Foundation Ledger Floor T0 Bridge Ledger Shadow Eq True Iff

The two-state shadow

In the Recognition Science framework, a ledger (a discrete record of posted recognition events) keeps a natural-number count for each kind of distinction it has recognized. The declaration ledgerShadow_eq_true_iff is a proved theorem about a simple projection of that ledger: the projection that answers only one question, whether the ledger is empty or not. The theorem states that this two-state shadow equals true if and only if the ledger is not the zero ledger, meaning at least one recognition has been posted. It is the formal way of saying that the shadow is a faithful emptiness test.

The theorem is one small piece of a larger identification. The framework distinguishes between an extensive ledger, which can hold any natural-number multiplicity of recognitions, and a two-state floor that only records whether any recognition has occurred. The declaration ledgerShadow_eq_true_iff establishes the exact condition under which the shadow reports true: precisely when the underlying ledger is nonzero. It also has a companion, ledgerShadow_eq_false_iff, which states the shadow is false exactly when the ledger is zero. Together they make the shadow a clean Boolean indicator of non-emptiness.

What this theorem does not claim is any information about how many recognitions the ledger holds. A shadow of true says only that the count is at least one; it says nothing about whether the count is one or a million. The theorem also does not address the cost of the recognitions, which is a separate quantity in the framework. It is purely a statement about the relationship between a ledger and its two-state projection, proved in the machine-checked library of formal theorems.

In the context of the framework's development, this theorem matters because it helps close a gap. Before this result, the two-state floor and the extensive ledger were built separately with no formal connection. This theorem, along with the surrounding identification bundle, provides the explicit truncation map that makes the floor the shadow of the ledger. The practical consequence is that the framework can now treat the two-state floor as a genuine quotient of the extensive ledger, not as an unrelated construction.

theorem ledgerShadow_eq_true_iff {I : Type v} {Γ : DefectLedger I} :
    ledgerShadow Γ = true ↔ Γ ≠ 0 := by
  unfold ledgerShadow
  by_cases hΓ : Γ = 0 <;> simp [hΓ]

theorem ledgerShadow_eq_false_iff {I : Type v} {Γ : DefectLedger I} :
    ledgerShadow Γ = false ↔ Γ = 0 := by
  unfold ledgerShadow
  by_cases hΓ : Γ = 0 <;> simp [hΓ]

What this page does not claim

The theorem says nothing about the number of recognitions, only that at least one exists. The theorem does not address the cost of the recognitions in the ledger. The theorem does not by itself establish the full identification between the floor and the 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/Foundation/LedgerFloorT0Bridge.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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