Holography Ledger State Horizon Context Induced Moving Step Heat Eq Posted Bit

The posted bit

In the Recognition Science (RS) framework, a ledger is a discrete record of events, and a legal atomic tick is the smallest possible update to that record. The theorem induced_movingStepHeat_eq_postedBit proves a tight identity: when a legal atomic tick moves the ledger from one state to the next, the heat-like step measured on the horizon is exactly one, and this value is the same as the posted bit computed from that tick. In plain terms, the act of posting a single unit to the ledger is not accompanied by a separate heat signal; the posting is the signal, and its magnitude is fixed at one.

The proof chain is direct. A legal atomic tick is defined to be a posting step that changes the ledger state, and its cost is forced to be exactly one unit. The horizon context, which is the boundary record derived from the ledger state, tracks the total committed units. The theorem then shows that the moving-step heat between the horizon cuts of the two states equals one, and that this equals the value of the posted bit, which is defined to be one for any legal tick. The identity holds for any nonnegative ledger state, meaning no account is ever negative before the tick.

What the theorem does not claim is just as important. It does not say that the posted bit can be zero, or that a tick with zero transfer is legal; in fact, a separate theorem proves that a zero-heat extension is never equal to the heat of a legal tick. It also does not claim that the posted bit is a free parameter that could vary from tick to tick. The bit is derived from the tick's unit commitment, not chosen. Finally, the theorem does not establish that all-ones exterior readings can distinguish between different posting steps that share the same pre-state; that separation remains a named residual in the framework.

The consequence is a clean identification: the boundary record of a ledger state is not an arbitrary decoration. It is a projection of the same event as the posting, and its heat content is the unit of the tick. This is what lets the framework treat the horizon context as a derived object rather than an input, and it is the step that makes a legal atomic tick a moving-cut extension of the ledger state.

/-- Unary packaging: heat of the induced ledger cut step equals 1. -/
theorem induced_movingStepHeat_eq_postedBit {d : Nat} {L L' : LedgerState d}
    (hn : NonnegLedger (d := d) L)
    (h : LegalAtomicTick (d := d) L L') :
    movingStepHeat (cutOfLedgerState L) (cutOfLedgerState L') =
      postedBitValue (postedBitOfLegalAtomicTick h) := by
  rw [induced_movingStepHeat_eq_one hn h, postedBitOfLegalAtomicTick, postedBitValue_one]

/-- 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⟩

/--
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

/-- Exposed bit of a legal atomic tick: computed as `1` from unit L1 commitment. -/
def postedBitOfLegalAtomicTick {d : Nat} {L L' : LedgerState d}
    (_h : LegalAtomicTick (d := d) L L') : ZMod 2 :=
  1

What this page does not claim

The theorem does not claim that the posted bit can be zero or that a tick with zero transfer is legal. The theorem does not claim that the posted bit is a free parameter that could vary from tick to tick. The theorem does not claim that all-ones exterior readings can distinguish between different posting steps that share the same 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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