Foundation Primitive Recognition Calculus Prcjcost Div To Rat

Division on ratio orbits

In the Recognition Science framework, a ratio orbit is a formal pair of counting numbers, a numerator and a nonzero denominator, that represents a rational number. The framework's machine-checked library of formal theorems defines division on these ratio orbits directly: to divide one ratio orbit by another, multiply the first by the reciprocal of the second. The declaration div_toRat proves that this formal operation agrees with ordinary rational division. For any two ratio orbits q and r, the rational number represented by div q r equals q.toRat / r.toRat.

This is a definitional consistency result, not a new mathematical discovery. It says that the framework's chosen encoding of rational numbers as ratio orbits behaves as expected under division. The proof is a short chain of simplifications: unfolding the division definition, applying the multiplication and reciprocal conversion lemmas, and refl. The declaration carries the @[simp] attribute, meaning the library's automated simplifier can use it to rewrite ratio-orbit division into ordinary rational division in later proofs.

The declaration does not claim anything about the cost function J(q) = ((q + q⁻¹) / 2) - 1 that the framework studies. Division is defined on ratio orbits before any cost is attached to them. The cost object onRatioOrbit uses division only inside its formula, and div_toRat merely supports that formula's conversion to rationals. The declaration also does not claim that every rational number has a unique ratio orbit representation; normalization is a separate theorem (normalized_invariant). Finally, div_toRat says nothing about the real-number uniqueness theorem for the cost function, which is bridged separately in the library.

What the declaration changes is practical: it makes ratio-orbit division computable and transparent. A reader who wants to verify that the framework's rational arithmetic matches ordinary arithmetic can check this lemma directly. It is a small but load-bearing piece of the framework's foundation, ensuring that the formal encoding does not silently diverge from the rational numbers it claims to represent.

theorem div_toRat (q r : RatioOrbit) :
    (div q r).toRat = q.toRat / r.toRat := by
  unfold div
  rw [RatioOrbit.mul_toRat, RatioOrbit.recip_toRat]
  rfl

What this page does not claim

The declaration does not prove any property of the cost function J. The declaration does not assert uniqueness of ratio orbit representations. The declaration does not address the real-number uniqueness theorem for the cost 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/Foundation/PrimitiveRecognitionCalculus/PRCJCost.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: