Holography Ledger State Horizon Context Legal Atomic Tick Nonzero Transfer

The forced transfer

In the Recognition Science framework, a ledger is a discrete record of events, a list of debits and credits that the universe keeps as its basic accounting. A legal atomic tick is the smallest possible entry in that record: a single, indivisible step that moves the ledger from one state to the next. The theorem legalAtomicTick_nonzero_transfer states the two properties such a tick must have. First, it must actually change something: the ledger state after the tick is different from the state before it. Second, it must be a PostingStep, which in this context means the change is a single posting to one account, either a debit or a credit, never a mixture of both.

The theorem is proved in the framework's machine-checked library of formal theorems, so it is not a convention or a modeling choice. It follows from the definition of a legal atomic tick, which requires that the tick has a non-zero cost in the ledger's L1 norm. The proof is short: if the state did not change, the cost would be zero, contradicting the definition. If the step were not a single posting, it would not be atomic. The theorem also has a companion result, legalAtomicTick_ne, which states the same non-identity property directly, and legalAtomicTick_is_PostingStep, which names the posting structure explicitly.

What the theorem does not claim is broader. It does not say that every legal atomic tick is a transfer of a specific amount, only that it is a single posting. The amount moved is not fixed by the theorem; it is the unit of the ledger's commitment. The theorem also does not say that the ledger state is always non-negative. That is a separate condition, NonnegLedger, which must be assumed for other results about how the ledger's committed units grow. Finally, the theorem does not say that the reverse of a legal atomic tick is also legal. In fact, a separate theorem, reverse_not_legalAtomicTick, proves the opposite: the reverse of a legal atomic tick is never legal. This is what makes the ledger's time direction meaningful.

The consequence for the framework is that a legal atomic tick is not a bookkeeping fiction. It is a real, forced change in the ledger's state, a step that cannot be undone and cannot be a no-op. This is the foundation for the framework's account of time and change: the universe's ledger is not static, and every entry is a genuine event.

/-- P3 bundle. -/
theorem legalAtomicTick_nonzero_transfer {d : Nat} {L L' : LedgerState d}
    (h : LegalAtomicTick (d := d) L L') :
    L ≠ L' ∧ PostingStep (d := d) L L' :=
  ⟨legalAtomicTick_ne h, legalAtomicTick_is_PostingStep h⟩

/-- P3 bundle. -/
theorem legalAtomicTick_nonzero_transfer {d : Nat} {L L' : LedgerState d}
    (h : LegalAtomicTick (d := d) L L') :
    L ≠ L' ∧ PostingStep (d := d) L L' :=
  ⟨legalAtomicTick_ne h, legalAtomicTick_is_PostingStep h⟩

/-- Reverse of a legal atomic tick is not a legal atomic tick (monotonicity). -/
theorem reverse_not_legalAtomicTick {d : Nat} {L L' : LedgerState d}
    (h : LegalAtomicTick (d := d) L L') :
    ¬ LegalAtomicTick (d := d) L' L := by
  intro hrev
  obtain ⟨k, side, rfl⟩ := legalAtomicTick_implies_PostingStep (d := d) h
  rcases hrev with ⟨⟨hmonoD, hmonoC⟩, _⟩
  cases side with
  | debit =>
    have hle : L.debit k + 1 ≤ L.debit k := by simpa [post] using hmonoD k
    omega
  | credit =>
    have hle : L.credit k + 1 ≤ L.credit k := by simpa [post] using hmonoC k
    omega

What this page does not claim

The theorem does not specify the amount transferred by a legal atomic tick. The theorem does not require the ledger state to be non-negative. The theorem does not state that a legal atomic tick is reversible.

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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