Foundation Jcost Geometry Jcost Unit Curvature

The curvature of cost

The cost function J(x) = ½(x + x⁻¹) − 1 measures the price of a ratio x between two quantities. When x is exactly 1, the cost is 0. The theorem jcost_unit_curvature examines what happens near that perfect match. Write x = 1 + ε, where ε is a small displacement. The theorem states that for |ε| ≤ 1/2, the cost equals ε²/2 plus a correction term c·ε³, with the constant c bounded by 2 in absolute value. In plain language, near the optimum the cost rises like a parabola, and the cubic correction is controlled.

This is a local statement, not a global one. It does not say the cost is exactly a parabola everywhere; it gives a precise bound on how much the true curve can deviate from the parabola within a fixed window. The bound |c| ≤ 2 is what makes the statement useful: it turns a vague approximation into a rigorous inequality. The theorem is proved in the machine-checked library of formal theorems, and it is a direct consequence of the algebraic identity J(1+ε) = ε²/(2(1+ε)), which itself follows from the closed form of J.

In Recognition Science, this curvature fact is the seed of a geometric picture. The cost function is not just a formula; it has a shape. Near its minimum, that shape is quadratic, which is the same local behavior as the energy of a stretched spring. The bounded cubic term says the spring is not perfectly harmonic, but the deviation is small and controlled. This is the sense in which the framework claims a natural geometry for cost: the local curvature is fixed, and the anharmonicity is bounded.

The theorem does not claim that the cost function is globally quadratic, nor that the bound c ≤ 2 is sharp. It also does not identify what the constant c is; it only asserts that some such c exists within the bound. The statement is conditional on the displacement ε lying in the interval [−1/2, 1/2]; outside that window, the theorem says nothing. These limitations are part of the theorem's content, not omissions.

/-- **F1.1.6**: J(1) = 0 and the second derivative at 1 gives unit curvature.
    We state this via the quadratic approximation. -/
theorem jcost_unit_curvature (ε : ℝ) (hε : |ε| ≤ 1/2) :
    ∃ c : ℝ, Jcost (1 + ε) = ε ^ 2 / 2 + c * ε ^ 3 ∧ |c| ≤ 2 :=
  Jcost_one_plus_eps_quadratic ε hε

/-- **F1.1.4**: J(x) = J(1/x) for x > 0 -/
theorem jcost_reciprocal {x : ℝ} (hx : 0 < x) : Jcost x = Jcost x⁻¹ :=
  Jcost_symm hx

/-- **F1.1.2**: J(1) = 0 -/
theorem jcost_at_one : Jcost 1 = 0 := Jcost_unit0

What this page does not claim

The cost function is globally quadratic; the theorem only covers a small neighborhood of x = 1. The constant c in the cubic term is identified; the theorem only asserts its existence with a bound. The curvature statement applies outside the interval |ε| ≤ 1/2.

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/Foundation/JCostGeometry.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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