Foundation Primitive Recognition Calculus Real Complete Ordered Field Prcjcost D

The translation-invariant cost

In ordinary geometry, the distance between two points on a line is unchanged if you slide both points by the same amount: the gap between 3 and 7 is the same as the gap between 10 and 14. This property, called translation invariance, is so basic that it often goes without saying. The declaration PRCJCostDistance_add_right states that the same holds for a particular cost function used in the Recognition Science framework. The cost function, which measures the price of recognizing one number from another, does not change when a fixed number is added to both arguments.

The statement is a theorem in the framework's machine-checked library of formal theorems. It says that for any three rational numbers a, b, and c, the cost of recognizing a plus c from b plus c equals the cost of recognizing a from b. The proof is short and direct: it unwraps the definition of the cost function, applies the same addition to both sides, and simplifies. The theorem also has a mirror version, PRCJCostDistance_add_left, which adds the same number on the left instead of the right. Together they show that the cost function is translation-invariant in both directions.

This property is not an afterthought. It is one of the conditions that the cost function must satisfy to be a sensible measure of recognition difficulty. Without it, the cost of recognizing two numbers would depend on where they sit on the number line, not just on how far apart they are. The theorem is a small but necessary step toward building a complete ordered field from the framework's primitive recognition calculus, a construction that aims to recover the real numbers from the cost function alone.

In Recognition Science, the framework models recognition events as a ledger, a discrete record of events, and the cost function is the forced price of those events. The theorem does not claim that the cost function is the only one with this property, nor does it say anything about what the cost function actually is. It only establishes a structural fact: the cost is invariant under translation. This invariance is what makes the cost function a candidate for a distance, and it is a prerequisite for the larger project of deriving the real numbers from recognition costs.

/-- J-cost distance is invariant under translating both endpoints on the
right. -/
theorem PRCJCostDistance_add_right (a b c : PRCRat) :
    PRCJCostDistance (a + c) (b + c) = PRCJCostDistance a b := by
  apply PRCRat.toRat_injective
  rw [PRCJCostDistance_toRat, PRCJCostDistance_toRat]
  simp [PRCJCostDistanceRatDisplay]

/-- J-cost distance is invariant under translating both endpoints on the left. -/
theorem PRCJCostDistance_add_left (a b c : PRCRat) :
    PRCJCostDistance (c + a) (c + b) = PRCJCostDistance a b := by
  apply PRCRat.toRat_injective
  rw [PRCJCostDistance_toRat, PRCJCostDistance_toRat]
  simp [PRCJCostDistanceRatDisplay]

What this page does not claim

This theorem does not identify the cost function or prove it is the unique function with translation invariance. It does not establish that the cost function is a metric in the full sense, only that it is invariant under addition. It does not show that the real numbers can be constructed from recognition costs; that is a larger project with many more steps.

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

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