consistent_cost
consistent_cost assigns to each ConsistentConfig the defect of its ratio. Researchers deriving logic from cost minimization in Recognition Science cite this when establishing that stable propositions reach zero cost. It is a direct one-line wrapper around the upstream defect function.
claimFor a consistent configuration $c$ with proposition $P$ and positive ratio $r$, the cost is $defect(r)$, where $defect(x) := J(x)$ and $J$ is the recognition cost functional.
background
ConsistentConfig is a structure pairing a proposition $P$ with a positive real ratio $r$, representing a configuration without its negation asserted. The defect function, imported from LawOfExistence, is defined as $defect(x) := J(x)$ for positive $x$, measuring deviation from unity. The module establishes that logical consistency emerges as the minimum-cost state in the Recognition Science framework, where cost minimization selects stable configurations. Upstream, defect_at_one shows $defect(1) = 0$, and defect_nonneg establishes non-negativity for positive arguments.
proof idea
This is a one-line wrapper that applies the defect function to the ratio field of the ConsistentConfig structure.
why it matters in Recognition Science
This definition supplies the cost measure used in consistent_minimum_cost, logic_from_cost, and why_logic_is_real, which establish that consistency achieves zero cost while contradictions do not. It fills the role of assigning costs in the cost-minimization derivation of logic, aligning with the framework claim that logic emerges from minimizing J-cost. It supports T5 J-uniqueness by providing the cost for consistent states.
scope and limits
- Does not prove existence of zero-cost configurations.
- Does not apply to contradictory configurations.
- Does not derive the J functional itself.
- Does not claim to found classical logic without metalanguage assumptions.
formal statement (Lean)
191noncomputable def consistent_cost (c : ConsistentConfig) : ℝ := defect c.ratio
proof body
Definition body.
192
193/-- **THEOREM 4**: Consistent configurations can have zero cost.
194
195 Unlike contradictions, a single proposition can stabilize at ratio = 1.
196 This is the minimum-cost state for a proposition. -/