OMEGA pair dynamics

Play the two clocks. Match the anchor to the world and the error stays at zero. Keep the OMEGA pair while the world is even slightly off, and the error grows. A random complementary pair (λ′+μ′=1) traces a different torus slope. Unconstrained frequencies generally drop λ+μ=1.

Write $\varphi=(1+\sqrt5)/2$, $\Omega_0=\varphi+\pi$, $\lambda=\varphi/\Omega_0$, $\mu=\pi/\Omega_0$. The experimental 4D map is

$R_\Omega=R(2\pi\lambda)\oplus R(2\pi\mu).$

A 4-vector $(x,y,z,w)$ rotates in the $xy$ plane by $2\pi\lambda t$ and in the $zw$ plane by $2\pi\mu t$. Because $\lambda+\mu=1$, the two angles sum to a full turn per unit time. On the torus of fractional windings the trajectory is a straight line of slope $\mu/\lambda=\pi/\varphi$.

That slope is the pair’s geometric signature. It is not a claim that random 4D dynamics prefer these frequencies.

Numbers below are from the published drift results (seed 20260831). This lab does not rerun those Monte Carlo trials.

• E1, matched OMEGA spiral: at 10⁵ steps the iterative float32 state error is 2.451e-3; the direct anchor stays at 3.798e-8. Direct reconstruction helps when the model is the true dynamics. A generic true-model renderer has the same advantage.
• E2, mismatch δᾱ=0.00001, δβ̄=-0.000017: at 10⁵ steps the OMEGA wrong-model error is 1.249, while the generic true-model anchor stays at 4.246e-8.
• E4, 2,000 random tasks and 200 random anchors: OMEGA mean state error 1.3340 versus random mean of means 1.3318. Fraction of random anchors better than OMEGA: 0.56. no universal privilege detected

Full tables and source: drift experiment.

Not a source paper, not a Paper VII theorem, not a memory-drift law, not a Navier–Stokes result, and not a claim that Ω₀ is a privileged dynamical anchor. Related: geometry, drift, exploration 02, sampling extension.