Foundation Primitive Recognition Calculus Grow Ratio Orbit Order Add Mono Le Q A

Order and addition

In mathematics, a total order is a way to arrange elements so that any two can be compared. An order is compatible with addition when adding the same amount to both sides of an inequality keeps the inequality true: if x is at most y, then x plus z is at most y plus z. This property, called translation invariance, is so familiar in ordinary arithmetic that it is easy to forget it can fail in other settings.

Recognition Science builds its account of growth on a discrete record of events, a ledger in which each entry is a ratio of two natural counts. The framework's library of formal theorems contains a declaration, leQ_add_right, which proves that this ratio order is translation invariant. The statement is precise: for any three ratio orbits p, q, and r, if p is at most q, then p added to r is at most q added to r. The proof is choice-free, meaning it does not rely on the axiom of choice, and it routes through a natural-number comparison bridge rather than a signed-integer one.

The theorem matters because it gives the order structure on ratio orbits a stability that later growth arguments depend on. Without translation invariance, adding a common step could reorder states, and the framework's monotone growth results would lose their footing. The declaration is a small but load-bearing piece of the primitive recognition calculus, the part of the framework that studies how a ledger grows one tick at a time.

What the theorem does not claim is broader. It does not assert that every order on ratio orbits is translation invariant, only that this particular one is. It does not say that addition is commutative or associative for ratio orbits; those are separate facts. And it does not establish that the ratio order is total, that any two orbits are comparable. The theorem is a monotonicity result, not a completeness result.

/-- Translation invariance of the delta-native ratio order: the order on `RatioOrbit`
    is compatible with addition. -/
theorem leQ_add_right (p q r : RatioOrbit) (h : leQ p q) :
    leQ (RatioOrbit.add p r) (RatioOrbit.add q r) := by
  unfold leQ at h ⊢
  rw [le_iff_toInt_le_cf] at h ⊢
  unfold RatioOrbit.add
  simp only [SignedOrbit.mul_toInt, SignedOrbit.ofOrbit_toInt, SignedOrbit.scaleByNat_toInt,
             SignedOrbit.add_toInt, DistinctionNat.toNat_mul] at h ⊢
  push_cast at h ⊢
  rw [← Int.sub_nonneg] at h ⊢
  have hc : (0:ℤ) ≤ (r.den.toNat : ℤ) * (r.den.toNat : ℤ) :=
    Int.mul_nonneg (by omega) (by omega)
  have hprod := Int.mul_nonneg h hc
  convert hprod using 1 <;> ring

/-- Choice-free Int order bridge, routed through the CF Nat bridge `le_iff_toNat_cf`
    (never through the choice-tainted `SignedOrbit.le_iff_toInt_le`). -/
private lemma le_iff_toInt_le_cf (a b : SignedOrbit) :
    SignedOrbit.le a b ↔ a.toInt ≤ b.toInt := by
  rw [le_iff_toNat_cf, toInt_eq a, toInt_eq b]
  constructor
  · intro hh; omega
  · intro hh; omega

What this page does not claim

The theorem does not assert that every order on ratio orbits is translation invariant. The theorem does not establish that addition on ratio orbits is commutative or associative. The theorem does not prove that the ratio order is total, only that it is monotone under addition.

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

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