Foundation Primitive Recognition Calculus Rigidity Ledger Transport Msat All Cov

Transporting truth

A ledger, in this framework, is a discrete record of recognition events, and the framework's core calculus is built on a canonical model of that record: the natural numbers 0, 1, 2, ... with their usual successor operation. The declaration msat_all_covers is a formal theorem about what happens when a formula is true in that canonical model. It proves that if a universally quantified statement holds in the canonical model, then it holds in any other model of the same kind, called a Peano model, and also that every element of that other model is actually covered by a witness from the canonical one. In plain terms, the truth of a universally quantified formula is not an accident of the particular carrier set; it transfers to every structurally equivalent model.

The proof works by constructing a transport map that sends each natural number n to the element obtained by applying the other model's successor operation n times, starting from its zero. The theorem msat_all_covers shows this map is surjective onto the other model's carrier, so the other model contains no elements that are unreachable from the canonical starting point. This is what makes the universal quantifier in the transported semantics genuinely range over all of the other model, rather than only over a subset. The framework's machine-checked library of formal theorems establishes this result with no choice principle: the construction is entirely explicit and forward-moving, never requiring an inverse or an arbitrary selection.

The significance is that the framework's kernel derivations, including the Gödel-test theorem that commutativity of distinction-composition is forced, hold in every Peano model at zero additional cost. A derivation accepted with an empty ledger is true in every such model, and a graded derivation is true under exactly the same gates it already required. The framework models this as a transport of certificates: the truth of a formula travels along the unique isomorphism between models, and the ledger, the record of recognition costs, is invariant under change of carrier. This is a formal theorem about the framework's own semantics, not a claim about any external physical system.

The declaration does not claim that the canonical model is the only model, nor that every model of the framework's axioms is a Peano model. It does not claim that the transport map is injective for models that fail the Peano conditions, and it does not establish that any particular formula is true in the canonical model in the first place. It only establishes the transfer property: if a formula is true in the canonical model, and the target model is a Peano model, then the formula is true there too. The theorem is a structural bridge, not a source of new truths.

/-- Non-vacuity of the transported universal: in a Peano model, EVERY element of
the carrier is reached by a witness, so `msat`'s guarded ∀ genuinely constrains
all of M. (Surjectivity is exactly what a Peano model adds here.) -/
theorem msat_all_covers (M : DeltaAlgebra) (hM : IsPeanoModel M)
    {a : DFormula} {ρ : Env} (h : msat M ρ (.all a)) (x : M.carrier) :
    ∃ n, natRec M n = x ∧ msat M (Env.cons n ρ) a := by
  obtain ⟨n, rfl⟩ := natRec_surjective M hM x
  exact ⟨n, rfl, h (natRec M n) n rfl⟩

/-- The transport map into a Peano model is surjective (the model's own
induction schema). Choice-free. -/
theorem natRec_surjective (M : DeltaAlgebra) (h : IsPeanoModel M) :
    Function.Surjective (natRec M) := by
  intro y
  refine h.induction (fun y => ∃ n, natRec M n = y) ⟨0, rfl⟩ ?_ y
  rintro x ⟨n, rfl⟩
  exact ⟨n + 1, rfl⟩

/-- **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)

/-- 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

What this page does not claim

The canonical model is the only model of the framework's axioms. Every model of the framework's axioms is a Peano model. The transport map is injective for models that fail the Peano conditions. Any particular formula is true in the canonical model.

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)

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