empty_ledger_phase
An empty ledger is assigned phase value 1 under the ledgerPhase function. Researchers constructing quantum states as ledger superpositions cite this base case when initializing the vacuum configuration. The proof is a direct simplification that unfolds the definitions of ledgerPhase and emptyLedger.
claimThe phase assigned to the empty ledger satisfies $ledgerPhase(emptyLedger) = 1$.
background
The Quantum Ledger module connects Recognition Science ledgers to quantum states by treating quantum states as superpositions over ledger configurations, with the Born rule emerging from J-cost minimization rather than being postulated. Ledger entries carry J-costs, and ledgerPhase extracts the associated phase from the eight-tick structure. The eight-tick phases are defined as $kπ/4$ for $k = 0,…,7$, periodic with period $2π$.
proof idea
The proof is a one-line simplification that applies the definitions of ledgerPhase and emptyLedger to obtain the equality directly.
why it matters in Recognition Science
This supplies the base case for the ledger-to-quantum mapping in the module, where the empty configuration corresponds to phase 1 before superpositions are formed. It anchors the eight-tick interference comment that the sum over all eight phases with equal amplitudes vanishes, modeling vacuum fluctuation cancellation. The result sits at the foundation of the ledger conservation and Born-rule derivations listed in the module documentation.
scope and limits
- Does not compute phases for any non-empty ledger.
- Does not derive the phase value from the J-cost function.
- Does not address measurement or collapse dynamics.
- Does not connect the phase to specific physical constants or tiers.
formal statement (Lean)
197theorem empty_ledger_phase : ledgerPhase emptyLedger = 1 := by
proof body
Term-mode proof.
198 simp [ledgerPhase, emptyLedger]
199
200/-- **8-TICK INTERFERENCE**: When summing over all 8 phase configurations
201 with equal amplitudes, the sum is zero.
202
203 This is the quantum version of vacuum fluctuation cancellation. -/