Digit-Polygon Fractals: Rotation, Memory, and Arithmetic Recovery

How much arithmetic information survives when a periodic digit word becomes a fractal? This working paper studies similarities based on consecutive digit pairs, with admissible compositions determined by observed blocks. Rotation, symbolic memory, and arithmetic coding provide complementary descriptions of the resulting sets. The geometric analysis gives dimension formulas under separation, classifies full-shift convexity, and determines connectedness and area in polygonal regimes. Disk certificates establish interior near the end of every nonreal parameter ray; the pair attractor is connected there except on the clockwise quarter-turn ray in bases two and three. The radius estimates are sufficient conditions, not sharp thresholds. A finite prefix–suffix tower reconstructs the cyclic word, while Fourier rank bounds the memory needed to recognize its phases. Compatible symmetries are cyclic, with exact alphabet costs both for arbitrary words and for words with a complementary half-period. Prime orbits in bases two and three admit complete symmetry classifications. Exact block frequencies can retain a prime denominator when support geometry is universal. Arithmetic recovery extends from Gaussian fractions to algebraic contractions. Leading coefficients certify periodic injectivity, resultants give explicit separation bounds, and modular division supplies exact decoders and period-bounded noisy recovery. A finite carry graph decides periodic injectivity whenever no algebraic conjugate lies on the unit circle. A separate endpoint-matched interpolation produces continuous closed curves through the indexed digit vertices. Their marked parametrizations retain the word and have exact length, Hölder, and p-variation regimes. These conclusions distinguish traversal regularity from the dimension of the unmarked image. Remaining questions concern sharp overlap boundaries, nonrectifiable image dimensions, recovery from unmarked curves, complete graph-profile collisions, prime-orbit symmetry realization, and algebraic parameters with unit-circle conjugates.