Holography Ledger State Horizon Context Posted Zero Heat Ne Legal Atomic Tick He

The posted zero and the real tick

A ledger is a discrete record of events, each entry a posting that moves value from one account to another. In the Recognition Science framework, a recognition event, the fundamental unit of change, is modeled as a legal atomic tick: a single posting step that is forced to transfer a nonzero amount. The theorem posted_zero_heat_ne_legalAtomicTick_heat proves a sharp separation between two kinds of records: a posted zero, which records no change, and a genuine tick, which records a real transfer.

The theorem states that the measured heat of a posted zero, defined as the change from a cut to itself extended by zero, is never equal to the measured heat of a legal atomic tick. In plainer terms, if you hold a boundary record still and post nothing to it, the resulting reading is always different from the reading produced by an actual event that moves one unit. The proof relies on two facts: a posted zero always yields a heat value of zero, while a legal atomic tick, by its definition, always yields a heat value of one. Since zero is not one, the two readings cannot coincide.

This result is part of a larger construction in the framework's machine-checked library of formal theorems. It builds a horizon context, a local description of the boundary of a ledger state, directly from the ledger's committed units, so that a legal atomic tick becomes a moving cut extension. The theorem same_event_projections then ties the story together: a legal atomic tick is a posting step, its induced heat is one, and its horizon record advances by exactly one unit. These are the projections of a single event onto three different views.

What the theorem does not claim is just as important. It does not say that all posted zeros are identical, nor that all legal atomic ticks are identical. It only proves that the heat reading alone can tell a genuine event apart from a non-event. The framework's own documentation flags a residual limitation: all-ones exterior readings from totals alone do not separate distinct posting steps that share a pre-state. So while the theorem cleanly separates event from non-event, it does not yet individuate one event from another of the same kind.

/--
Posted-zero decoy: exposing a zero bit grows capacity with heat 0, while every
legal tick from a nonnegative ledger induces heat 1. Therefore a posted-zero
extension is never the heat signature of a legal tick.
-/
theorem posted_zero_heat_ne_legalAtomicTick_heat {d : Nat} {L L' : LedgerState d}
    (hn : NonnegLedger (d := d) L)
    (h : LegalAtomicTick (d := d) L L')
    (c : LocalCut (contextOfLedgerState L)) :
    movingStepHeat c (extendCut c 0) ≠
      movingStepHeat (cutOfLedgerState L) (cutOfLedgerState L') := by
  rw [movingStepHeat_extendCut, postedBitValue_zero,
    induced_movingStepHeat_eq_one hn h]
  decide

/--
Posted-zero decoy: exposing a zero bit grows capacity with heat 0, while every
legal tick from a nonnegative ledger induces heat 1. Therefore a posted-zero
extension is never the heat signature of a legal tick.
-/
theorem posted_zero_heat_ne_legalAtomicTick_heat {d : Nat} {L L' : LedgerState d}
    (hn : NonnegLedger (d := d) L)
    (h : LegalAtomicTick (d := d) L L')
    (c : LocalCut (contextOfLedgerState L)) :
    movingStepHeat c (extendCut c 0) ≠
      movingStepHeat (cutOfLedgerState L) (cutOfLedgerState L') := by
  rw [movingStepHeat_extendCut, postedBitValue_zero,
    induced_movingStepHeat_eq_one hn h]
  decide

/-- Induced moving-cut heat of a legal atomic tick is exactly one. -/
theorem induced_movingStepHeat_eq_one {d : Nat} {L L' : LedgerState d}
    (hn : NonnegLedger (d := d) L)
    (h : LegalAtomicTick (d := d) L L') :
    movingStepHeat (cutOfLedgerState L) (cutOfLedgerState L') = 1 := by
  unfold movingStepHeat cutOfLedgerState
  rw [activeExteriorBits_mkLedgerCut, activeExteriorBits_mkLedgerCut]
  have hs := ledgerCommittedUnits_legalAtomicTick_succ hn h
  rw [hs]
  norm_num

/--
Same-event packaging: posting step, unit heat, and context-record extension are
projections of one legal tick. Cut equality does **not** recover the changed
account (see `allOnes_cut_does_not_separate_debit_credit`).
-/
theorem same_event_projections {d : Nat} {L L' : LedgerState d}
    (hn : NonnegLedger (d := d) L)
    (h : LegalAtomicTick (d := d) L L') :
    PostingStep (d := d) L L' ∧
      movingStepHeat (cutOfLedgerState L) (cutOfLedgerState L') = 1 ∧
      (contextOfLedgerState L').horizonRecord =
        (extendContext (contextOfLedgerState L)).horizonRecord :=
  ⟨legalAtomicTick_implies_PostingStep h,
    induced_movingStepHeat_eq_one hn h,
    T1 hn h⟩

What this page does not claim

The theorem does not claim that all posted zeros are identical. The theorem does not claim that all legal atomic ticks are identical. The theorem does not claim that all-ones exterior readings can separate distinct posting steps that share a pre-state.

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/Holography/LedgerStateHorizonContext.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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