Foundation Ledger Floor T0 Bridge Ledger Shadow Single

The single posting

In the Recognition Science framework, a ledger is a discrete record of recognition events, where each event is a posting to a named account. The declaration ledgerShadow_single concerns the simplest possible case: a ledger with exactly one posting, carrying a natural-number multiplicity n. The shadow of a ledger is a two-state Boolean value: false when the ledger is empty, true as soon as any recognition has been posted. The declaration proves that for a single posting with multiplicity n, the shadow equals the Boolean truncation of n: it is false exactly when n = 0, and true exactly when n > 0.

This is a formal theorem in the framework's machine-checked library of formal theorems, carrying no unproved assumptions beyond the standard logical axioms. It is a lemma within a larger identification: the two-state T0 floor, a Boolean indicator of whether any recognition has occurred, is shown to be the shadow of the extensive ledger, which counts recognitions as natural numbers. The single-posting case is the atomic step of that identification. It shows that the Boolean floor does not lose information about whether anything happened, only about how many times it happened.

The declaration does not claim that the shadow preserves the full multiplicity n, only its zero-versus-positive status. It does not claim that a single posting with multiplicity 2 is distinguishable from one with multiplicity 3 at the floor level; both map to true. It also does not claim that the shadow is a homomorphism for addition in general; that is a separate theorem. The single-posting lemma is a precise, limited statement: the Boolean truncation of a natural number is exactly the shadow of a ledger with that multiplicity.

In plain terms, the declaration establishes the boundary of what the two-state floor can see. It sees whether recognition happened, not how much. This is the foundation for the framework's claim that the T0 floor is a genuine quotient of the extensive ledger, not an arbitrary Boolean choice. The single posting is the unit of that quotient, and its shadow is the unit of the floor.

/-- On a single primitive distinction the shadow is exactly the Boolean
truncation of the natural-number multiplicity. -/
theorem ledgerShadow_single {I : Type v} (i₀ : I) (n : ℕ) :
    ledgerShadow (Finsupp.single i₀ n) = booleanTruncation n := by
  by_cases hn : n = 0
  · subst hn
    rw [Finsupp.single_zero, ledgerShadow_zero, booleanTruncation_zero]
  · have hne : Finsupp.single i₀ n ≠ 0 := by
      rw [Ne, Finsupp.single_eq_zero]; exact hn
    rw [ledgerShadow_eq_true_iff.mpr hne, booleanTruncation_pos hn]

/-- The Boolean two-state **shadow** of a ledger: `false` on the empty ledger,
`true` as soon as any recognition has been posted. This is the truncation of the
extensive `ℕ`-valued multiplicity to the two-state floor. -/
noncomputable def ledgerShadow {I : Type v} (Γ : DefectLedger I) : Bool :=
  if Γ = 0 then false else true

What this page does not claim

The shadow preserves the full multiplicity n, only its zero-versus-positive status. The shadow is a homomorphism for ledger addition in general; that is a separate theorem. The single-posting lemma establishes the full identification of the T0 floor with the extensive ledger, which requires the surjectivity and kernel theorems.

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)

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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