empty_ledger_cost
The total cost of the empty ledger configuration equals zero under the J-cost functional. Researchers deriving the ledger forcing principle or explicit cost formulas for finite Euler ledgers cite this as the base case for conservation. The proof is a direct simplification that unfolds the definitions of totalCost and emptyLedger.
claimThe total cost functional applied to the empty ledger configuration equals zero: $totalCost(emptyLedger) = 0$.
background
In the Quantum Ledger module the ledger is a collection of recognition events, each carrying a J-cost derived from the Recognition Composition Law. The empty ledger is the configuration containing no events. The totalCost functional sums these J-costs over the ledger entries, as introduced in the CostAlgebra and LambdaRecDerivation modules. Upstream results establish the reciprocal automorphism on positive reals and the active edge count A per fundamental tick, both of which enter the cost definitions used here.
proof idea
This is a one-line wrapper that applies the simp tactic to the definitions of totalCost and emptyLedger, reducing the goal to a trivial identity.
why it matters in Recognition Science
This theorem supplies the zero-cost base case for the ledger forcing principle, which derives double-entry conservation from J-symmetry and reciprocity. It is invoked in the empty-support case of the finite Euler ledger cost formula. Within the Recognition Science framework it anchors the vacuum-like state whose superpositions yield quantum states and the emergent Born rule from J-cost minimization.
scope and limits
- Does not compute costs for any non-empty ledger.
- Does not derive the explicit expression for totalCost.
- Does not address quantum measurement or Born-rule probabilities.
Lean usage
| [] => by simp [finiteEulerLedger, empty_ledger_cost]formal statement (Lean)
103theorem empty_ledger_cost : totalCost emptyLedger = 0 := by
proof body
Term-mode proof.
104 simp [totalCost, emptyLedger]
105
106/-! ## Ledger Updates -/
107
108/-- A ledger update is a pair of entries that cancel (reciprocal ratios). -/