Cost Derivative Lin J Unit

The unit multiplier

The declaration linJ_unit establishes a single, narrow fact about a function called the linearized per-bond delta, written linJ. That function measures the first-order change in a recognition cost when a bond's multiplier moves from a base value x by a logarithmic strain L. The lemma states that when the base value is exactly 1, this linear term is zero for any strain L. In symbols: linJ 1 L = 0.

The proof is immediate from the definition: linJ x L = ((x - x⁻¹) / 2) * L, and at x = 1 the factor (1 - 1)/2 vanishes. This is a statement about the derivative of the J-cost function, which is J(x) = (x + x⁻¹)/2 - 1. The derivative is J'(x) = (1 - x⁻²)/2, and at x = 1 it is zero. The lemma is the special case of the more general identity linJ x L = J'(x) * x * L when x = 1.

What this establishes in plain terms: at the unit multiplier, the linear approximation to the cost change is flat. A small strain around x = 1 produces no first-order change in cost; the leading behavior is quadratic, not linear. This is why the remainder term remJ 1 L equals the full cost J(e^L), which is nonnegative by the AM-GM inequality.

What the lemma does not claim: it says nothing about the sign or magnitude of the cost change for finite strains, only about the linear term. It does not assert that the cost is minimized at x = 1, though the derivative being zero is consistent with that. It does not claim anything about the behavior of linJ at other base values, nor about the global properties of J beyond its differentiability. The lemma is a local, first-order statement, and its proof uses only the definition of linJ and the algebraic fact that 1 - 1 = 0.

In the framework's library, this lemma is a building block: it is the unit case that connects the linearization to the derivative identity and, through that, to the harm linearization theorem. But as a standalone result, it is a single line of calculus, not a claim about ethics, harm, or the structure of reality.

/-- At unit multiplier (x=1), the linear term vanishes. -/
lemma linJ_unit (L : ℝ) : linJ 1 L = 0 := by simp [linJ]

/-- The linearized per-bond delta for J under log-strain L at base x. -/
noncomputable def linJ (x L : ℝ) : ℝ := ((x - x⁻¹) / 2) * L

What this page does not claim

The lemma does not claim that the cost is minimized at x = 1, only that the derivative is zero there. It does not claim anything about the behavior of linJ at base values other than 1. It does not claim any global property of the J-cost function beyond its differentiability at positive x.

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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