Cost Aczel Classification H One Of Normalized

The normalization step

The declaration H_one_of_normalized establishes a small but load-bearing fact inside the Recognition Science framework's proof that a single cost function is forced. The framework models cost as a function F that assigns a nonnegative number to a recognition event, where the event is a ratio x of two quantities being compared. The declaration concerns a helper function H, defined from F by a logarithmic change of variables, and it proves: if F is normalized, meaning F(1) = 0, then H(0) = 1. In plain terms, when the two quantities being compared are equal, the cost is zero, and the helper function that the proof works with takes the value one at that point.

This fact is not a conclusion about the physical world by itself. It is a step in a chain. The chain begins with five plain conditions on the cost function: reciprocal symmetry, zero cost at unity, a forced composition law, calibration, and continuity. The framework's library, a machine-checked collection of formal theorems, proves that any cost function satisfying those conditions must equal J(x) = (x + 1/x)/2 - 1. The declaration H_one_of_normalized is one of the early links in that proof: it supplies the starting value H(0) = 1 that later steps use to derive the differential equation H'' = H, which in turn leads to the uniqueness result.

The declaration does not claim that normalization alone forces the cost function, nor that it establishes the full uniqueness theorem. It claims only the specific equality H(0) = 1, under the single assumption that F is normalized. The full theorem requires all five conditions together. The declaration also does not claim anything about what the cost function is for any particular physical system; it is a statement about the abstract framework's axioms, not about measurements.

private theorem H_one_of_normalized (F : ℝ → ℝ)
    (hNorm : IsNormalized F) : H F 0 = 1 := by
  have h0 : F 1 = 0 := by simpa [IsNormalized] using hNorm
  simp [H, G, h0]

What this page does not claim

Normalization alone does not force the cost function; all five primitive hypotheses are needed. The declaration does not establish the uniqueness theorem J(x) = (x + 1/x)/2 - 1 by itself. No claim is made about the cost function for any specific physical system.

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

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