Foundation Primitive Recognition Calculus Grow Ratio Orbit Le Neg Le Q Neg Neg I

Order reversal under negation

A ratio orbit is a discrete sequence of ratios generated by repeated application of a fixed growth rule. The framework's library of machine-checked theorems records an order relation, written leQ, on these orbits: one orbit is at most another when it sits earlier in the growth ordering. The declaration leQ_neg_neg_iff establishes a symmetry: negating both orbits reverses the comparison. If orbit p is at most orbit q, then the negation of q is at most the negation of p, and conversely. In symbols, leQ (negate q) (negate p) holds exactly when leQ p q holds.

This is a theorem in the framework's library, proved for the specific definition of ratio orbit and the specific order relation leQ. It is not a claim about numbers in the ordinary sense. Negation here is an operation on orbits, not on real numbers, and the order relation is the framework's growth order, not the usual less-than-or-equal on the real line. The theorem holds for all ratio orbits p and q, with no additional conditions stated in the declaration.

The practical content is a reversal law. In any ordered structure, negating both sides of a comparison often flips the direction, as with real numbers where x ≤ y implies -y ≤ -x. The framework's ratio orbits carry an analogous property, despite the order being defined by growth rather than by magnitude. This symmetry is a building block for further results about the structure of growth sequences, though the declaration itself only states the equivalence, not any consequence about specific orbits.

In Recognition Science, this theorem supports the internal consistency of the growth order. The framework models recognition events as a discrete ledger, and ratio orbits describe how ratios evolve under repeated application of a rule. The reversal law ensures that the order relation behaves coherently under the negation operation, a property that later results about comparison and scaling can rely on. The theorem is proved in the framework's library, so its status is that of a derived result within the framework, not an empirical observation.

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

The theorem does not claim that negation preserves the order, only that it reverses it. The theorem does not claim anything about the usual order on real numbers, since ratio orbits are not real numbers. The theorem does not establish any property of specific ratio orbits, only a general equivalence.

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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