Foundation Primitive Recognition Calculus Grow Signed Orbit Le Product Right Factor Iff Of Balanced Choice Free

A structural lemma about signed orbits

A signed orbit is a pair of natural numbers, one positive and one negative, written as a single object. Think of it as a tally sheet with two columns: how many recognition events count for a thing, and how many count against it. The framework's ledger, a discrete record of events, keeps these tallies as its basic entries.

Two signed orbits are called balanced when their difference is zero: the positive column of one plus the negative column of the other equals the negative column of the first plus the positive column of the second. This is an equivalence relation, and it is the natural notion of sameness for these tallies.

The module proves a structural fact about multiplication. If you multiply a fixed signed orbit on the left by two balanced partners, the two resulting products are again balanced. The proof is a direct computation on the natural-number entries, using distributivity and the defining balance equation.

From that, the module derives an ordering lemma. Given a fixed left factor and a bound, the product with one balanced partner lies at or below the bound exactly when the product with the other balanced partner does. In symbols: if b and b' are balanced, then (a·b) ≤ c holds if and only if (a·b') ≤ c holds.

In plain language, this means the ordering on products is insensitive to replacing a right factor by a balanced equivalent. The lemma is a congruence property: balance is preserved under left multiplication, and the ≤ relation respects that preservation. It is a building block for larger structural results in the framework's calculus of recognition, though it does not by itself establish any claim about the physical world.

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

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)

What this page does not claim

This module does not establish any physical claim about recognition or the universe. The lemma does not show that balance is preserved under right multiplication by a fixed factor. The ordering property does not hold for arbitrary right factors, only balanced ones.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: