Foundation Jcost Convexity In Log Space H Pos Off Zero

The log-ratio function

The function h(t) = t²/2 is one of the simplest nonlinear functions in mathematics: it takes a number, squares it, and halves the result. In the Recognition Science framework, this function appears as a log-ratio cost, a measure of how far a recognition event, a discrete record of an event in the framework's ledger, sits from a reference point. The framework's machine-checked library of formal theorems proves a basic fact about it: h(t) is positive for every nonzero t, and h(0) = 0. In plain words, the cost of being away from the reference point is always strictly positive, and the cost of being exactly at the reference point is zero.

The theorem h_pos_off_zero states this precisely: if t is not zero, then 0 < h(t). The proof is a one-line algebraic argument, since a square of a nonzero real number is positive. The companion theorem h_at_zero gives h(0) = 0. Together they show that h has a unique global minimum at t = 0, the only place where the cost vanishes. This is the same fixed-point behavior that the framework's central cost function J(x) = (x + 1/x)/2 - 1 exhibits at x = 1, and the same symmetry: h(t) = h(-t), meaning the cost depends only on the size of the deviation, not its direction.

The framework uses h as a simplified stand-in for J in log coordinates, where t = ln(x). Near t = 0, the two functions agree to second order: J(e^t) ≈ t²/2. The library proves they share the same fixed point and the same even symmetry, but it does not prove they are equal. The quadratic h is a local approximation, not a global identity. This distinction matters for the framework's closed-loop control results, which rely on the positivity and symmetry of the cost, but do not require the full functional form of J.

What h_pos_off_zero does not claim is just as important as what it proves. It does not say that h is the only cost function with these properties, nor that h equals J anywhere except near zero. It does not establish that the framework's recognition process actually uses h as its cost; h is a definitional choice, a model, not a derived necessity. The theorem is a fact about a specific quadratic function, proved in the framework's library, and nothing more.

theorem h_pos_off_zero {t : ℝ} (ht : t ≠ 0) : 0 < h t := by
  unfold h; positivity

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

theorem h_even (t : ℝ) : h t = h (-t) := by unfold h; ring

What this page does not claim

h equals J globally; they agree only near t = 0. h is the unique cost function with a positive-off-zero property. The framework derives h from first principles; it is a definitional choice.

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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