Delta Kernel Examples Forced Route Forced

A concrete proof route

The declaration forcedRoute_forced is a worked example inside a machine-checked library of formal theorems. It concerns the statement 0 = 0 ∨ ¬(0 = 0), a disjunction that is true in classical logic by the law of excluded middle. The example constructs a proof of this disjunction without invoking that law as a postulate. Instead, it builds the proof from a more basic rule: if you already have a proof of 0 = 0, you can introduce a disjunction by proving its left side. The declaration records that this construction is accepted by the framework's kernel checker, and that the resulting proof carries no special axiom flag.

This is one of several examples in the same file that illustrate different proof routes. A companion declaration, emRoute, builds the same disjunction by explicitly posting the excluded-middle posit, and the checker records that this route does carry the em flag. The contrast is the point: the ledger of assumptions is sensitive to how a proof is built, not just to what it proves. The forcedRoute example shows that for this concrete instance, the schematic posit was not needed.

In Recognition Science, this kind of example supports the idea that the cost of recognition is forced, not chosen. The framework's kernel accepts certain derivation trees and rejects others, and the ledger records which postulates were actually used. The example does not show that excluded middle is false, nor that it is always avoidable. It shows only that for this particular disjunction, a proof exists without it.

What the example does not claim is broader. It does not prove that the law of excluded middle is dispensable in general, nor that every classical theorem has a constructive proof. It does not establish anything about the framework's main forcing chain, which derives constants and dimensions from a cost function. It is a small, concrete demonstration of a proof-construction route, not a general philosophical or mathematical result.

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

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

What this page does not claim

This example does not show that the law of excluded middle is false or always avoidable. It does not prove that every classical theorem has a constructive proof. It does not establish anything about the framework's main forcing chain that derives constants and dimensions.

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

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: