Delta Kernel Sound Sound Forced

What sound_forced proves

In formal logic, a derivation is a step-by-step argument that a conclusion follows from premises. The Recognition Science framework includes a small checker for such derivations, and the declaration sound_forced is a theorem about that checker. It states: if the checker accepts a derivation with an empty ledger, meaning no special logical principles were used, then the conclusion is true in the framework's canonical model. The proof is machine-checked in the framework's library of formal theorems, so the claim rests on verified steps, not on an author's assurance.

The point of the empty ledger is that it records which extra logical principles a derivation needed. Three such principles exist: excluded middle, the limited principle of omniscience, and Markov's principle. Each is a classical or omniscience-style assumption that some constructive logicians avoid. When the ledger is empty, none of these principles were needed, and sound_forced proves the conclusion holds without any of them. This is a constructive soundness result: the derivation is valid even in a setting that rejects those three principles.

The theorem does not claim that every accepted derivation is constructive. A separate corollary, sound_classical, covers any derivation the checker accepts, but it supplies all three principles from the ambient classical metatheory. That is where the choice axiom enters, by design and by ledger. The framework also has a third result, sound_isForced, which accepts full induction while still avoiding the three principles; this matches intuitionistic Heyting Arithmetic, not just a weaker fragment.

What sound_forced does not claim is broader than it might appear. It does not say the framework's physics follows constructively. The theorem is about the kernel's own soundness, not about the forcing chain that derives constants or dimensions. It also does not claim that the empty ledger means the derivation used no assumptions at all; it means no special logical principles beyond the base rules. Finally, it does not claim that the canonical model is the only model, or that truth in that model settles physical truth. The theorem is a precise, narrow result about one checker's behavior, and its strength is exactly that precision.

/-- FORCED soundness: a derivation the kernel accepts with the EMPTY ledger is
true in the canonical model, with NONE of the three metatheoretic principles.
The gates are discharged vacuously (`Gated.empty`), so the proof term never
touches EM/LPO/MP or `Classical.choice`. `#print axioms sound_forced` is the
kernel's own σ0 / DELTA_FORCED self-audit. -/
theorem sound_forced {d : Deriv} {φ : DFormula} (h : Forced [] d φ) :
    ∀ ρ : Env, DFormula.sat ρ φ := by
  intro ρ
  refine sound_cond d [] φ Ledger.empty h Gated.empty ρ ?_
  intro ψ hψ
  cases hψ

/-- FORCED soundness: a derivation the kernel accepts with the EMPTY ledger is
true in the canonical model, with NONE of the three metatheoretic principles.
The gates are discharged vacuously (`Gated.empty`), so the proof term never
touches EM/LPO/MP or `Classical.choice`. `#print axioms sound_forced` is the
kernel's own σ0 / DELTA_FORCED self-audit. -/
theorem sound_forced {d : Deriv} {φ : DFormula} (h : Forced [] d φ) :
    ∀ ρ : Env, DFormula.sat ρ φ := by
  intro ρ
  refine sound_cond d [] φ Ledger.empty h Gated.empty ρ ?_
  intro ψ hψ
  cases hψ

/-- CLASSICAL soundness: ANY accepted derivation is true, supplying the three
principles from the ambient classical metatheory. This is the "run the kernel
over classical metatheory" reading, and it is the ONE place `Classical.choice`
legitimately enters, recorded, by design, exactly as the ledger would demand of
a maximally posit-heavy derivation. -/
theorem sound_classical {Γ : Ctx} {d : Deriv} {φ : DFormula} {O : Ledger}
    (h : check Γ d = some (φ, O)) (ρ : Env) (hΓ : CtxSat ρ Γ) :
    DFormula.sat ρ φ := by
  refine sound_cond d Γ φ O h ?_ ρ hΓ
  refine ⟨fun _ P => Classical.em P, fun _ P _ => ?_, fun _ P _ hnn => ?_⟩
  · exact (Classical.em (∃ n, P n)).elim Or.inl
      (fun hne => Or.inr (fun n hn => hne ⟨n, hn⟩))
  · exact Classical.byContradiction (fun hne => hnn hne)

What this page does not claim

The framework's physical derivations, such as the forcing chain for constants, are constructive. An empty ledger means the derivation used no assumptions whatsoever. Truth in the canonical model settles physical truth. The theorem applies to any derivation the checker accepts, only those with an empty ledger.

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/Sound.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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