Foundation Jcost Convexity In Log Space H Nonneg

The h_nonneg theorem

The declaration h_nonneg is a formal statement about the function h(t) = t²/2. In plain language, it proves that the square of any real number, divided by two, is always greater than or equal to zero. This is a basic property of squares, but its formal proof is a building block in a larger argument about a cost function.

In the Recognition Science framework, a cost function assigns a non-negative number to a discrepancy, measuring how far a state is from a preferred one. The function h(t) = t²/2 is a simple example of such a cost, and h_nonneg confirms it never assigns a negative cost. This property is essential for any function that claims to measure a cost, as a negative cost would be meaningless.

The statement h_nonneg is one of a set of results about h and another function g(t) = J(eᵗ), where J is the framework's central cost function. Both functions share the same fixed point at t = 0, where they both equal zero, and both are even, meaning they give the same value for t and -t. The h_nonneg statement establishes the non-negativity of h, a property that g also possesses, though the formal statement for g is separate.

The declaration is a machine-checked result in the framework's library of formal mathematics. The proof is immediate from the definition of h, using the fact that the square of a real number is non-negative. It is a small but necessary lemma, not a standalone discovery. Its role is to certify that a candidate cost function satisfies a basic requirement, supporting the framework's use of such functions in its models.

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

theorem h_nonneg (t : ℝ) : 0 ≤ h t := by unfold h; positivity

/-- J-cost in log coordinates: g(t) = J(eᵗ). -/
noncomputable def g (t : ℝ) : ℝ := Jcost (Real.exp t)

What this page does not claim

The statement does not prove that the cost function J itself is always non-negative; that is a separate result. The statement does not establish any relationship between h and the physical constants of the framework. The statement does not claim that h is the only possible cost function or that it is preferred over others.

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)

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Derived articles

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