Foundation Primitive Recognition Calculus Grow Signed Orbit Mul Balanced Zero Of Balanced Zero Right Choice Free

The closed ledger

A signed orbit is a discrete record of events, each marked as positive or negative, together with a rule for combining two such records. The framework's ledger, a discrete record of recognition events, uses these orbits as its basic objects. A signed orbit is balanced when its positive and negative counts are equal, so it sums to zero. The theorem states that if two signed orbits are each balanced, then their product, defined by the framework's multiplication rule, is also balanced. In plain terms, the balanced property is closed under multiplication: combining two balanced ledgers yields another balanced ledger.

The proof is short and mechanical. It rewrites the balanced condition in terms of natural-number counts, expands the multiplication rule into addition and multiplication of those counts, and then applies the hypothesis that the second orbit is balanced. The result follows directly from arithmetic. This is a small lemma, but it is load-bearing for the framework's arithmetic of recognition events: it guarantees that the product operation never leaves the balanced class, so the framework can build larger structures without stepping outside its own ledger discipline.

In Recognition Science, this closure property is one of the elementary steps in a larger forcing chain. The framework models physical structure as arising from a cost function on recognition events, and the signed orbits are a primitive layer beneath that cost. This result establishes a consistency condition: if the ledger starts balanced, multiplication preserves that balance. It is not a claim about physics directly; it is a formal guarantee about the arithmetic of the framework's own primitives. The theorem is checked in the framework's machine-checked library of formal theorems, so the closure holds by construction, not by convention.

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

This proof does not establish that the multiplication rule is the only possible one. The theorem does not claim that physical recognition events are balanced in nature. The result does not derive any physical constant or empirical prediction.

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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