Foundation Primitive Recognition Calculus Grow Signed Orbit Mul Balanced Zero Of

Multiplication and balance

A signed orbit is a discrete record of events in the framework's primitive recognition calculus, where each event is marked as positive or negative. A balanced orbit is one whose positive and negative events cancel in a precise arithmetic sense: the count of positive events equals the count of negative events. The lemma mul_balanced_zero_of_balanced_zero_right_cf states that if you multiply any signed orbit by a balanced one, the result is again balanced. In symbols: if w is balanced, then (z.mul w) is balanced for every z.

The proof is a direct computation using the definitions of multiplication and balance. It rewrites balance as an equality of natural-number counts, expands the multiplication into positive and negative parts, and then applies the hypothesis that w's counts are equal. The argument is short and purely algebraic; it does not depend on any special properties of the orbits beyond the definitions.

In Recognition Science, this lemma is a small structural fact about the algebra of orbits. It shows that the balanced orbits form a kind of closed set under right multiplication: multiplying anything by a balanced orbit stays inside the balanced class. This is a stepping stone for larger results about the calculus, but the lemma itself says nothing about which orbits are balanced or how balance arises from the forcing chain.

theorem mul_balanced_zero_of_balanced_zero_right_cf
    (z w : SignedOrbit)
    (hw : w.balanced SignedOrbit.zero) :
    (z.mul w).balanced SignedOrbit.zero := by
  rw [SignedOrbit.balanced_iff_toNat_eq] at hw ⊢
  simp only [SignedOrbit.mul_pos, SignedOrbit.mul_neg, SignedOrbit.zero,
    DistinctionNat.toNat_add, DistinctionNat.toNat_mul, DistinctionNat.toNat_zero,
    Nat.add_zero, Nat.zero_add] at hw ⊢
  rw [hw]

theorem mul_balanced_zero_of_balanced_zero_right_cf
    (z w : SignedOrbit)
    (hw : w.balanced SignedOrbit.zero) :
    (z.mul w).balanced SignedOrbit.zero := by
  rw [SignedOrbit.balanced_iff_toNat_eq] at hw ⊢
  simp only [SignedOrbit.mul_pos, SignedOrbit.mul_neg, SignedOrbit.zero,
    DistinctionNat.toNat_add, DistinctionNat.toNat_mul, DistinctionNat.toNat_zero,
    Nat.add_zero, Nat.zero_add] at hw ⊢
  rw [hw]

What this page does not claim

The lemma does not claim that every orbit is balanced or that balance is common. It does not claim that multiplication is commutative or that the balanced set forms a group. It does not claim any connection to the fine-structure constant or other physical constants.

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

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