Cost Derivative

Cost Derivative

In Recognition Science, the cost of a recognition event is measured by the J-cost function, J(x) = (x + x⁻¹)/2 - 1. The cost derivative is the ordinary calculus derivative of this function with respect to its argument, taken for positive x. The module establishes that J is differentiable for all positive x, and that its derivative is exactly (1 - x⁻²)/2. This is a established theorem in Lean, not an assumption.

The derivative has a direct structural meaning. Consider a small multiplicative change to the recognition scale, written as a log-strain L applied at a base value x. The linearized change in cost, called linJ(x, L), is ((x - x⁻¹)/2) · L. The module proves the key identity linJ(x, L) = J'(x) · x · L, which shows that this linearization is precisely the first-order term of the cost change along an exponential path. At the unit multiplier x = 1, the linear term vanishes, and the remainder of the cost change is exactly J(e^L).

The importance of this result is that it justifies a linear approximation to cost changes. The remainder after linearization is quadratic in L, so for small strains the linear term dominates. This is what allows the framework to replace a global axiom about harm with a local, derivative-based statement: the linearized bond delta is the correct directional derivative of the cost function. The module proves this connection explicitly, showing that the harm linear term used in the Ethics/Harm module is exactly this derivative.

THEOREM deriv_Jcost_eq · IndisputableMonolith/Cost/Derivative.lean

THEOREM linJ_eq_derivative_times_x · IndisputableMonolith/Cost/Derivative.lean

THEOREM linJ_unit · IndisputableMonolith/Cost/Derivative.lean

THEOREM harm_linearization_correct · IndisputableMonolith/Cost/Derivative.lean

What this page does not claim

Not a derivation of the J-cost function itself, which is established elsewhere. Not a statement about the physical interpretation of the log-strain L. Not a claim that the remainder is zero; it is only bounded as quadratic.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cost/Derivative.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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