Foundation Primitive Recognition Calculus Grow Signed Orbit Order Choice Free No

The choice-free sign test

In ordinary arithmetic, deciding whether a number is negative or nonnegative seems trivial. The ledger, a discrete record of events, tracks each signed quantity as a pair of natural-number positions: one for the positive part, one for the negative part. The theorem nonnegFlag_iff_nonneg_cf states that a signed orbit is nonnegative exactly when its negative position is at most its positive position. This is a purely structural fact, proved by induction on the positions themselves.

The proof avoids a subtle logical dependency. Earlier versions of the order relation routed through the integer type, which carries a choice axiom, an assumption that a selection can be made without specifying how. The new theorem rebuilds the same fact using only the natural-number order and a Boolean flag, so its axiom audit reduces to just two standard logical principles: propositional extensionality and quotient soundness. The machine-checked library of formal theorems records this as a choice-free foundation.

What the theorem does not claim is broader significance. It does not assert that the signed-orbit order is the only possible order, nor that it has any physical meaning. It establishes only a clean logical reduction: the sign test collapses to a comparison of two natural numbers. This matters because downstream theorems about ratios and ordered fields can now build on a foundation free of an unnecessary choice assumption.

For a reader, the practical consequence is confidence. When the framework later proves properties of ratios or comparisons, those proofs inherit a cleaner logical pedigree. The theorem itself is a small but precise step: it removes a hidden assumption from a basic operation, making the framework's logical foundations more transparent and more reliable.

/-- The structural nonnegative flag agrees with internal nonnegativity, proved
choice-free via `leq_eq_true_iff_cf` (NOT `nonnegFlag_eq_true_iff_nonneg`, which goes through `ℤ`). -/
theorem nonnegFlag_iff_nonneg_cf (z : SignedOrbit) :
    z.nonnegFlag = true ↔ SignedOrbit.nonneg z := by
  rw [nonneg_iff_toNat_le]
  unfold SignedOrbit.nonnegFlag
  rw [leq_eq_true_iff_cf]

/-- Choice-free restatement of the structural Boolean order's agreement with the ℕ order.
The original `DistinctionNat.leq_eq_true_iff` is choice-TAINTED (its `simp [leq]` proof pulls
`Classical.choice` through a classical `Bool`/`Decidable` simp lemma). Here the same fact is
reproved by bare structural induction on the δ-orbit positions: every step is a constructor
match plus `omega` over ℕ, so the closure stays inside `{propext, Quot.sound}`. -/
theorem leq_eq_true_iff_cf (a b : DistinctionNat) :
    DistinctionNat.leq a b = true ↔ a.toNat ≤ b.toNat := by
  induction a generalizing b with
  | zero =>
      cases b with
      | zero => exact ⟨fun _ => Nat.le_refl _, fun _ => rfl⟩
      | succ b => exact ⟨fun _ => Nat.zero_le _, fun _ => rfl⟩
  | succ a ih =>
      cases b with
      | zero =>
          constructor
          · intro h
            exact absurd h Bool.false_ne_true
          · intro h
            rw [DistinctionNat.toNat_succ, DistinctionNat.toNat_zero] at h
            exact absurd h (Nat.not_succ_le_zero _)
      | succ b =>
          have hstep : DistinctionNat.leq (DistinctionNat.succ a) (DistinctionNat.succ b)
              = DistinctionNat.leq a b := rfl
          rw [hstep, ih, DistinctionNat.toNat_succ, DistinctionNat.toNat_succ]
          exact Nat.succ_le_succ_iff.symm

/-- The structural nonnegative flag agrees with internal nonnegativity, proved
choice-free via `leq_eq_true_iff_cf` (NOT `nonnegFlag_eq_true_iff_nonneg`, which goes through `ℤ`). -/
theorem nonnegFlag_iff_nonneg_cf (z : SignedOrbit) :
    z.nonnegFlag = true ↔ SignedOrbit.nonneg z := by
  rw [nonneg_iff_toNat_le]
  unfold SignedOrbit.nonnegFlag
  rw [leq_eq_true_iff_cf]

What this page does not claim

The theorem does not assert that the signed-orbit order is unique or physically meaningful. It does not claim that all choice axioms are eliminable from the framework, only from this specific order foundation. It does not establish any empirical or experimental content about signed orbits.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: