Foundation Primitive Recognition Calculus Integer Order Recip Num Balanced Negat

The reciprocal's numerator

In ordinary arithmetic, the reciprocal of a number is 1 divided by that number. The reciprocal of 2 is 1/2, and the reciprocal of -3 is -1/3. A basic question about this operation is how it interacts with the sign of the original number. The answer is simple: the reciprocal of a negative number is negative, and the reciprocal of a positive number is positive. The exception is zero, which has no reciprocal at all.

The Recognition Science framework builds its own version of the integers, called signed orbits, which are a discrete record of counting steps that can go forward or backward. Within this framework, the declaration recip_num_balanced_negate_ofOrbit_den_iff_negativeFlag_of_not_balanced_zero is a proved theorem about the numerator of a reciprocal. It states a precise condition: if the numerator of a signed orbit is not balanced against zero (meaning it is not zero in the framework's sense), then the numerator of its reciprocal is the negation of the original denominator exactly when the original numerator has a negative flag, which is the framework's way of marking a number as negative.

This theorem is a formal, machine-checked statement within the framework's library of proofs. It does not say anything about the ordinary integers or about fractions as they are usually written. It is a statement about the framework's own internal objects, the signed orbits and their ratios. The theorem does not claim that the reciprocal of a negative number is always negative in the usual sense, nor does it claim anything about the reciprocal of zero, which is undefined in the framework as it is in ordinary arithmetic.

What the theorem does establish is a consistency property of the framework's construction. It shows that the framework's way of representing negative numbers and reciprocals behaves in a way that mirrors the familiar arithmetic fact. This is not a new discovery about mathematics, but a verification that the framework's internal definitions are coherent. It is a small piece of the larger project of showing that the framework can reproduce the structure of ordinary mathematics from its own starting points.

theorem recipNonzero_num_negativeFlag_eq (a : RatioOrbit)
    (h : ¬ SignedOrbit.balanced a.num SignedOrbit.zero) :
    (RatioOrbit.recipNonzero a h).num.negativeFlag = a.num.negativeFlag := by
  by_cases hneg : a.num.negativeFlag = true
  · rw [RatioOrbit.recipNonzero_num_eq_of_negativeFlag (a := a) (h := h) hneg,
      SignedOrbit.negativeFlag_negate_ofOrbit_of_ne_zero a.den a.den_ne_zero,
      hneg]
  · have hnegFalse : a.num.negativeFlag = false := by
      cases hflag : a.num.negativeFlag with
      | false => rfl
      | true =>
          exfalso
          exact hneg hflag
    have hnonneg : a.num.nonnegFlag = true :=
      (SignedOrbit.negativeFlag_eq_false_iff_nonnegFlag_eq_true a.num).mp hnegFalse
    rw [RatioOrbit.recipNonzero_num_eq_of_nonnegFlag (a := a) (h := h) hnonneg,
      SignedOrbit.negativeFlag_ofOrbit, hnegFalse]

What this page does not claim

This theorem does not claim anything about the ordinary integers or fractions as usually written. It does not claim that the reciprocal of a negative number is always negative in the usual sense. It does not claim anything about the reciprocal of zero.

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

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