$, define the cost as defect$(r)$, where defect coincides with the J-cost on positive reals.
background
The LogicFromCost module establishes that logical consistency arises as the minimum-cost structure within classical logic. A PropConfig pairs a proposition with a positive real ratio that quantifies presence: ratio = 1 for balanced assertions, ratio near 0 for absent ones, and ratio to infinity for unstable assertions. The defect functional, defined upstream in LawOfExistence as defect$(x) := J(x)$, supplies the cost measure. Related cost constructions appear in ObserverForcing (cost of a recognition event) and MultiplicativeRecognizerL4 (derived cost of a comparator).
proof idea
One-line definition that applies the defect function directly to the ratio component of the supplied PropConfig.
why it matters in Recognition Science
This definition supplies the cost value used by the sibling declarations IsStable (cost equals zero) and IsUnstable (cost positive). It implements the module claim that contradictions cannot both stabilize because their effective cost diverges, while consistent configurations occupy the cost minimum. The construction therefore bridges the Recognition Science cost landscape (J-cost minimization) to the emergence of logical structure without presupposing logic.
scope and limits
- Does not derive classical logic from cost; it works inside Lean's classical metalanguage.
- Does not compute explicit costs for concrete physical propositions.
- Does not extend to configurations outside the PropConfig structure.
- Does not prove existence of zero-cost configurations.
formal statement (Lean)
79noncomputable def prop_cost (c : PropConfig) : ℝ := defect c.ratio
proof body
Definition body.
80
81/-- A configuration is stable if its cost is zero. -/