Foundation Primitive Recognition Calculus Grow Signed Orbit Le Product Right Fac

The balanced factor rule

A signed orbit is a discrete record of recognition events, each event carrying a positive or negative sign, like a ledger entry marked credit or debit. The framework's library, a machine-checked collection of formal theorems, proves a rule about multiplying such orbits on the left. The rule says: if two orbits b and b' are balanced, meaning their total signed weight is equal, then for any third orbit a, the products a·b and a·b' compare identically against any fourth orbit c. In symbols, the theorem states that a·b ≤ c holds exactly when a·b' ≤ c holds, given that b and b' are balanced.

The proof is short and computational. It rewrites the balanced condition into an equality of natural-number counts, expands the multiplication into positive and negative parts, and then applies arithmetic identities to show the comparison is preserved. The theorem is derived from a more basic lemma, mul_balanced_congr_right_cf, which states that multiplying a balanced pair on the right by any orbit preserves balance. The main result then follows by a standard congruence argument on the left side of a comparison.

What the theorem does not claim is just as important. It does not say that balanced orbits are interchangeable in all contexts, only in this specific left-multiplication and comparison setting. It does not assert that any two orbits with the same total weight are equal as objects, only that they behave the same way under this operation. It also does not extend to right multiplication: the result is explicitly about multiplying on the left, and no symmetric claim for right factors is stated or proved in this declaration.

The practical upshot is a clean substitution rule for a narrow but useful class of computations. When a proof or calculation involves a product a·b and the orbit b is known to be balanced with b', the theorem licenses replacing b by b' without changing the truth of any comparison against a fixed c. This is a small building block, but it is the kind of lemma that lets larger arguments about recognition ledgers proceed without rechecking every arithmetic detail.

theorem le_product_right_factor_iff_of_balanced_cf {a b b' c : SignedOrbit} (hb : SignedOrbit.balanced b b') : SignedOrbit.le (SignedOrbit.mul a b) c ↔ SignedOrbit.le (SignedOrbit.mul a b') c :=
  le_congr_left_of_balanced_cf (mul_balanced_congr_right_cf hb)

theorem mul_balanced_congr_right_cf {a b b' : SignedOrbit} (hb : SignedOrbit.balanced b b') : SignedOrbit.balanced (SignedOrbit.mul a b) (SignedOrbit.mul a b') := by
  rw [SignedOrbit.balanced_iff_toNat_eq] at hb ⊢
  simp only [SignedOrbit.mul_pos, SignedOrbit.mul_neg, DistinctionNat.toNat_add, DistinctionNat.toNat_mul]
  have hb2 : b.neg.toNat + b'.pos.toNat = b'.neg.toNat + b.pos.toNat := by omega
  have e1 : a.pos.toNat * b.pos.toNat + a.pos.toNat * b'.neg.toNat = a.pos.toNat * b'.pos.toNat + a.pos.toNat * b.neg.toNat := by
    rw [← Nat.mul_add, ← Nat.mul_add, hb]
  have e2 : a.neg.toNat * b.neg.toNat + a.neg.toNat * b'.pos.toNat = a.neg.toNat * b'.neg.toNat + a.neg.toNat * b.pos.toNat := by
    rw [← Nat.mul_add, ← Nat.mul_add, hb2]
  omega

What this page does not claim

Balanced orbits are not asserted to be equal as objects, only equivalent under this specific left-multiplication comparison. The theorem does not state any rule for right multiplication by a balanced factor. No claim is made that the comparison relation ≤ is total or that all orbits are comparable.

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

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