Delta Kernel Ledger Union Eq Empty

The union ledger

A ledger, in the Recognition Science framework, is a discrete record of which named logical principles a derivation consumed. The record is just four Boolean flags: one for the full excluded middle schema, one for the limited principle of omniscience, one for Markov's principle, and one tier marker for whether induction was used on a formula with quantifiers. Each flag is either true or false, so a ledger is a tiny piece of bookkeeping, not a statement about the world.

The declaration union_eq_empty proves a structural fact about combining ledgers. When you take the union of two ledgers, you get a new ledger whose flags are the logical OR of the two originals: a posit appears in the union if it appears in either input. The theorem says that this union equals the empty ledger if and only if both input ledgers are themselves empty. In other words, combining two records of assumptions leaves no trace exactly when neither record had any trace to begin with. The proof is a routine case analysis on the sixteen possible flag combinations, checked mechanically in the framework's machine-checked library of formal theorems.

What the theorem does not claim is just as important. It says nothing about whether excluded middle, the limited principle of omniscience, or Markov's principle are true or false. It only records whether a derivation used them. A ledger with the excluded-middle flag set to true is not a claim that excluded middle holds; it is a claim that the derivation consumed that principle. The theorem also does not say that an empty ledger means a derivation is philosophically pure or foundationally privileged. It only says the ledger has no posits recorded in it.

The payoff is a clean accounting rule. If you want to know whether a combined derivation used any named classical principle, you do not need to inspect the derivation's history. You only need to check whether each component ledger is empty. The union of two empty ledgers is empty, and any nonempty component makes the union nonempty. That is the whole content of union_eq_empty, and it is a fact about bookkeeping, not about mathematics.

/-- A union is empty iff both parts are: the FORCED verdict is inherited by
every sub-derivation. -/
theorem union_eq_empty {a b : Ledger} :
    a.union b = empty ↔ a = empty ∧ b = empty := by
  cases a; cases b
  simp only [union, empty, Ledger.mk.injEq, Bool.or_eq_false_iff]
  constructor
  · rintro ⟨⟨h1, h2⟩, ⟨h3, h4⟩, ⟨h5, h6⟩, ⟨h7, h8⟩⟩
    exact ⟨⟨h1, h3, h5, h7⟩, ⟨h2, h4, h6, h8⟩⟩
  · rintro ⟨⟨h1, h3, h5, h7⟩, ⟨h2, h4, h6, h8⟩⟩
    exact ⟨⟨h1, h2⟩, ⟨h3, h4⟩, ⟨h5, h6⟩, ⟨h7, h8⟩⟩

/-- Merge two ledgers (a derivation consumes the posits of all its parts). -/
def union (a b : Ledger) : Ledger :=
  ⟨a.em || b.em, a.lpo || b.lpo, a.mp || b.mp, a.indFull || b.indFull⟩

/-- The posit-and-tier ledger. Fields `em`, `lpo`, `mp` are POSITS (named
classical/omniscience principles consumed). Field `indFull` is a TIER marker:
the derivation used induction on a formula with quantifiers. Extending the
posit alphabet = adding a field. -/
structure Ledger where
  /-- Full excluded middle schema `φ ∨ ¬φ`. -/
  em : Bool
  /-- Limited principle of omniscience (arithmetical form):
  pointwise decidability of `φ` yields `(∃x φ) ∨ (∀x ¬φ)`. -/
  lpo : Bool
  /-- Markov's principle for quantifier-free matrices:
  `¬¬(∃x φ) → ∃x φ`. -/
  mp : Bool
  /-- TIER (not a posit): induction was used on a non-quantifier-free
  formula. `FORCED @ FULL-IND` vs `FORCED @ QF-IND`. -/
  indFull : Bool

What this page does not claim

The theorem does not assert that excluded middle, LPO, or Markov's principle are true or false. An empty ledger does not imply a derivation is foundationally privileged or philosophically pure. The theorem does not describe how the ledger is used in the broader forcing-spectrum accounting.

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/Ledger.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.

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