Cost Cont Diff Reduction Cont Diff Two Differentiable Deriv

The regularity bridge

In calculus, a function is called twice continuously differentiable, often written C², when it, its first derivative, and its second derivative all exist and are continuous. The lemma contDiffTwo_differentiable_deriv states a simple consequence: if a real-valued function is C², then its first derivative is itself differentiable. This is a standard fact from analysis, not a new discovery; the framework's contribution is to formalize it in its machine-checked library of formal theorems and use it as a bridge in a longer argument.

The argument concerns the cost, the framework's term for the price reality pays to recognize an event. The central result of Recognition Science is that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. One of those conditions is a composition law, an equation that constrains how costs combine when two events are recognized together. Proving the uniqueness theorem requires differentiating that equation. The lemma contDiffTwo_differentiable_deriv supplies the needed regularity: it guarantees that the derivative of a C² cost function can itself be differentiated, so the composition law can be differentiated twice.

The lemma does not, by itself, establish anything about costs, recognition, or the uniqueness theorem. It is a purely analytic statement about C² functions, with no framework-specific content. Its role is enabling, not substantive. The theorem that uses it, law_of_logic_forces_jcost_of_contDiff, combines the lemma with normalization, the composition law, calibration, and C² regularity to conclude that the cost must equal J(x). The lemma is one link in that chain, not the chain itself.

What the lemma does not claim is equally precise. It does not claim that every differentiable function is C², only the one-way implication. It does not claim that the cost function itself is C²; that is a separate hypothesis supplied by the user of the theorem. And it does not claim that the uniqueness theorem holds without the C² assumption. The regularity condition is essential: without it, the differentiation step that the proof relies on is not justified, and the conclusion may fail. The lemma is a tool, and its scope is exactly the scope of the implication it states.

private lemma contDiffTwo_differentiable_deriv {Hf : ℝ → ℝ}
    (h_diff : ContDiff ℝ 2 Hf) : Differentiable ℝ (deriv Hf) := by
  have h_diff' := h_diff
  rw [show (2 : WithTop ℕ∞) = 1 + 1 from rfl] at h_diff'
  rw [contDiff_succ_iff_deriv] at h_diff'
  exact h_diff'.2.2.differentiable (by decide : (1 : WithTop ℕ∞) ≠ 0)

What this page does not claim

The lemma does not establish anything about costs or recognition by itself. The lemma does not claim that every differentiable function is C². The lemma does not prove the uniqueness theorem without the C² assumption.

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

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