Delta Kernel Godel Test Succ Add Forced

The priced proof

The declaration succAdd_forced is a machine-checked result about a tiny formal system. The system is a ledger: a discrete record of derivation steps, where each step is a rule application that a checker program audits. The theorem states that a specific derivation, named succAdd, passes the checker with an empty ledger. The derivation proves the formula Sx + y = S(x+y), which says that adding one to a number before adding another number is the same as adding the second number first and then adding one. This is the first stepping stone toward commutativity, the fact that x + y = y + x.

The empty ledger is the point. The system prices derivations by what they require. A derivation that uses only quantifier-free induction formulas posts no tier flag. The succAdd_forced theorem, verified by the executable checker with decide, shows that this particular derivation needs nothing beyond that base level. A second theorem, addCommFull_tier, shows the contrast: a different derivation of commutativity that uses a quantified induction formula posts the ofIndFull tier flag. The pricing therefore discriminates between proof routes, not between truths. Both routes export through soundness to the same host theorem about natural numbers, add_comm_certified for the careful route and add_comm_full_certified for the convenient one.

This is the pricing half of a larger program. The full arithmetization, encoding the derivation relation inside the natural numbers and proving the kernel's own soundness at a measured tier, remains open and is pre-registered as future work. Cantor pairing over the distinction signature is the intended sequence encoding. Nothing in succAdd_forced claims that arithmetization. The theorem also does not claim that the empty ledger means the proof is conceptually simpler; it claims only that the derivation avoids the quantified induction formula, nothing more.

/-- The kernel accepts `succAdd` with the EMPTY ledger: FORCED @ QF-IND. -/
theorem succAdd_forced :
    check [] succAdd = some (.all (.all succAddFormula), .empty) := by
  decide

/-- The kernel accepts `succAdd` with the EMPTY ledger: FORCED @ QF-IND. -/
theorem succAdd_forced :
    check [] succAdd = some (.all (.all succAddFormula), .empty) := by
  decide

/-- 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

/-- Commutativity of `Nat` addition, certified through the δ-kernel's
CAREFUL route (`FORCED @ QF-IND`). -/
theorem add_comm_certified (n m : Nat) : n + m = m + n :=
  sound_forced addComm_forced (fun _ => 0) n m

/-- 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

What this page does not claim

The empty ledger does not imply the proof is conceptually simpler, only that it avoids the quantified induction formula. The theorem does not claim the full arithmetization of the kernel's soundness, which remains open. The tier flag measures proof-route strength, never the truth of the underlying arithmetic statement.

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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