omega
plain-language theorem explainer
The eight-tick phase factor is the primitive eighth root of unity $\omega=e^{2\pi i/8}$. Recognition-first quantum work cites it as the braiding phase by which the cost-rate clock advances once per recognition tick on $\mathrm{ZMod}\,8$. The declaration is a direct complex-exponential definition, not a derived identity.
Claim. Define $\omega := \exp(2\pi i/8)\in\mathbb{C}$, the primitive eighth root of unity. On the eight-tick recognition cycle this is the phase by which the cost-rate advances once per recognition tick.
background
The module develops the eight-tick Weyl relation as the recognition root of canonical non-commutativity. Conventional QM postulates $[x,p]=i\hbar$; RS derives the discrete precursor on the cycle $\mathrm{ZMod},8$, where occupation and cost-rate are the shift and clock operators of the finite Heisenberg–Weyl group.
The fundamental RS time quantum is one tick ($\tau_0=1$). One octave is eight ticks, the fundamental evolution period (forcing landmark T7). The cyclic shift advances the reading index by one tick mod 8 and is the discrete time-evolution generator. The companion clock multiplies by successive powers of the phase defined here.
The continuum limit $[x,p]=i\hbar$ and the magnitude $\hbar=\varphi^{-5}$ are left open (node D6); only the discrete braiding is closed in this file.
proof idea
Pure definition: $\omega$ is introduced as Complex.exp (2 * Real.pi * Complex.I / 8). No lemmas are applied; noncomputability is inherited from the complex exponential. Downstream facts such as $\omega^8=1$ and $\omega\neq 1$ are proved separately.
why it matters
This phase is the only ring-specific content of the eight-tick Weyl relation $\mathrm{clock}\circ\mathrm{shift}=\omega\cdot(\mathrm{shift}\circ\mathrm{clock})$. Sibling results (omega_pow_eight, omega_ne_one, eightTick_weyl, canonical_noncommutativity) use it to show that occupation and cost-rate do not commute, so canonical non-commutativity is cyclic recognition structure rather than an axiom.
It sits on forcing landmark T7 (eight-tick octave, period $2^3$). The continuum commutator and the tie of its magnitude to $\hbar=\varphi^{-5}$ through the J-cost quantum remain open derive-tick work (nodes D3/D6), not asserted here.
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