Digit polygons and their higher-dimensional lifts: Geometry, reconstruction, and prime denominators

This mathematical working paper studies the geometry and arithmetic of periodic digit words. Consecutive digits form a planar polygon; longer consecutive blocks form delay lifts in higher-dimensional spaces. For a prime denominator q and a base of period k, the central arithmetic question is whether some numerator attains the largest dimension permitted by parity: k − 1 when k is odd and k/2 when k is even. The paper connects this question to simultaneous nonvanishing of Fourier orders, cyclotomic divisibility, and multiplicative residue orbits. It develops the geometric theory, proves reconstruction and existence theorems, gives exact constructions and density bounds, and identifies precise unresolved arithmetic obstructions. Principal results Geometry and dimension. Fourier support determines the affine dimension and the affine type of the time-labelled configuration of a full delay lift. Fourier magnitudes determine its time-labelled Euclidean geometry. Maximal prime-denominator lifts are simplexes or affine crosspolytopes according to parity. The paper gives exact face and volume formulas and classifies a family of deficient lifts as free sums of simplexes. Fourth-order reconstruction. The mean, power spectrum, and complete fourth-order cyclic correlation recover every even-period prime-denominator digit word up to cyclic shift, including words with missing Fourier modes. The proof identifies the exact modulo-three obstruction for symmetric odd supports containing all primitive frequencies and shows that arithmetic excludes it. A shorter fourth-order slice and an explicit stability bound apply under complete odd support. A uniform period-fifteen theorem. Every available period 15·2ˢ admits a maximal numerator at every admissible prime denominator. Fourteen explicit multiplier candidates suffice from any starting numerator, and at least 1/14 of all numerator classes are maximal. This theorem uses exact computer-assisted covering certificates and dyadic decimation; it has no denominator threshold. Explicit difference numerators. Let m ≥ 3 and h ≥ 1 both be odd, and let b ≥ 2 be an integer. Put k = mh and Q = (b^k − 1)/(b^h − 1). For an integer L with h ≤ L ≤ k − h, the numerator b^L − 1 has exact period k and dimension k − gcd(k,L). Choosing L = (k − 1)/2 gives maximality. When gcd(k,L) = 1, cancellation is exactly gcd(Q,b^L − 1) = gcd(m,b − 1). A prime-power specialization realizes an entire ladder of dimensions. These statements allow composite Q; prime-denominator consequences require the stated prime or reduced-prime hypotheses. Sharp limiting prime-power density. At every odd prime-power period pᵉ, the limiting proportion Δ(pᵉ) of maximal numerator classes is at least 3/4. Equality occurs exactly at period nine. These densities tend to one as pᵉ grows through all odd prime powers. The proof uses independent Eulerian block sums and an antichain concentration inequality. Explicit finite density bounds. For k = pᵉ with p an odd prime and e ≥ 2, more than q/50 nonzero numerators are maximal whenever q > (128k)^φ(k). For distinct odd primes p < r, more than 13q/512 are maximal at period pr whenever q > (256r)^φ(pr), with stronger proportions under the further stated bounds. These conclusions transfer to available dyadic extensions. The symbol φ denotes Euler’s totient function. Fixed-period existence and semiprime densities. Every fixed period has maximal numerators for all sufficiently large admissible prime denominators, uniformly in the base, and a positive rational limiting density. The paper also gives quantitative conditioning bounds. At semiprime periods, Δ(pr) ≥ 65/144, and the density tends to one with the larger prime factor. All density limits first let the denominator tend to infinity with the period fixed. Exact arithmetic criteria. Missing Fourier orders are detected by cyclotomic divisibility, integral grouped sums, prime-adic content, and nonnegative energies. Joint character identities express simultaneous survival through sums of squared Bernoulli convolutions. At period twenty-seven, maximality is equivalent to at least one of two residue moments modulo three being nonzero; failure for all numerators is equivalent to a single explicit polynomial divisibility over the field with three elements. Minimum height and arithmetic constraints. A finite-orbit minimum principle, signed spacing, carry relations, and cubic norm constraints reduce possible failures to exact feasibility conditions. Period twenty-one has one excluded extreme profile and eleven remaining anchored cases. Period twenty-seven has five cyclic defect families, or eight profiles when the arithmetic anchor is fixed. An explicit nonarithmetic configuration shows why the stated spacing relaxation alone cannot prove descent. Applications include prime-ratio arrays, synchronized bases, and simplex completion. Methods and verification Methods include finite Fourier analysis, cyclotomic polynomial division and norms, character sums and generalized Bernoulli numbers, Dedekind-sum identities, prime-adic valuations, exact linear arithmetic, Fejér smoothing, transportation polytopes, and elementary concentration inequalities. Analytic proofs, computer-assisted theorems, conditional criteria, and finite evidence are distinguished explicitly. The period-fifteen covering theorem has exact solver certificates; the bounded-index semiprime theorem combines an analytic tail with a complete finite certificate. The accompanying project retains exact programs, input formulas, recorded results, proof artifacts, and reproduction instructions. An exhaustive anchored-candidate census covers 3,361,516 pairs with q ≤ 20,000,000 and odd composite k ≤ 3000. It is finite evidence, not a proof for unrestricted parameters. The difference-numerator formulas also pass 1,680 independent exact checks; their general validity follows from the symbolic proof. Scope and open questions The unrestricted maximal-numerator conjecture remains open. The paper does not prove that a maximal numerator exists for every prime denominator and every period. The remaining period-twenty-one and period-twenty-seven cases, the general semiprime orbit-sum dichotomy, finite prime-power denominators below the explicit thresholds, and interactions among several prime factors require further arguments. Individual character nonvanishing does not rule out cancellation in the joint convolutions. Limiting density one does not imply existence at every finite denominator. The explicit numerator constructions do not transfer automatically to arbitrary prime factors of their denominators. No infinitude theorem for primes of the form n² + 1, or for the displayed cyclotomic prime values, is claimed.