Foundation Primitive Recognition Calculus Grow Ratio Orbit Mul Pos

The ratio orbit and its multiplication

A ratio orbit is a pair of counting numbers, one for the negative side and one for the positive side, written as (neg, pos). It records a state in a discrete ledger: a running tally of events that can be marked down or up. The ledger is a ledger, a discrete record of events, and each orbit is one snapshot of its two-sided balance. The positive count is the number of upward marks, the negative count the number of downward marks. A ratio orbit is called strictly positive when the positive count is larger than the negative count, meaning the ledger has net upward motion.

The central result is that multiplication respects this positivity. If you take two strictly positive ratio orbits and multiply them, the product is again strictly positive. The multiplication rule is not component-wise; it mixes the counts. The positive part of the product is pos_a * pos_b + neg_a * neg_b, and the negative part is pos_a * neg_b + neg_a * pos_b. This is the same algebraic shape as multiplying two complex numbers or two hyperbolic rotations, but here the entries are natural numbers and the result is a theorem about the ledger, not an assumption.

The proof is a short chain of arithmetic facts. It first rewrites the product's positive and negative parts using the definitions of the multiplication, then uses the hypothesis that each input has a larger positive count. The key step is a standard natural-number identity: if a < b and c < d, then a*d + c*b < b*d + a*c. The proof obtains this by writing b = a + s and d = c + t, substituting, and simplifying with the ring and omega tactics. The final theorem, ltQ_mul_pos, packages this into the statement that the product of two strictly positive ratio orbits is strictly positive.

In Recognition Science, this is one of the primitive steps in the grow calculus, the part of the framework that studies how ledgers expand. The result matters because it shows that the property of being net-positive is closed under the ledger's own multiplication. It is a structural guarantee: once a ledger is growing, multiplying it by another growing ledger keeps it growing. This is not an empirical observation; it is a theorem proved in the machine-checked library of formal theorems, with no hidden assumptions beyond the definitions.

The practical consequence is that the framework can build larger composite ledgers from smaller positive ones without losing the growth property. This closure is the kind of fact that later stages of the framework rely on when they construct more complex objects from ratio orbits. For a reader, the takeaway is simple: in this calculus, growth is contagious under multiplication, and the proof is a matter of arithmetic, not of interpretation.

theorem ltQ_mul_pos (p q : RatioOrbit) (hp : ltQ RatioOrbit.zero p) (hq : ltQ RatioOrbit.zero q) : ltQ RatioOrbit.zero (RatioOrbit.mul p q) := by
  rw [zero_ltQ_iff_num] at hp hq ⊢
  have h : (RatioOrbit.mul p q).num = SignedOrbit.mul p.num q.num := rfl
  rw [h]
  exact mul_strictpos_cf p.num q.num hp hq

theorem mul_strictpos_cf (a b : SignedOrbit) (ha : a.neg.toNat < a.pos.toNat) (hb : b.neg.toNat < b.pos.toNat) : (SignedOrbit.mul a b).neg.toNat < (SignedOrbit.mul a b).pos.toNat := by
  have hpos : (SignedOrbit.mul a b).pos.toNat = a.pos.toNat * b.pos.toNat + a.neg.toNat * b.neg.toNat := by
    show (a.pos * b.pos + a.neg * b.neg).toNat = _
    rw [DistinctionNat.toNat_add, DistinctionNat.toNat_mul, DistinctionNat.toNat_mul]
  have hneg : (SignedOrbit.mul a b).neg.toNat = a.pos.toNat * b.neg.toNat + a.neg.toNat * b.pos.toNat := by
    show (a.pos * b.neg + a.neg * b.pos).toNat = _
    rw [DistinctionNat.toNat_add, DistinctionNat.toNat_mul, DistinctionNat.toNat_mul]
  rw [hpos, hneg]; obtain ⟨s, hs⟩ := Nat.exists_eq_add_of_lt ha; obtain ⟨t, ht⟩ := Nat.exists_eq_add_of_lt hb; rw [hs, ht]; ring_nf; omega

theorem mul_strictpos_cf (a b : SignedOrbit) (ha : a.neg.toNat < a.pos.toNat) (hb : b.neg.toNat < b.pos.toNat) : (SignedOrbit.mul a b).neg.toNat < (SignedOrbit.mul a b).pos.toNat := by
  have hpos : (SignedOrbit.mul a b).pos.toNat = a.pos.toNat * b.pos.toNat + a.neg.toNat * b.neg.toNat := by
    show (a.pos * b.pos + a.neg * b.neg).toNat = _
    rw [DistinctionNat.toNat_add, DistinctionNat.toNat_mul, DistinctionNat.toNat_mul]
  have hneg : (SignedOrbit.mul a b).neg.toNat = a.pos.toNat * b.neg.toNat + a.neg.toNat * b.pos.toNat := by
    show (a.pos * b.neg + a.neg * b.pos).toNat = _
    rw [DistinctionNat.toNat_add, DistinctionNat.toNat_mul, DistinctionNat.toNat_mul]
  rw [hpos, hneg]; obtain ⟨s, hs⟩ := Nat.exists_eq_add_of_lt ha; obtain ⟨t, ht⟩ := Nat.exists_eq_add_of_lt hb; rw [hs, ht]; ring_nf; omega

What this page does not claim

This module does not define what a ratio orbit represents physically. The theorem does not claim that all ratio orbits are strictly positive. The multiplication rule is not claimed to be commutative or associative in this module.

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

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