Foundation Primitive Recognition Calculus Rigidity Ledger Transport Add Comm Tra

Transporting a proof

The natural numbers, 0, 1, 2, and so on, are the familiar counting numbers. The statement that addition commutes, that a + b always equals b + a, is one of the first facts a student meets. Recognition Science begins with a more primitive object: a ledger, a discrete record of distinction events, where each step marks a new difference. Its formal library, a machine-checked collection of formal theorems, proves commutativity for this primitive structure, and the declaration addComm_transported extends that proof to any structure satisfying the Peano axioms, the standard rules that characterize the natural numbers.

The transport works through a map that unrolls each natural number into the corresponding sequence of distinction steps in the target structure. The library proves this map is a bijection, a perfect one-to-one pairing, for any Peano model. The key lemma, the Transport Lemma, shows that a formula true in the canonical model remains true after transport, formula by formula. The proof is choice-free: it never invokes the axiom of choice, a principle sometimes needed in mathematics to make selections without a rule. This matters because the framework aims to show its results rest on a minimal set of postulates.

In Recognition Science, the declaration establishes a specific instance: the commutativity of distinction-composition, the Gödel-test theorem, holds in every Peano model at zero transported cost. The ledger, the record of which derivation steps were used, is invariant under this change of carrier. A derivation accepted with the empty ledger, meaning no assumptions, remains true in every Peano model. A derivation accepted with a nonempty ledger remains true under exactly the same gates, no new ones added.

The declaration does not claim that addition commutes in every conceivable algebraic structure. It applies only to Peano models, structures satisfying the Peano axioms. It does not claim that the transport map works without the Peano property; injectivity and surjectivity, the two halves of bijectivity, each rely on it. It does not claim that the framework has proved the Riemann Hypothesis or any other unproved conjecture; it is a theorem about a specific formula, commutativity, in a specific class of models.

/-- Instance: the Gödel-test theorem (commutativity of distinction-composition,
FORCED @ QF-IND) holds in every Peano model, at zero transported cost. -/
theorem addComm_transported (M : DeltaAlgebra) (hM : IsPeanoModel M) (ρ : Env) :
    msat M ρ (.all (.all GodelTest.commFormula)) :=
  transport_forced GodelTest.addComm_forced M hM ρ

/-- The kernel's canonical model IS the δ-base up to the (unique) iso: the base
recursor into ℕ is bijective. Reading ℕ as "the" model is licensed by rigidity,
not by an encoding choice. -/
theorem base_to_nat_bijective : Function.Bijective (baseRec natAlgebra) :=
  ⟨baseRec_injective natAlgebra natAlgebra_peano,
   baseRec_surjective natAlgebra natAlgebra_peano⟩

/-- **The Transport Lemma.** For a Peano model M, satisfaction transported into
M coincides with satisfaction in the canonical model, formula by formula.
Injectivity of the transport map handles the atoms; quantifier witnesses travel
forward. Choice-free. -/
theorem msat_iff_sat (M : DeltaAlgebra) (hM : IsPeanoModel M) :
    ∀ (φ : DFormula) (ρ : Env), msat M ρ φ ↔ DFormula.sat ρ φ := by
  intro φ
  induction φ with
  | eq t s =>
      intro ρ
      simp only [msat, DFormula.sat]
      exact ⟨fun h => natRec_injective M hM h, fun h => congrArg (natRec M) h⟩
  | fls => intro ρ; exact Iff.rfl
  | conj a b iha ihb =>
      intro ρ
      simp only [msat, DFormula.sat]
      exact and_congr (iha ρ) (ihb ρ)
  | disj a b iha ihb =>
      intro ρ
      simp only [msat, DFormula.sat]
      exact or_congr (iha ρ) (ihb ρ)
  | impl a b iha ihb =>
      intro ρ
      simp only [msat, DFormula.sat]
      exact imp_congr (iha ρ) (ihb ρ)
  | all a ih =>
      intro ρ
      simp only [msat, DFormula.sat]
      constructor
      · intro h n
        exact (ih (Env.cons n ρ)).mp (h (natRec M n) n rfl)
      · intro h x n _
        exact (ih (Env.cons n ρ)).mpr (h n)
  | ex a ih =>
      intro ρ
      simp only [msat, DFormula.sat]
      constructor
      · rintro ⟨_, n, _, h⟩
        exact ⟨n, (ih (Env.cons n ρ)).mp h⟩
      · rintro ⟨n, h⟩
        exact ⟨natRec M n, n, rfl, (ih (Env.cons n ρ)).mpr h⟩

/-- **Kill test, FORCED case: PASS.** A derivation the kernel accepts with the
EMPTY ledger is true in the transported semantics of EVERY Peano model, with no
metatheoretic principle beyond the forced fragment. Transport along the unique
iso costs zero posits. -/
theorem transport_forced {d : Deriv} {φ : DFormula} (h : Forced [] d φ)
    (M : DeltaAlgebra) (hM : IsPeanoModel M) (ρ : Env) : msat M ρ φ :=
  (msat_iff_sat M hM φ ρ).mpr (sound_forced h ρ)

/-- The transport map into a Peano model is injective. Choice-free. -/
theorem natRec_injective (M : DeltaAlgebra) (h : IsPeanoModel M) :
    Function.Injective (natRec M) := by
  intro a
  induction a with
  | zero =>
      intro b he
      cases b with
      | zero => rfl
      | succ b => exact absurd he.symm (h.zero_not_succ (natRec M b))
  | succ a ih =>
      intro b he
      cases b with
      | zero => exact absurd he (h.zero_not_succ (natRec M a))
      | succ b => exact congrArg Nat.succ (ih (h.succ_injective he))

What this page does not claim

The declaration does not claim commutativity holds in structures that are not Peano models. It does not claim the transport map is a bijection without the Peano property. It does not claim the framework has proved the Riemann Hypothesis or any other unproved conjecture.

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/Rigidity/LedgerTransport.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.

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