Digit polygons and their higher-dimensional lifts

Problem and framework A periodic base-b digit word defines a polygon by taking consecutive digits as coordinates and a family of higher-dimensional delay lifts by taking longer blocks. This working paper studies the affine dimension of those lifts and the arithmetic effect of changing the numerator of a fraction with prime denominator. Its central question asks whether, for every odd prime q and base b coprime to q, some numerator attains the largest dimension allowed by parity: k − 1 for odd period k and k/2 for even period k, where k is the multiplicative order of b modulo q. Principal results Exact dimension formulas. The affine dimension of an m-coordinate lift is min(m, D), where D is the number of nonzero Fourier coefficients of the centered periodic word. A family with the same square trace has arbitrarily large full delay dimension and crosspolytope hulls. The paper also classifies decimal three-coordinate lifts at prime denominators. Arithmetic of the numerator. Cancellation at nonconstant frequencies is equivalent to cyclotomic divisibility of a least-residue polynomial and is detected by exact integer energies. Primitive frequencies always survive, and every parity-allowed frequency occurs in some numerator class. Structural existence results include odd-prime square periods p² and their power-of-two multiples. Orbit-sum divisibility, modular multiplicity, prime-adic content, and sparse grouped words give further certificates. Explicit constructions. Prescribed digit words yield exact dimension families at odd periods, including composite denominators. Maximal numerators are constructed whenever the relevant prime-power or semiprime cyclotomic value is prime. For semiprime values Φₚᵣ(b), the construction uses a numerator bᴸ − 1, with one explicitly handled binary exception. No infinitude of prime cyclotomic values is assumed. Semiprime criteria and bounded-index families. For k = pr, with distinct odd primes p < r, row and column energies detect the two possible proper-order defects. Second moments, quadratic-residue tests, and energy-support bounds provide sufficient conditions for simultaneous survival. The orbit-sum dichotomy is established when p is 3, 5, or 7 and the order-pr subgroup has index at most 64. Its proof combines an analytic tail with an exhaustive, computer-assisted certificate for 34,812 finite parameter pairs. Decimation transfers maximality to available periods 2ˢpr. Effective existence and limiting density. At each fixed period, every sufficiently large admissible prime denominator has maximal numerators, uniformly over the base. It suffices that q exceed (8k⁹) raised to the power φ(k), where φ is Euler’s totient. The maximal proportion converges to a positive rational density Δₖ. Exact values include Δ₉ = 3/4, Δ₂₇ = 225/256, and Δ₂₅ = 318355/331776. An explicit stronger range for odd periods guarantees both central orbit sums (k − 1)/2 and (k + 1)/2. Semiprime density bounds. A rectangular torus model gives Δₚᵣ ≥ 65/144 and Δₚᵣ → 1 as r grows, uniformly over p < r. Exact Haar energy moments and Fejér approximation give a separate finite result: if q exceeds (2¹⁷r²) raised to the power φ(pr), more than 1/50 of the numerator classes are maximal, simultaneously for every available power-of-two extension. Prime arrays and synchronization. The paper classifies cyclic types in prime-ratio columns, derives exact digit-block histograms and combined winding profiles, and describes prime sampling for fixed denominators. For synchronized expansions in several bases, it determines dimension and phase equivalence, proves exact orthogonality in distinct two-adic period classes, and constructs simplex completions. At denominator 73, seven base orders are proved necessary and sufficient for a collection that works for every numerator. Methods and scope The arguments combine finite Fourier analysis, cyclotomic divisibility and norms, character nonvanishing, Dedekind sums, exact integer and rational arithmetic, explicit digit subtraction, rational-polytope decompositions, Eulerian distributions, and trigonometric approximation on a cyclotomic torus. Limiting density statements first let q tend to infinity at a fixed period. Their probability models do not assert independence of coordinates on each finite residue orbit. Computer-assisted theorems, conditional criteria, conjectures, and exploratory finite evidence are identified separately. Open questions The unrestricted maximal-numerator conjecture and the semiprime orbit-sum dichotomy remain open. The uniform strict energy-support inequality is also unproved. Further questions concern arithmetic descriptions of grouped residue words, the minimum number of bases needed for simplex completion, and prime sampling when denominators or observation lengths grow. Counterexamples delimit stronger proposed criteria and restricted numerator constructions; in particular, the prime-power coprime-orbit-sum assertion is false in general. The results do not establish infinitely many primes of the form n² + 1. Contents of the deposit The project contains the manuscript PDF, editable LaTeX source, supplementary mathematical notes, exact verification programs, recorded datasets, reproduction instructions, and file checksums. A separate report documents the revision, verification scope, reference checks, and unresolved issues.