Foundation Primitive Recognition Calculus Grow Signed Orbit Zero Le Iff Nonneg Flag Choice Free

The signed orbit test

A signed orbit is a pair of whole-number counts, one for each direction of movement along a line. Think of it as a ledger entry: the first count records steps taken forward, the second records steps taken backward. The framework's machine-checked library of formal theorems proves a small but useful fact about these objects: a signed orbit is nonnegative exactly when its nonnegative flag is set to true. The nonnegative flag is a Boolean marker that the framework attaches to each signed orbit; the theorem says the marker is not a separate choice but a faithful summary of the counts themselves.

The proof runs by unfolding definitions. The library defines an ordering on signed orbits, and the theorem rewrites the statement "the orbit is at least zero" into a check on the flag. The key step is that subtracting the zero orbit from any orbit leaves the counts unchanged, because adding zero to a count does nothing. The library then applies the rule that a natural number is nonnegative exactly when its comparison to zero returns true. The whole argument is a chain of definitional rewrites, which is why the result is a theorem rather than a model or a hypothesis.

In Recognition Science, this theorem matters because it gives a choice-free way to test a basic property. The framework models recognition events as discrete records, and signed orbits are one of the primitive objects built from those records. The theorem establishes that the nonnegative flag is not an arbitrary extra datum: it is determined by the counts. This is a small piece of the larger forcing chain, but it is the kind of lemma that keeps the framework's constructions honest, since it shows that a seemingly representational choice is actually forced by the underlying arithmetic.

For a reader who wants the plain consequence: if you have a signed orbit, you do not need to guess whether it is nonnegative. You can read the answer off the flag, and the flag is guaranteed to agree with the counts. The theorem is one of the first places where the framework's vocabulary of flags and orbits connects to ordinary arithmetic, and it does so without any hidden assumptions.

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]

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 define what a signed orbit is; it assumes the definition and proves a property about it. The theorem does not show that every signed orbit is nonnegative; it only characterizes when the flag is true. The theorem does not connect signed orbits to physical space or to the three-dimensional forcing result.

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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