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

The right-factor lemma

In the recognition calculus, the framework's discrete record of events, a signed orbit tracks a quantity that can increase or decrease in whole steps. Think of a balance sheet where each entry is either a credit or a debit, and the net position is what matters. Two signed orbits are balanced when they have the same net position, even if the individual credits and debits differ.

The module proves a congruence lemma: if two signed orbits are balanced, then multiplying each on the right by the same third orbit preserves that balance. In symbols, if b and b' are balanced, then a · b and a · b' are balanced for any a. The proof is a direct calculation with natural-number arithmetic, using the definition of balance as equality of net positions.

From that, the module derives the main result: for any c, the order relation c ≤ a · b holds if and only if c ≤ a · b', whenever b and b' are balanced. In plain words: when comparing a product against a fixed threshold, only the balanced class of the right factor matters, not its internal composition. This is a monotonicity or congruence property for the order under right multiplication.

This lemma is a small but load-bearing step in the framework's growth calculus, the part that studies how signed orbits evolve under multiplication. It guarantees that order comparisons are stable under replacing a factor by a balanced equivalent, which is exactly the kind of invariance the framework needs when it reasons about recognition costs and derived structure. The theorem is machine-checked in the framework's library of formal theorems, with no unproved assumptions beyond the standard logical axioms.

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_of_product_right_factor_iff_of_balanced_cf {c a b b' : SignedOrbit} (hb : SignedOrbit.balanced b b') : SignedOrbit.le c (SignedOrbit.mul a b) ↔ SignedOrbit.le c (SignedOrbit.mul a b') :=
  le_congr_right_of_balanced_cf (mul_balanced_congr_right_cf hb)

What this page does not claim

This module does not define the full recognition calculus or the signed orbit type; it only proves a lemma about them. The lemma does not say that order is preserved under arbitrary replacement of factors, only under balanced replacement. No claim is made about how this lemma connects to the forcing chain or 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/SignedOrbitLeOfProductRightFactorIffOfBalancedChoiceFree.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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