Finite Prime Phase Recurrence

Abstract

Every finite set of prime phases returns arbitrarily close to coherent phase.

Theorem 1.1 (Finite prime phases recur above every bound).

$$\begin{gathered} \forall P:\mathrm{Finset}\left(Primes\right),\epsilon ,B\in \mathbb{R}, \\ 0<\epsilon \Rightarrow \exists \xi \in \mathbb{R},B<\xi \wedge \forall p\in P,\Vert \exp (i\xi \log p)-1\Vert <\epsilon \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Weil/PrimeAddress/FinitePrimePhaseRecurrence.finite_prime_phase_recurrence (✓ std3). ∎

Source. Repository-derived.

Commentary.

Compactness of the finite product of unit circles gives a convergent subsequence of sampled prime-phase vectors. Quotients of consecutive subsequence terms converge to the coherent phase. Sampling with a step larger than the requested bound makes the resulting recurrence time larger than that bound.

References

• Truth anchor: D5/S3/Weil/PrimeAddress/FinitePrimePhaseRecurrence.finite_prime_phase_recurrence
• Dependency: D5/S3/Weil/PrimeAddress/PrimeLogIndependence