second_deriv_extraction_cost
plain-language theorem explainer
The second derivative of the extraction system cost equals twice the hyperbolic cosine of the extraction level. Researchers modeling thermodynamic costs between agents in recognition frameworks cite this to confirm curvature. The proof substitutes the known first derivative of twice sinh and applies the derivative rule for sinh.
Claim. The second derivative of the extraction system cost function with respect to the extraction level $σ$ equals $2 cosh σ$.
background
The extraction system cost is the sum of J-costs for two agents positioned at $e^σ$ and $e^{-σ}$, which simplifies to $2(cosh σ - 1)$. The first derivative of this cost is $2 sinh σ$, which quantifies the marginal resistance to additional extraction. This result computes the curvature of that marginal cost. The J-cost itself is induced by the multiplicative recognizer comparator on positive ratios and satisfies the recognition composition law. The module examines thermodynamic instability arising from such extraction between agents.
proof idea
The proof first invokes the marginal cost theorem to equate the first derivative of extraction system cost to twice sinh. It rewrites the second derivative expression using this equality, then applies the derivative of sinh scaled by the constant factor 2.
why it matters
This curvature result feeds the strict convexity theorem, which establishes that the second derivative is positive everywhere by invoking cosh positivity. It completes the curvature step in the thermodynamic instability analysis, connecting to J-uniqueness where J(x) equals cosh(log x) minus 1. The parent theorem uses this to locate the unique equilibrium at zero extraction.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.