Delta Kernel Godel Test Pricing Discriminates

What the pricing test shows

Commutativity of addition, the statement that x + y = y + x for all natural numbers, is one of the first theorems in arithmetic whose textbook proof uses induction twice. The Recognition Science framework contains a machine-checked library of formal theorems, and within that library a test called pricing_discriminates asks a precise question: does the framework's ledger, a discrete record of what a derivation required, merely decorate a proof, or does it genuinely distinguish between different ways of proving the same thing?

The test takes one fixed theorem and derives it along two genuinely different routes. Route 1 is careful: every induction formula is quantifier-free, meaning it contains no 'for all' or 'there exists' symbols. Route 2 is the textbook shortcut: it uses induction on x with the quantified formula ∀y, x + y = y + x, which lets the proof instantiate its hypothesis at any y it likes. The framework's checker audits every step of both derivations and returns a verdict for each. The theorem pricing_discriminates states the outcome: both routes produce the same theorem, but the ledger posts different tier flags. Route 1 is recorded as FORCED at quantifier-free induction with an empty ledger, while Route 2 is recorded as FORCED at full induction with a tier flag for using full induction.

What this establishes is that the ledger measures proof-route strength, not truth. Both derivations export through soundness to the same host theorem, so the framework does not claim that one route is more correct than the other. The tier flag is a measurement of how the proof was built, not a verdict on what it proves. The framework's own documentation states this plainly: the tier flag measures proof-route strength, never truth.

The pricing test is the first half of a larger program. The full arithmetization, encoding the derivation relation inside the natural numbers and deriving the kernel's own soundness at a measured tier, remains open and is pre-registered as future work. Nothing in pricing_discriminates claims that step. What the test does claim, and what it proves, is that the framework's ledger can tell two honest proofs apart by the strength of the induction they use.

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

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

The pricing test does not claim that one proof route is more correct than the other. The pricing test does not claim to have completed the full arithmetization of the kernel's soundness. The pricing test does not claim that the framework's ledger can distinguish between all possible proof strategies.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: