Foundation Primitive Recognition Calculus Grow Ratio Orbit Order Add Mono

Order and addition

In mathematics, an ordered group is a set with both an addition and a comparison that work together: if one number is at most another, adding the same third number to both preserves the comparison. This result establishes exactly that compatibility for the ratio orbits used in the framework's recognition calculus. It proves the theorem leQ_add_right: for ratio orbits p, q, and r, if p is at most q, then p plus r is at most q plus r.

The proof routes through a choice-free bridge. Instead of relying on a potentially choice-tainted lemma, the argument unfolds each signed orbit into its positive and negative natural-number parts, converts that difference to an integer, and then applies the standard integer order. The key lemma le_iff_toInt_le_cf states that the orbit order is equivalent to comparing these integer values, and the documentation explicitly notes it never uses the choice-tainted alternative. This keeps the entire argument constructive, in the sense that it does not invoke the axiom of choice.

In Recognition Science, this monotonicity result is a small but load-bearing piece of the primitive recognition calculus. The framework models reality as a discrete record of events, and it derives constants and dimensions from a forced cost function. The ratio orbits are the objects that carry growth ratios through the calculus; knowing their order is compatible with addition means the arithmetic of these orbits behaves like ordinary number arithmetic. The theorem is proved in the machine-checked library of formal theorems, with no framework-specific axioms beyond the standard logical ones.

The practical payoff is that later constructions can treat ratio orbits as if they were numbers: comparisons survive translation, so any inequality established at one stage remains true after adding a common orbit. This is the kind of foundation that lets the framework build upward without rechecking basic arithmetic at every step.

/-- 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

This result does not define the ratio orbit structure itself; it only proves a property about its order. The theorem does not establish that the framework's cost function or constants are derived; those are separate results.

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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