A 2-adic Bridge Between the Chinese Rings Puzzle, Cubic Residues mod 2ⁿ , and the Golden Book Sequence via Self-Base Digit Expansions

For an odd integer x ≥ 3, I study the base-2x digit expansion of xⁿ as n → ∞. Using the Chinese Remainder Theorem, I show that for n large enough the digit at any fixed position stabilizes into a purely periodic sequence whose period is the multiplicative order of x modulo a power of 2. I give a closed formula for this order in terms of the 2-adic valuations v₂(x-1) and v₂(x+1), and use it to derive an exact recurrence for the number of zero digits appearing within one period at each depth. The correction term of this recurrence is periodic with period ord(2 mod x), and the resulting sequence satisfies a linear recurrence with characteristic polynomial (t-2)(t^(ord(2,x))-1). As direct corollaries, the cases x=3,5,7 reproduce three previously unrelated integer sequences: A005578 (the Chinese Rings / Pisot-type sequence), A256494 (the "Golden Book" leveling sequence), and A046630 (the count of cubic residues modulo 2ⁿ). I also establish the asymptotic digit density 1/(2x).