Cost Functional Equation Has Log Curvature Full Filter Forces Zero

The curvature filter

The declaration hasLogCurvature_full_filter_forces_zero is a technical lemma in the framework's machine-checked library of formal theorems. It concerns the behavior of a function near a point, a standard topic in calculus. The statement is this: if a function Hf satisfies the limit condition that 2 times (Hf(t) minus 1) divided by t squared tends to a number κ as t approaches 0, then κ must be 0. In plainer terms, if you look at how fast a function's value deviates from 1 near zero, and that deviation grows no faster than t squared, the only possible limiting ratio is zero.

This is not a statement about any particular function. It is a constraint on what the limit can be. The proof in the library is short: it compares the limit along the whole neighborhood of zero with the limit along the single point zero itself. At the point zero, the expression evaluates to a fixed number, and the two limits must agree. The only number that can equal its own value at that point is itself, which forces κ to be zero. The lemma is a filter property, a way of saying that the limit is uniquely determined by the function's behavior at the point.

In Recognition Science, this lemma acts as a calibration check. The framework models a ledger, a discrete record of recognition events, and assigns a cost, a number measuring the price of each recognition, to every positive input. The central theorem of the framework, law_of_logic_forces_jcost, proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. One of those conditions is calibration, and the curvature lemma helps ensure that the calibration condition is consistent. It shows that the log-curvature, the second-order behavior of the function in log coordinates, cannot be an arbitrary number; it is forced to be zero at the point where the function equals one.

What this lemma does not claim is more limited than it might appear. It does not by itself prove the uniqueness of the cost function. The full proof requires the composition law, the reciprocal symmetry, and the other conditions. It also does not say anything about the value of the cost function itself; it only constrains the limit of a particular ratio. The lemma is a supporting actor in a larger proof, not the main event. It is a precise, narrow statement about limits, and its role is to make the framework's calibration condition well-defined.

/-- **Tripwire.** The full-filter reading of log curvature forces `κ = 0`, so at
the calibrated value `κ = 1` it is satisfied by no function at all. Kept as a
theorem so the defect cannot be reintroduced without a failing build. -/
theorem hasLogCurvature_full_filter_forces_zero (Hf : ℝ → ℝ) (κ : ℝ)
    (h : Filter.Tendsto (fun t => 2 * (Hf t - 1) / t^2) (nhds 0) (nhds κ)) :
    κ = 0 := by
  have h1 : Filter.Tendsto (fun t : ℝ => 2 * (Hf t - 1) / t ^ 2)
      (pure 0) (nhds κ) := h.mono_left (pure_le_nhds 0)
  have h2 : Filter.Tendsto (fun t : ℝ => 2 * (Hf t - 1) / t ^ 2)
      (pure 0) (nhds (2 * (Hf 0 - 1) / (0 : ℝ) ^ 2)) := tendsto_pure_nhds _ _
  have h3 := tendsto_nhds_unique h2 h1
  simpa using h3.symm

/-- **Tripwire.** The full-filter reading of log curvature forces `κ = 0`, so at
the calibrated value `κ = 1` it is satisfied by no function at all. Kept as a
theorem so the defect cannot be reintroduced without a failing build. -/
theorem hasLogCurvature_full_filter_forces_zero (Hf : ℝ → ℝ) (κ : ℝ)
    (h : Filter.Tendsto (fun t => 2 * (Hf t - 1) / t^2) (nhds 0) (nhds κ)) :
    κ = 0 := by
  have h1 : Filter.Tendsto (fun t : ℝ => 2 * (Hf t - 1) / t ^ 2)
      (pure 0) (nhds κ) := h.mono_left (pure_le_nhds 0)
  have h2 : Filter.Tendsto (fun t : ℝ => 2 * (Hf t - 1) / t ^ 2)
      (pure 0) (nhds (2 * (Hf 0 - 1) / (0 : ℝ) ^ 2)) := tendsto_pure_nhds _ _
  have h3 := tendsto_nhds_unique h2 h1
  simpa using h3.symm

/-- **Tripwire.** The full-filter reading of log curvature forces `κ = 0`, so at
the calibrated value `κ = 1` it is satisfied by no function at all. Kept as a
theorem so the defect cannot be reintroduced without a failing build. -/
theorem hasLogCurvature_full_filter_forces_zero (Hf : ℝ → ℝ) (κ : ℝ)
    (h : Filter.Tendsto (fun t => 2 * (Hf t - 1) / t^2) (nhds 0) (nhds κ)) :
    κ = 0 := by
  have h1 : Filter.Tendsto (fun t : ℝ => 2 * (Hf t - 1) / t ^ 2)
      (pure 0) (nhds κ) := h.mono_left (pure_le_nhds 0)
  have h2 : Filter.Tendsto (fun t : ℝ => 2 * (Hf t - 1) / t ^ 2)
      (pure 0) (nhds (2 * (Hf 0 - 1) / (0 : ℝ) ^ 2)) := tendsto_pure_nhds _ _
  have h3 := tendsto_nhds_unique h2 h1
  simpa using h3.symm

What this page does not claim

The lemma does not prove the uniqueness of the cost function by itself. The lemma does not specify what the cost function J is. The lemma does not claim that the limit condition holds for any particular function.

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/FunctionalEquation.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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