empty_ledger_balance
The empty ledger carries zero balance by construction in the quantum ledger model. Researchers deriving conservation laws and Born-rule emergence from J-cost minimization cite this base case. The proof reduces directly to the definition of emptyLedger via reflexivity.
claimThe balance of the ledger with no entries is zero: $balance(emptyLedger) = 0$.
background
The QuantumLedger module formalizes the link between Recognition Science ledgers and quantum states. A ledger entry is a recognition event carrying J-cost; quantum states appear as superpositions over ledger configurations, with the Born rule emerging from cost minimization rather than being postulated. Balance conservation under updates is a core derived property.
proof idea
One-line wrapper applying reflexivity to the definition of emptyLedger, whose balance field is set to 0 and whose balance_eq is already proved by simplification.
why it matters in Recognition Science
The result is invoked inside quantum_ledger_fundamentals to anchor the zero-balance clause among the listed fundamentals (non-negative costs, conservation, superposition structure). It supplies the base case for ledger conservation in the Recognition Science framework, supporting the claim that balance is preserved under evolution and that eight-tick phases enable interference.
scope and limits
- Does not establish balance for ledgers containing entries.
- Does not compute or bound J-cost values.
- Does not address superpositions or the Born rule.
- Does not reference the phi-ladder or forcing-chain steps T0-T8.
formal statement (Lean)
101theorem empty_ledger_balance : emptyLedger.balance = 0 := rfl
proof body
102