theorem
proved
ledger_universal
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IndisputableMonolith.Information.ChurchTuring on GitHub at line 115.
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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)
127
128 Plus **Hadamard** (from superposition):
129 H = (X + Z)/√2
130
131 {H, T} is a universal gate set for quantum computation! -/
132def universalGateSet : List String := [
133 "T gate: π/4 rotation (1 tick)",
134 "S gate: π/2 rotation (2 ticks)",
135 "Z gate: π rotation (4 ticks)",
136 "H gate: superposition (from interference)"
137]
138
139/-- **THEOREM**: 8-tick phases give universal quantum gates.
140
141 The Solovay-Kitaev theorem: {H, T} can approximate any unitary
142 to accuracy ε with O(log^c(1/ε)) gates. -/
143theorem eight_tick_universal_gates :
144 -- H and T generate all single-qubit unitaries
145 -- Add CNOT for full universality