Foundation Jcost Convexity In Log Space H At Zero

The log-ratio function

The declaration h_at_zero is a small, precise fact about a function called h, defined as h(t) = t²/2. In plain language, this function takes a number t, squares it, and divides by two. The theorem proves that when t is zero, h is zero: h(0) = 0. This is not a deep claim about the universe; it is a direct consequence of the definition, since zero squared is zero. The proof in the machine-checked library of formal theorems is a single step, using the definition of h.

The function h is the log-ratio form of a cost function. In the Recognition Science framework, a cost measures the price of a recognition event, a discrete record of a comparison. The framework's central cost function J(x) is defined for positive numbers x, and it has a unique minimum at x = 1, where J(1) = 0. When you change coordinates by taking the logarithm, t = ln(x), the cost J becomes a new function g(t) = J(eᵗ). The function h is a simpler cousin: it is the quadratic approximation of g near t = 0, and it shares key properties with g, namely the same fixed point at zero and the same even symmetry, meaning h(t) = h(-t).

The theorem h_at_zero is a foundational building block, not a standalone result. It is one of several facts that together certify that g and h behave alike: both are zero at t = 0, both are even, and both are positive for any t that is not zero. The framework's library packages these shared properties into a single structure called JCostLogSpaceCert, and h_at_zero is one of the fields that fills it. This certification matters for the framework's closed-loop control result, which uses the log-space picture to reason about recognition costs.

What h_at_zero does not claim is just as important as what it proves. It does not say that h is the same function as g, only that they agree at one point and share structural symmetries. It does not establish the full uniqueness theorem for J, which is proved elsewhere. It does not assert anything about the physical meaning of the cost, only a formal property of a definition. The theorem is a lemma, a step in a larger machine-checked argument, and its value lies in being exactly that: a verified, reusable piece of a formal proof.

theorem h_at_zero : h 0 = 0 := by simp [h]

/-- The log-ratio function h(t) = t²/2 has the same fixed point and sign. -/
noncomputable def h (t : ℝ) : ℝ := t ^ 2 / 2

What this page does not claim

h_at_zero does not claim that h equals the cost function J in log coordinates, only that they share a fixed point and symmetry. It does not prove the uniqueness theorem for the cost function J. It does not assign any physical meaning to the cost; it only records a formal property of a definition.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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