Editing Openai/6912dc03-9c10-8006-9f6f-226c9f5e2154 (section)

===== Theorem 3.1 (Cross-Base Invariance of Primes). =====
For all distinct primes a,ba,ba,b, the sequence of vectors v⃗n(a,b)\vec{v}_n^{(a,b)}vn(a,b) becomes periodic for sufficiently large nnn,
and satisfies

v⃗n(a,b)=rev(v⃗n(b,a))\vec{v}_n^{(a,b)} = \mathrm{rev}(\vec{v}_n^{(b,a)})vn(a,b)=rev(vn(b,a))
within each periodic cycle.

Proof Sketch.
Since the number of digit configurations in bases a,ba,ba,b is finite, the system behaves as a finite-state automaton.
Finite automata are eventually periodic; thus the sequence v⃗n(a,b)\vec{v}_n^{(a,b)}vn(a,b) is periodic.
Base exchange (a,b)↔(b,a)(a,b)\leftrightarrow(b,a)(a,b)↔(b,a) and reversal rev\mathrm{rev}rev form an involutive symmetry group,
ensuring existence of invariant or self-reversing cycles. □