extractionSystemCost
extractionSystemCost defines the total recognition cost for an extraction of magnitude σ between two agents at reciprocal scales e^σ and e^{-σ}. Researchers modeling thermodynamic limits in recognition-based ethics cite this as the base cost function for extraction processes. The definition is introduced directly as the sum of the individual J-costs on those states.
claimThe system cost of extraction σ is defined by $J(e^σ) + J(e^{-σ})$, where $J$ is the J-cost function satisfying $J(x) = (x + x^{-1})/2 - 1$.
background
The Ethics.ThermodynamicInstabilityOfExtraction module develops consequences of the Recognition Composition Law for ethical extraction processes. J-cost is the cost function induced by the multiplicative recognizer on positive ratios, satisfying the functional equation J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). The definition assumes the non-negativity of costs from ObserverForcing.cost and the identity property that self-comparison costs zero from LogicAsFunctionalEquation.Identity. This places the extraction cost in the context of the phi-ladder and forcing chain T5-T8.
proof idea
The definition is a direct one-line expression that adds the J-cost of the extracted state to that of its reciprocal. No lemmas are applied in the definition body itself; subsequent theorems such as extraction_cost_eq_cosh reduce it to 2(cosh σ - 1) by invoking the relation between J and the hyperbolic cosine from DAlembert.LedgerFactorization.
why it matters in Recognition Science
This definition anchors the theorems on marginal costs, non-negativity, and strict convexity in the extraction module, which demonstrate that any nonzero extraction creates a positive surcharge that grows with σ. It connects to the broader Recognition Science claim that stable equilibria occur only at σ = 0, consistent with J-uniqueness in T5 and the RCL. It leaves open whether alternative cost functions could stabilize extraction.
scope and limits
- Does not extend to multi-agent systems beyond the two-agent pair.
- Does not specify the physical units or scaling of σ.
- Does not prove the convexity or minimum properties itself.
- Does not address quantum or relativistic corrections to the cost.
formal statement (Lean)
70def extractionSystemCost (σ : ℝ) : ℝ :=
proof body
Definition body.
71 Jcost (Real.exp σ) + Jcost (Real.exp (-σ))
72
73/-- **Theorem (Extraction Cost Identity)**: The system cost of extraction σ
74 is exactly 2(cosh(σ) − 1). -/
used by (13)
-
deriv_extraction_cost -
extraction_cost_eq_cosh -
extraction_cost_eq_zero_iff -
extraction_cost_minimum_at_zero -
extraction_cost_nonneg -
extraction_cost_strictly_convex -
extraction_cost_strict_minimum -
extraction_creates_surcharge -
extraction_unique_equilibrium -
force_always_toward_balance -
restoring_force_negative -
restoring_force_positive -
second_deriv_extraction_cost