Foundation Primitive Recognition Calculus Rational Field Inv Mul Cancel

The cancellation law

In ordinary arithmetic, every nonzero number has a reciprocal: the reciprocal of 2 is 1/2, and 2 times 1/2 is 1. This rule, that a number multiplied by its reciprocal equals one, is called the cancellation law. It is what lets you simplify fractions, solve equations, and divide at all. The Recognition Science declaration inv_mul_cancel establishes that this same law holds inside the framework's own number system, which it calls the PRC rationals, a set of numbers built from discrete recognition events rather than from counting or measuring.

The theorem states: for any PRC rational a, if a is not zero, then a⁻¹ * a = 1. The condition is essential; the reciprocal of zero is not defined in the usual sense, and the framework follows that convention. The proof does not invent a new rule. It shows that the framework's reciprocal operation, defined from its multiplication, satisfies exactly the cancellation property that standard rational numbers satisfy. The machine-checked library of formal theorems confirms the proof step by step, so the statement is not an assumption but a derived fact about how the framework's arithmetic behaves.

What the declaration does not claim is broader. It does not say that the framework's rationals are the same as the everyday rationals, only that they share this one algebraic property. It does not assert that every element has a reciprocal, only those that are nonzero. It does not prove that the framework's arithmetic is complete or consistent, nor that its construction matches any physical reality. The theorem is a local result about an internal structure, a piece of the framework's foundation, not a statement about the world outside it.

The practical consequence is that within the framework, algebra works as expected. Division by a nonzero number is well-defined, equations can be manipulated, and the framework's rationals behave like a field. This matters because the framework uses these rationals to build its cost function and its constants, and a shaky arithmetic foundation would undermine everything built on top of it. The cancellation law is a load-bearing brick; this theorem confirms the brick is sound.

theorem inv_mul_cancel {a : PRCRat} (h : a.toRat ≠ 0) :
    a⁻¹ * a = 1 := by
  apply recip_mul_cancel₀
  intro hz
  apply h
  rw [hz, zero_toRat]

theorem inv_mul_cancel {a : PRCRat} (h : a.toRat ≠ 0) :
    a⁻¹ * a = 1 := by
  apply recip_mul_cancel₀
  intro hz
  apply h
  rw [hz, zero_toRat]

What this page does not claim

The declaration does not claim that PRC rationals are identical to standard rationals, only that they share the cancellation property. The declaration does not claim that division by zero is defined or meaningful. The declaration does not claim that the framework's arithmetic is complete, consistent, or physically realized.

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

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