universalGates
Recognition Science equates the universal quantum gate set cardinality to the spatial dimension D by direct assignment of 3. Researchers modeling quantum circuit depth via J-cost minimization cite this when linking gate counts to the forcing chain. The definition is a constant assignment with no lemmas or reductions applied.
claimThe universal gate set has cardinality $3$, equal to the spatial dimension $D$.
background
The module frames quantum computation as sequences of J-cost-minimizing recognition operations, with J the functional obeying the Recognition Composition Law. Five canonical gate types (Pauli, Clifford, T, CNOT, Toffoli) match the configuration dimension, while the single-qubit Pauli group has size $2^3$. Universal sets such as {H, T, CNOT} are assigned size 3, again equal to D from the eight-tick octave.
proof idea
This is a direct definition that assigns the natural number 3 to universalGates.
why it matters in Recognition Science
It supplies the universal_D field inside the QuantumComputingDepthCert structure, which certifies RS-derived quantum computation depth. The assignment closes the link between gate-set cardinality and D = 3 forced at T8 of the UnifiedForcingChain. The module states that such sets generate all unitaries through J-cost minimization.
scope and limits
- Does not prove that any concrete set such as {H, T, CNOT} is universal.
- Does not derive the value 3 from the forcing chain or RCL.
- Does not address multi-qubit or continuous-parameter gate families.
formal statement (Lean)
32def universalGates : ℕ := 3