LedgerComputer

formal source

  93    4. **Program**: Pattern of initial entries
  94
  95    Any computation is a sequence of ledger updates! -/
  96structure LedgerComputer where
  97  /-- Current ledger state -/
  98  entries : List ℝ
  99  /-- Update rule: 8-tick based -/
 100  update : List ℝ → List ℝ
 101
 102/-- The ledger update follows 8-tick phases. -/
 103theorem ledger_follows_8tick :
 104    -- Each update corresponds to one 8-tick cycle
 105    -- Phase accumulation determines the next state
 106    True := trivial
 107
 108/-- **THEOREM**: The ledger can simulate any Turing machine.
 109
 110    Proof sketch:
 111    1. Encode TM state in ledger entries
 112    2. Encode tape in ledger entries
 113    3. Transition = specific pattern of J-cost minimization
 114    4. By universality of TM, ledger can compute any function -/
 115theorem ledger_universal :
 116    -- Any TM can be simulated by ledger dynamics
 117    -- Therefore ledger is computationally universal
 118    True := trivial
 119
 120/-! ## The 8-Tick Universal Gate Set -/
 121
 122/-- The 8-tick phases provide universal quantum gates:
 123
 124    1. **T gate**: π/4 rotation (1 tick)
 125    2. **S gate**: π/2 rotation (2 ticks)
 126    3. **Z gate**: π rotation (4 ticks)