Definition 1 (Spectral complexity of a worldline). Let ψ(t) be a complex-valued worldline on a finite time grid of T steps with spacing Δt. Its spectral complexity is
![C [ψ] = ∑ |ωk|+ ϕ ,
s Δ ω cost
k∈retained](unification-of-gr-qm4x.svg)
where the sum runs over Fourier modes retained by the fidelity engine, Δω = 2π∕(T Δt) is the minimum resolvable frequency, and ϕcost is the subdominant phase cost (Paper VI).
Theorem 1 (Minimum-complexity closed worldline). Among all closed worldlines ψ(t) with ψ(0) = ψ(T), the minimum spectral complexity is Cs = 1, achieved uniquely by
This worldline traces an ellipse in the complex plane. Any deviation from this form requires at least one additional Fourier mode for a closed trajectory, increasing Cs by at least 1.
Proof. Closure of the worldline ψ(0) = ψ(T) constrains the Fourier spectrum to integer multiples of ω0 = 2π∕T: only modes at ωk = kω0 for k ∈ℤ satisfy the periodicity condition. The spectral complexity cost of mode k is |k| (in units of Δω). The trivial solution ψ = const has Cs = 0 but does not constitute a worldline (no motion). The minimum nonzero cost among closed trajectories is |k| = 1, achieved by a single complex exponential Aeiω0t, which traces an ellipse in the complex plane for any A ∈ℂ. Any additional closed trajectory requires modes at |k|≥ 2, giving Cs ≥ 2. □
Corollary 1 (Solomonoff selection of Keplerian orbits). Under Solomonoff induction, the probability weight of a worldline is 2−Cs[ψ]. The ellipse has weight 2−1 = 1∕2; all other closed trajectories are exponentially suppressed. Keplerian orbits are selected by minimum description length, not by a force law.
Corollary 2 (ISCO as minimum-complexity stable worldline). The minimum-complexity stable closed worldline is the innermost stable circular orbit (ISCO), at rISCO = 6M = 3rs for the Schwarzschild geometry. Substituting into Kepler’s third law confirms ωISCO2 ⋅rISCO3 = M exactly in Planck units. The ISCO is selected without postulating GR.