Foundation Primitive Recognition Calculus Grow Ratio Orbit Le Trans Antisymm Le

Ordering the orbits

In mathematics, an antisymmetry law is the precise way of saying that two things cannot each be smaller than or equal to the other unless they are actually the same thing. For ordinary numbers, if a ≤ b and b ≤ a, then a = b. The Recognition Science declaration leQ_antisymm establishes exactly this kind of law for a special ordering called leQ, which compares growth ratios in the framework's primitive recognition calculus.

The objects being ordered are RatioOrbit elements, which represent possible growth ratios arising in the framework's discrete recognition ledger, a record of events where the cost of recognition is forced. The ordering leQ compares two ratios by cross-multiplying their numerator and denominator pairs. The theorem states that if leQ p q and leQ q p both hold, then the two ratios are equivalent under the relation crossEq, which identifies ratios whose cross products are equal. In plain terms, mutual ordering forces equality of the underlying ratios.

The proof is choice-free, meaning it does not rely on the axiom of choice. It proceeds by converting the cross-multiplication inequalities into integer comparisons, then applying the standard antisymmetry law for integers. The result is a clean structural fact: the ordering on growth ratios behaves like a partial order up to the intended equivalence, not a weaker or loopier relation.

What leQ_antisymm does not claim is that the ordering is total, that every pair of ratios is comparable. It also does not claim that crossEq is equality of the ratios themselves as distinct objects, only equivalence under the defined relation. The theorem is a local structural result about the ordering, not a statement about which growth ratios actually occur or what their numerical values are.

/-- Antisymmetry of `leQ` up to the rational equivalence `crossEq`: mutual `leQ`
    forces the cross products to be equal, which is exactly `crossEq` through the
    purified `crossEq_iff_toIntCross` bridge. Choice-free. -/
theorem leQ_antisymm (p q : RatioOrbit) (hpq : leQ p q) (hqp : leQ q p) :
    RatioOrbit.crossEq p q := by
  unfold leQ at hpq hqp
  rw [SignedOrbit.le_iff_toInt_le] at hpq hqp
  simp only [SignedOrbit.mul_toInt, SignedOrbit.ofOrbit_toInt] at hpq hqp
  rw [RatioOrbit.crossEq_iff_toIntCross]
  exact Int.le_antisymm hpq hqp

/-- Antisymmetry of `leQ` up to the rational equivalence `crossEq`: mutual `leQ`
    forces the cross products to be equal, which is exactly `crossEq` through the
    purified `crossEq_iff_toIntCross` bridge. Choice-free. -/
theorem leQ_antisymm (p q : RatioOrbit) (hpq : leQ p q) (hqp : leQ q p) :
    RatioOrbit.crossEq p q := by
  unfold leQ at hpq hqp
  rw [SignedOrbit.le_iff_toInt_le] at hpq hqp
  simp only [SignedOrbit.mul_toInt, SignedOrbit.ofOrbit_toInt] at hpq hqp
  rw [RatioOrbit.crossEq_iff_toIntCross]
  exact Int.le_antisymm hpq hqp

What this page does not claim

leQ is a total order on RatioOrbit elements. crossEq is identity of the RatioOrbit objects themselves. The theorem determines which growth ratios occur in the framework.

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

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