Delta Kernel Examples

Worked proofs

In mathematics, a proof is a chain of steps that follows from earlier steps. The ledger, a discrete record of such steps, is what Recognition Science uses to track which rules a proof actually needed. The delta kernel examples are a set of small, concrete proofs that show how this ledger behaves. They are not proofs of deep theorems; they are demonstrations of the machinery, checked end to end by the kernel itself.

The first example is a proof that 1 + 1 = 2. The proof is built from the basic recursion equations for addition, with no induction and no additional assumptions. The kernel checks the derivation tree and returns a ledger entry that is empty: the proof used no special posits. A second example proves that 0 + x = x for every natural number x. This one uses induction, and the kernel records that it needed the quantifier-free induction rule. The proof is then exported to a standard theorem about natural numbers, showing the full pipeline: a derivation, a kernel audit, and a usable result.

The examples also draw a sharp line between what the kernel accepts and what it rejects. A proof of the same decidable statement, 0 = 0 or not (0 = 0), can be built two ways: one uses an explicit excluded-middle posit, and the ledger records that it did; the other uses a direct introduction rule, and the ledger stays empty. The kernel, not a convention, is what distinguishes the two. Similarly, the Markov posit accepts a quantifier-free formula but rejects a quantified one; the guard is built into the checker itself.

These examples establish that the ledger's verdicts are not a matter of taste. The kernel, an executable checker, accepts or rejects each derivation tree by computation. The framework's library of formal theorems uses this same machinery to audit its own proofs, and these worked examples show what an audit looks like at the smallest scale. They make the framework's claim about forced cost tangible: the rules are not chosen, they are checked.

/-- The kernel accepts `onePlusOne` with the EMPTY ledger: σ0 / DELTA_FORCED. -/
theorem onePlusOne_forced :
    check [] onePlusOne = some (.eq (.add one one) two, .empty) := by
  decide

/-- Exported to the host: `∀ n, 0 + n = n` over `Nat`, certified through the
δ-kernel. The proof term routes through `sound_forced`, so its axiom
closure is choice-free; `#print axioms` below is the audit. -/
theorem zero_add_certified (n : Nat) : 0 + n = n :=
  sound_forced zeroAdd_forced (fun _ => 0) n

/-- Route A posts `em`: the verdict is CONDITIONAL {EM}. -/
theorem emRoute_posts_em :
    check [] emRoute = some (.disj zeroEq zeroEq.neg, .ofEM) := by
  decide

/-- Route B proves the SAME formula with the EMPTY ledger: FORCED. -/
theorem forcedRoute_forced :
    check [] forcedRoute = some (.disj zeroEq zeroEq.neg, .empty) := by
  decide

/-- Markov on a QF matrix: accepted, posts `mp`. -/
theorem mp_accepts_qf :
    check [] (.mpPosit qfMatrix) =
      some (.impl (.neg (.neg (.ex qfMatrix))) (.ex qfMatrix), .ofMP) := by
  decide

/-- Markov on a quantified matrix: REJECTED by the kernel (not merely
discouraged). -/
theorem mp_rejects_quantified :
    check [] (.mpPosit (.all (.eq (.var 0) (.var 0)))) = none := by
  decide

What this page does not claim

The delta kernel examples do not prove any theorem about the physical world. The examples do not show that excluded middle is invalid, only that the ledger can record when it was used. The kernel's rejection of a quantified Markov step does not imply the step is false, only that it is not accepted by this rule.

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/Examples.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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