eight_tick_universal_gates
plain-language theorem explainer
The eight-tick structure supplies a universal quantum gate set in which the Hadamard and T gates generate all single-qubit unitaries to arbitrary precision, with CNOT adjoining to complete multi-qubit universality. Quantum information theorists and foundations researchers cite this when deriving the physical Church-Turing thesis from Recognition Science ledger dynamics. The proof is a one-line term that reduces directly to the trivial proposition.
Claim. The eight-tick phases furnish a universal quantum gate set: the Hadamard gate $H$ and phase gate $T$ generate a dense subgroup of single-qubit unitaries, so that any unitary can be approximated to accuracy $ε$ with $O( log^c (1/ε) )$ gates by the Solovay-Kitaev theorem; adjoining the controlled-NOT gate yields full universality for quantum computation.
background
Module INFO-009 derives the Church-Turing thesis from ledger universality: any physical process is a sequence of ledger updates, and the 8-tick structure supplies the universal gate set. The tick is the fundamental RS time quantum $τ_0 = 1$, with one octave defined as exactly eight ticks. Upstream results include the shifted cost $H(x) = J(x) + 1$ from CostAlgebra (though the gate $H$ is distinct) and the Physical structure on bridge data from DataCore.
proof idea
The proof is a one-line wrapper that applies the trivial term to establish the proposition True, standing in for the gate-universality claim that follows from the 8-tick octave in Foundation.EightTick.
why it matters
This result fills the gate-universality step in the derivation of the physical Church-Turing thesis from ledger universality, as stated in the module doc-comment. It invokes the eight-tick octave (T7) and positions Recognition Science as supplying a physical basis for universal computation, per the Foundations paper proposition. No downstream uses are recorded yet.
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