Foundation Primitive Recognition Calculus Rigidity Ledger Transport Base To Nat

The canonical model

The natural numbers, 0, 1, 2, and so on, are the counting numbers of ordinary arithmetic. Peano arithmetic, the standard set of axioms for them, says that zero is a number, every number has a successor, and that if two numbers have the same successor they are the same number. These axioms are enough to prove statements by induction, the method that establishes a fact for all numbers at once.

In Recognition Science, the framework begins with a ledger, a discrete record of events, built from a primitive operation of distinction: starting from zero, each step adds one new distinction. The framework's machine-checked library of formal theorems shows that this primitive structure is not merely analogous to the natural numbers. It proves that the ledger's own recursor, the function that maps each ledger element to its place in the count, is a bijection onto the natural numbers. That is, every ledger element corresponds to exactly one natural number, and every natural number is reached by exactly one ledger element. The two structures are the same up to a unique isomorphism.

This identification is a theorem, not a choice. The framework proves that the natural numbers, equipped with zero and successor, form a Peano model of the ledger's distinction operation. The bijection then follows from the fact that the ledger's recursor is both injective and surjective onto the natural numbers. The proof is choice-free, meaning it does not rely on the axiom of choice or any other non-constructive principle beyond the framework's own basis.

In Recognition Science, this result licenses reading the natural numbers as the canonical model of the ledger. It is the foundation for transporting kernel certificates, the framework's accepted derivations, into any other Peano model. The transport lemma shows that a formula true in the canonical model is true in every Peano model, with the same ledger conditions and no additional cost. This is what makes the framework's arithmetic statements portable across all its models.

The theorem does not claim that the natural numbers are the only possible model of the ledger. Other Peano models exist, and the bijection is to the natural numbers specifically. It does not claim that the framework derives the natural numbers from nothing; the natural numbers are taken as given in the ambient type theory. It also does not claim that the ledger's distinction operation is the same as ordinary addition or multiplication; those operations are defined separately and their properties, such as commutativity, are proved as further theorems.

/-- 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⟩

/-- ℕ is a Peano model of distinction. -/
theorem natAlgebra_peano : IsPeanoModel natAlgebra where
  succ_injective := fun _ _ h => Nat.succ.inj h
  zero_not_succ := fun x h => Nat.succ_ne_zero x h
  induction := fun _ h0 hs x => Nat.rec h0 (fun n ih => hs n ih) x

/-- **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⟩

What this page does not claim

The natural numbers are the only model of the ledger. The framework derives the natural numbers from nothing; they are taken as given in the ambient type theory. The ledger's distinction operation is the same as ordinary arithmetic operations.

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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