Foundation Primitive Recognition Calculus Grow Ratio Orbit Le Neg

The negation rule

The ratio orbit, a discrete sequence of ratios generated by a fixed growth rule, carries an ordering that compares two orbits term by term. The classical question is simple: if you reverse the sign of every ratio in two orbits, does the ordering between them reverse as well? For ordinary numbers, negating reverses order: if a is less than b, then negative b is less than negative a. The framework asks whether the same reversal holds for its entire orbit structure, not just for single values.

The module RatioOrbitLeNeg answers exactly this. Its theorem leQ_neg_neg_iff states that for any two ratio orbits p and q, the ordering leQ of the negated orbits (negate q) and (negate p) holds if and only if the original ordering leQ p q holds. In plain language: flipping the sign of every ratio in both orbits flips the ordering between them. This is the orbit-level analogue of the familiar number-line rule, and it holds without exception across the entire discrete structure.

Why does this matter? The framework's growth calculus builds larger claims from such local symmetries. A reversal rule like this one is a consistency check: it ensures that the ordering relation behaves predictably under the fundamental operation of negation. Without it, comparisons between orbits could shift arbitrarily when signs flip, and any derived inequality would carry an unstated caveat. With it, the ordering is stable under sign reversal, so later theorems can safely invert orbit comparisons as needed.

The proof is direct: it unfolds the definition of the ordering and the negation operation, then applies the corresponding reversal property for the underlying numeric comparison. The result is a machine-checked theorem in the framework's library of formal proofs, which means the claim is verified by a computer program that checks every logical step against the axioms. This is not a heuristic or a hand-waved symmetry; it is a formal guarantee that the reversal rule holds for the entire orbit type.

In practice, this lemma functions as a small but load-bearing pivot. When later work in the framework needs to compare orbits after a sign flip, it can cite this theorem instead of re-deriving the reversal from first principles. The framework's growth calculus thus gains a reusable tool: a single, verified rule that makes sign-flipped comparisons safe. This is how the framework builds its larger results: not by assuming symmetries, but by proving each one and then reusing it.

theorem leQ_neg_neg_iff (p q : RatioOrbit) :
    leQ (RatioOrbit.negate q) (RatioOrbit.negate p) ↔ leQ p q := by
  unfold leQ
  rw [SignedOrbit.le_iff_toInt_le, SignedOrbit.le_iff_toInt_le]
  simp only [RatioOrbit.negate, SignedOrbit.mul_toInt, SignedOrbit.ofOrbit_toInt,
    SignedOrbit.negate_toInt]
  constructor
  · intro h
    have e1 : -(q.num.toInt) * ↑(p.den.toNat) = -(q.num.toInt * ↑(p.den.toNat)) := by ring
    have e2 : -(p.num.toInt) * ↑(q.den.toNat) = -(p.num.toInt * ↑(q.den.toNat)) := by ring
    rw [e1, e2] at h
    omega
  · intro h
    have e1 : -(q.num.toInt) * ↑(p.den.toNat) = -(q.num.toInt * ↑(p.den.toNat)) := by ring
    have e2 : -(p.num.toInt) * ↑(q.den.toNat) = -(p.num.toInt * ↑(q.den.toNat)) := by ring
    rw [e1, e2]
    omega

What this page does not claim

This module does not define the ratio orbit or the growth rule; it assumes them. The theorem does not claim that negation preserves ordering, only that it reverses it consistently.

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

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