Foundation Ledger Floor T0 Bridge Ledger Shadow Eq False Iff

The ledger's shadow

The declaration ledgerShadow_eq_false_iff is a small but load-bearing result in the Recognition Science framework's formal library. It states, in plain terms, that a ledger's two-state shadow is false if and only if the ledger itself is the empty ledger. The shadow here is a simple projection: it takes a ledger that records how many times each kind of recognition event has occurred, and collapses that whole record into a single Boolean value, false if nothing has been posted and true if anything has. The theorem's content is that this collapse loses no information about emptiness: the shadow says "empty" exactly when the underlying ledger really is empty, with no hidden postings that the projection fails to see.

This result is proved in the framework's machine-checked library of formal theorems, with no unproved assumptions and no new axioms. Its role is to close a gap in the framework's construction: earlier work had built a two-state "floor" that records whether any recognition has occurred, and separately an extensive ledger that counts how many times each event occurred, but the two were not formally connected. This theorem, together with its companion ledgerShadow_eq_true_iff (shadow is true exactly when the ledger is nonempty), supplies that connection. It shows the floor is the shadow of the ledger, a genuine projection rather than an unrelated choice.

What the theorem does not claim is just as important. It does not say that the shadow determines the full contents of the ledger; two quite different ledgers, one with a single posting and one with a thousand, both cast the same true shadow. It does not say anything about the cost of recognition, only about the emptiness of the record. And it does not, by itself, establish that the framework's recognition cycle or its constants follow; that is the work of other theorems in the chain. The theorem is a precise, narrow fact: the shadow is false exactly when the ledger is zero.

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]

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]

/-- The shadow lift surjects onto the T0 floor: every floor state is the shadow
of some ledger, so the floor is a genuine quotient (shadow) of the ledger. The
single primitive distinction `i₀` witnesses the marked state. -/
theorem ledgerToFloor_surjective {K : Type*} (h : ∃ x y : K, x ≠ y)
    {I : Type v} (i₀ : I) :
    Function.Surjective (ledgerToFloor (I := I) h) := by
  intro q
  by_cases hq : forcedQuotientBoolEquiv h q = false
  · refine ⟨0, ?_⟩
    unfold ledgerToFloor
    rw [ledgerShadow_zero]
    exact (Equiv.symm_apply_eq _).mpr hq.symm
  · have hqt : forcedQuotientBoolEquiv h q = true := by
      cases hb : forcedQuotientBoolEquiv h q
      · exact absurd hb hq
      · rfl
    refine ⟨Finsupp.single i₀ 1, ?_⟩
    unfold ledgerToFloor
    have hne : Finsupp.single i₀ (1 : ℕ) ≠ 0 := by
      rw [Ne, Finsupp.single_eq_zero]; exact one_ne_zero
    rw [ledgerShadow_eq_true_iff.mpr hne]
    exact (Equiv.symm_apply_eq _).mpr hqt.symm

What this page does not claim

The shadow does not determine the full contents of the ledger. The theorem does not address the cost of recognition, only the emptiness of the record. The theorem does not, by itself, derive the framework's constants or the recognition cycle.

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.

Derived articles

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