Delta Kernel Godel Test Add Comm Full Certified

The certified commutativity theorem

Addition is commutative: for any two natural numbers n and m, n + m = m + n. This is one of the first facts a student proves about arithmetic, and its textbook proof typically uses induction twice. The declaration add_comm_full_certified is a machine-checked theorem that establishes this commutativity inside a small, custom-built logical kernel. The kernel is a tiny checker for formal derivations; it is not a general-purpose proof assistant but a focused tool that audits every step of a proof against a fixed set of rules.

The theorem's proof follows a specific route. It uses a convenient form of induction: the induction hypothesis is a quantified statement, ∀y, x + y = y + x, which the step can instantiate at any value of y it chooses. This is the standard shortcut that full induction permits. The kernel records this choice by attaching a tier flag, ofIndFull, to the derivation. The declaration add_comm_full_certified then exports this derivation, through a soundness theorem, to the host system's own natural numbers, yielding the ordinary theorem that n + m = m + n.

What the declaration does not claim is just as important. It does not claim that this is the only way to prove commutativity, nor that the full-induction route is the strongest. In fact, the same kernel also contains a separate derivation, add_comm_forced, that proves the same theorem using only quantifier-free induction formulas, and the kernel's pricing function assigns that route an empty ledger, meaning no special tier flag. The two routes produce the same host theorem, but they carry different proof-route strengths. The tier flag measures the proof route, never the truth of the statement.

In Recognition Science, this distinction matters. The framework models proof effort as a ledger, a discrete record of the resources a derivation consumes. The add_comm_full_certified declaration is a concrete instance of that model: it shows that the ledger can price a proof route without changing the mathematical conclusion. The full arithmetization of the kernel's own soundness remains an open target, but this theorem demonstrates the pricing mechanism on a familiar, foundational fact.

/-- Commutativity of `Nat` addition, certified through the δ-kernel's
CONVENIENT route (`FORCED @ FULL-IND`). Still choice-free: the tier flag
measures proof-route strength, not metatheoretic consumption. -/
theorem add_comm_full_certified (n m : Nat) : n + m = m + n := by
  have h := sound_cond addCommFull [] (.all (.all commFormula)) Ledger.ofIndFull
    addCommFull_tier Gated.ofIndFull (fun _ => 0) (fun ψ hψ => by cases hψ)
  exact h n m

/-- ROUTE 2 VERDICT (pre-registered): the convenient route proves the SAME
formula but the ledger posts the TIER flag: `FORCED @ FULL-IND`. The
kernel priced the shortcut. -/
theorem addCommFull_tier :
    check [] addCommFull = some (.all (.all commFormula), .ofIndFull) := by
  decide

/-- ROUTE 1 VERDICT (pre-registered): the careful route is accepted with
the EMPTY ledger. `FORCED @ QF-IND`, the strongest verdict the kernel
issues, for full commutativity of addition. -/
theorem addComm_forced :
    check [] addComm = some (.all (.all commFormula), .empty) := by
  decide

theorem pricing_discriminates :
    (check [] addComm = some (.all (.all commFormula), .empty)) ∧
    (check [] addCommFull = some (.all (.all commFormula), .ofIndFull)) :=
  ⟨addComm_forced, addCommFull_tier⟩

What this page does not claim

This declaration does not claim that the full-induction route is the only way to prove commutativity. It does not claim that the tier flag indicates the strength of the theorem itself, only the proof route. It does not claim the kernel's own soundness is fully arithmetized; that remains an open target.

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/DeltaKernel/GodelTest.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

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