Foundation Primitive Recognition Calculus Grow Signed Orbit Zero Le Iff Nonneg F

The zero test

A signed orbit is a record of two counting numbers, one for positive and one for negative steps, kept as a discrete ledger of events. The theorem zero_le_iff_nonnegFlag_cf says that such an orbit is at least zero exactly when its nonnegative flag is true. In plain terms: the ledger is not negative precisely when its marker says so. This is a proved equivalence, not a definitional choice.

The proof runs by rewriting the order relation in terms of subtraction and then unfolding the flag. Subtracting zero from an orbit leaves both counts unchanged, so the comparison reduces to checking the flag directly. The argument is short because the framework has already established that order and flags are interchangeable for signed orbits.

What this theorem does not claim is broader. It does not say that every nonnegative orbit is positive, nor that the flag is the only way to test order. It also does not assert anything about how signed orbits relate to the cost function or the golden ratio; those are separate results in the framework's library. The theorem is a local bridge between two representations of the same fact.

The consequence is practical for the framework: when a proof needs to know whether an orbit is nonnegative, it can use the flag as a computational handle. The equivalence lets the machine-checked library switch between a relational statement and a boolean test without loss. That is the kind of small, exact link that makes larger derivations tractable.

theorem zero_le_iff_nonnegFlag_cf (z : SignedOrbit) :
    SignedOrbit.le SignedOrbit.zero z ↔ z.nonnegFlag = true := by
  rw [le_iff_nonnegFlag_sub_cf]
  unfold SignedOrbit.nonnegFlag
  rw [leq_eq_true_iff_cf, leq_eq_true_iff_cf]
  have hp : (SignedOrbit.sub z SignedOrbit.zero).pos = z.pos + DistinctionNat.zero := rfl
  have hn : (SignedOrbit.sub z SignedOrbit.zero).neg = z.neg + DistinctionNat.zero := rfl
  rw [hp, hn, DistinctionNat.toNat_add, DistinctionNat.toNat_add,
      DistinctionNat.toNat_zero, Nat.add_zero, Nat.add_zero]

What this page does not claim

The theorem does not prove that a nonnegative orbit is positive. The theorem does not relate signed orbits to the golden ratio or the cost function. The theorem does not define the flag; it only links the flag to the order relation.

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

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