extraction_cost_eq_zero_iff
plain-language theorem explainer
Extraction system cost vanishes exactly when the extraction parameter σ is zero. Researchers establishing thermodynamic equilibria in Recognition Science cite this equivalence to confirm uniqueness of the zero-extraction state. The proof applies the strict surcharge inequality in one direction and the cosh reduction in the other.
Claim. Let $C(σ)$ be the total J-cost of agents at scales $e^σ$ and $e^{-σ}$. Then $C(σ)=0$ if and only if $σ=0$.
background
The extractionSystemCost function sums the J-costs of two agents differing by extraction level σ, with one at $e^σ$ and the other at $e^{-σ}$. J-cost is defined via the multiplicative recognizer as $J(x)=(x+x^{-1})/2-1$, equivalent to cosh(log x)-1. The module imports the cost functional equation and d'Alembert factorization to ground these definitions in the Recognition framework.
proof idea
The term proof constructs the biconditional. One leg assumes cost zero and derives a contradiction from the surcharge theorem when σ is nonzero. The other leg rewrites the cost via the cosh identity at σ=0 and simplifies with cosh zero equal to one.
why it matters
Downstream minimum theorems rewrite the zero case using this equivalence to establish that σ=0 is the global and strict minimum. It closes the step showing cost vanishes only at equilibrium, aligning with the J-uniqueness in the forcing chain T5 and the RCL composition law. The result reinforces that nonzero extraction is always costly in the Recognition Science model.
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