extraction_cost_eq_cosh
plain-language theorem explainer
The extraction cost identity equates the combined J-cost of two agents at reciprocal exponential positions e^σ and e^{-σ} to twice the hyperbolic cosine of σ minus two. Researchers deriving marginal costs or non-negativity bounds in Recognition Science ethics models cite this result to obtain an explicit closed form. The proof is a direct algebraic reduction that unfolds the system cost definition, applies the exponential inverse rule, and invokes the cosh identity twice before combining with linarith.
Claim. Let $J(x) = (x + x^{-1})/2 - 1$. The extraction system cost, defined as the sum $J(e^σ) + J(e^{-σ})$, satisfies $J(e^σ) + J(e^{-σ}) = 2(ℝ.cosh σ - 1)$.
background
In the module on thermodynamic instability of extraction the system cost is introduced as the sum of J-costs for a pair of agents where one has extracted parameter σ from the other, placing them at e^σ and e^{-σ}. J-cost is the recognition cost function J(x) = (x + x^{-1})/2 - 1 that matches the T5 J-uniqueness form cosh(log x) - 1. The local setting uses this pair to model extraction as a log-exchange whose total J-cost is strictly positive for nonzero σ, consistent with the Recognition Composition Law.
proof idea
The proof first shows Jcost(exp σ) = cosh σ - 1 by unfolding Jcost, rewriting the inverse exponential via simp, and applying the cosh definition. It repeats the argument for Jcost(exp (-σ)), using cosh(-σ) = cosh σ after ring simplification. It then unfolds the system cost definition, substitutes both equalities, and closes with linarith.
why it matters
This supplies the closed form required by the marginal cost theorem (deriv_extraction_cost), the zero-equivalence theorem (extraction_cost_eq_zero_iff), the non-negativity theorem (extraction_cost_nonneg), and the surcharge theorem (extraction_creates_surcharge). It fills the algebraic step that converts the two-agent J-sum into hyperbolic functions, anchoring the ethics module to the J-uniqueness and eight-tick octave landmarks of the forcing chain.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.