Foundation Inevitability Equivalence Loglift Cont Diff Of Cost Cont Diff

The smoothness bridge

In mathematics, a function is smooth when it can be differentiated as many times as needed; the second derivative is what controls curvature and acceleration. The declaration loglift_contDiff_of_cost_contDiff, from the framework's machine-checked library of formal theorems, establishes a bridge: if a cost function is twice continuously differentiable in its original variable, then the same function, viewed in logarithmic coordinates, is also twice continuously differentiable. In symbols, if cost(x) is C², then the composite function t ↦ cost(eᵗ) is C² as well.

This is not a deep result. It follows from the chain rule, since the exponential map eᵗ is itself infinitely differentiable. The theorem's proof is one line: compose a smooth function with another smooth function. What matters is why the framework bothers to state it. In Recognition Science, the central object is a cost function (a rule assigning a nonnegative number to each ratio, measuring the price of a recognition event) and the framework's uniqueness theorem works most naturally in logarithmic coordinates, where multiplicative symmetry becomes additive translation. The smoothness bridge guarantees that switching coordinates does not quietly discard any regularity that the original cost function possessed.

The theorem does not claim that every smooth cost function satisfies the framework's axioms, nor that the cost function equals the specific J(x) = (x + 1/x)/2 − 1. It only preserves a technical hypothesis: if the original cost is smooth, the log-lifted version is smooth too. The deeper conclusion, that any cost obeying the full axiom bundle must equal J, is a separate theorem (the inevitability chain), not this declaration. This bridge is a necessary ingredient, not the meal.

What the bridge changes for a reader is the shape of the proof strategy. Because the log-lift preserves smoothness, the framework can state its calibration condition, that the second derivative of the log-lift at zero equals 1, without worrying that the coordinate change introduces singularities. The smoothness bridge is what makes that calibration condition well-posed, and it is one of the small, unglamorous steps that let the larger uniqueness argument stand on checked ground.

/-- Smoothness bridge: direct smoothness of `cost` implies smoothness of its log-lift. -/
theorem loglift_contDiff_of_cost_contDiff (cost : ℝ → ℝ)
    (hSmooth : ContDiff ℝ 2 cost) :
    ContDiff ℝ 2 (fun t => cost (Real.exp t)) :=
  hSmooth.comp Real.contDiff_exp

What this page does not claim

The theorem does not prove that any smooth cost function equals J(x) = (x + 1/x)/2 − 1. The theorem does not establish that the framework's axioms are satisfied by any particular cost function. The theorem does not address whether the cost function is unique or whether alternatives exist.

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

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