clock
plain-language theorem explainer
Defines the recognition cost-rate clock on the eight-tick cycle: pointwise multiplication of a complex amplitude on Z/8Z by the k-th power of a primitive eighth root of unity. Anyone citing the finite Weyl relation or the RS derivation of [x,p] uses this operator as the cost-rate half of the pair. The body is a one-line pointwise definition, not a proof.
Claim. On complex amplitudes $\psi:\mathbb{Z}/8\mathbb{Z}\to\mathbb{C}$, the cost-rate clock acts by $(C\psi)(k)=\omega^{k}\,\psi(k)$, where $\omega$ is a fixed primitive eighth root of unity and $k$ is read as its integer representative in $\{0,\ldots,7\}$.
background
The module develops recognition-first quantum kinematics on the eight-tick cycle $\mathbb{Z}/8\mathbb{Z}$. Conventional QM postulates $[x,p]=i\hbar$; here occupation and cost-rate are realized as the shift and clock operators of the finite Heisenberg–Weyl group on that cycle, so non-commutativity is cyclic structure rather than an axiom.
The companion root $\omega$ is a primitive eighth root of unity (with $\omega^8=1$ and $\omega\neq 1$ recorded as sibling facts). The fundamental RS time quantum is one tick $\tau_0=1$, and one octave is eight ticks; the geometric phases $k\pi/4$ for $k=0,\ldots,7$ are the real angles underlying the same discrete clock. The continuum limit $[x,p]=i\hbar$ with magnitude $\hbar=\varphi^{-5}$ is left open (node D6).
proof idea
Pure definition: no tactics. The operator is the diagonal multiplication map $k\mapsto \omega^{k.\mathrm{val}}\cdot\psi(k)$ on functions $\mathbb{Z}/8\mathbb{Z}\to\mathbb{C}$. Noncomputable only because complex exponentials enter the ambient $\omega$. Downstream lemmas (Weyl braiding, non-commutativity) do the work; this declaration only names the clock half of the pair.
why it matters
This is the cost-rate operator in the eight-tick Weyl pair. Together with the shift, it yields the Weyl relation $C\circ S=\omega\cdot(S\circ C)$ and thus canonical non-commutativity on $\mathbb{Z}/8\mathbb{Z}$ (sibling canonical_noncommutativity), tying T7's eight-tick octave to the root of $[x,p]$.
Downstream, the same clock language feeds horizon clock rates (Euclidean angle advance), seam/trace readings, black-hole charges (mass from time-translation / 8-tick clock), pointer-state sieves, complex-structure forcing on the 8-cycle, and cosmology's global 8-tick synchronization story for the horizon problem. The continuum commutator and the $\hbar=\varphi^{-5}$ magnitude remain open; only the finite cyclic skeleton is closed here.
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