Foundation Quantum Ledger Ledger Balance Conserved

The conserved balance

A ledger, in this framework, is a discrete record of recognition events. Each event has a ratio, a positive real number, and a cost computed from that ratio. The ledger's balance is the sum of the logarithms of all entry ratios. The empty ledger has balance zero, and each entry contributes its logarithm to the total.

The key theorem, ledger_balance_conserved, states that a specific kind of update leaves the balance unchanged. An update adds two entries whose ratios are reciprocals of each other, such as 2 and 1/2. Because log(2) + log(1/2) = 0, the new entries cancel in the sum, and the balance stays exactly as it was. The proof is a direct computation from the definitions, and the theorem is checked in the machine-checked library of formal theorems.

This conservation is a structural fact about the ledger's arithmetic, not a physical law imported from outside. It holds for any ledger and any update of that reciprocal form. It does not say that costs are conserved; costs are nonnegative and generally increase when entries are added. It does not say that balance is conserved under arbitrary changes, only under the reciprocal-pair updates the framework defines.

The result matters because it gives the ledger a stable quantity across evolution. In the framework's account, quantum states are superpositions of ledger configurations, and the conserved balance is part of what makes those configurations comparable over time. The theorem is one of several that connect the ledger to quantum mechanics, alongside results on the Born rule and on eight-tick interference.

/-- **CONSERVATION THEOREM**: Applying an update preserves balance. -/
theorem ledger_balance_conserved (L : Ledger) (u : LedgerUpdate) :
    (applyUpdate L u).balance = L.balance := rfl

/-- A ledger is a collection of entries with conservation constraint. -/
structure Ledger where
  /-- The entries in the ledger -/
  entries : List LedgerEntry
  /-- Total balance (sum of log-ratios) -/
  balance : ℝ
  /-- Balance equals sum of log-ratios -/
  balance_eq : balance = (entries.map (fun e => Real.log e.ratio)).sum

theorem empty_ledger_balance : emptyLedger.balance = 0 := rfl

/-- The J-cost of an entry is non-negative. -/
theorem entry_cost_nonneg (e : LedgerEntry) : 0 ≤ e.cost := by
  rw [e.cost_eq]
  exact Jcost_nonneg e.ratio_pos

What this page does not claim

Costs are conserved under updates; they are nonnegative and generally increase. The balance is conserved under arbitrary ledger changes, only under reciprocal-pair updates. The theorem establishes a physical conservation law independent of the framework's definitions.

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

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