Digit Polygons and Prime Signatures for n^2 + 1

Problem and scope This mathematical working paper studies the arithmetic information carried by digit polygons: the closed ordered walks formed from consecutive digits of repeating expansions. Its central family is N = n² + 1 with even n ≥ 2. The paper separates exact geometric criteria for the primality of an individual denominator from estimates that would count prime denominators. It develops rigorous identities, almost-prime results, and conditional prime-counting criteria; it does not prove that n² + 1 is prime infinitely often. Principal results Quarter-turn geometry. Every nonzero numerator with denominator n² + 1 gives a four-digit square in base n, including unreduced fractions. Thus the square shape alone does not characterize primality. Complementary-square bases correspond exactly to square roots of −1 modulo N, and their number is 2 raised to the number of distinct prime divisors of N. Together with the classical exclusion of proper powers n² + 1, this gives an exact primality criterion. A sharp area separation. A normalized mean H(n), taken over the complete set of complementary-square bases, equals (n − 1)/n at primes. For composites its minimum is exactly 8886839/6193536, attained only at n = 12. The roots also form a Boolean lattice; a weighted metric and its spectrum recover the complete prime-power factors. Parity and prime-counting reductions. The centered reduced-numerator signed-area mean records prime-factor parity on squarefree denominators. Positive sieve weights yield an exact prime projection. A translated Chebyshev polynomial of degree eight has certified ordinary mass and a controlled odd-composite penalty. Degree eight is minimal within the stated interval bounds and mass calibration. Reduced-residue harmonics and denominator-chain identities represent the same signed observable with explicit errors. Sparse area counts. For every fixed bound on the complementary-area score, there are only O(log X) composite inputs up to X. At threshold two, Pell numbers give a finer classification and the bound O(log X/(log log X)²), together with an exact area–parity identity accounting for the exceptional input n = 70. Signed sums and their limitations. Gaussian factorization, smooth sieve weights, cofactor energies, local correlations, and matching capacities give exact decompositions and sufficient inequalities for prime counts. Continuous semiprime models and Fourier examples show why the specified unsigned estimates and operator bounds do not by themselves provide the required sign information. Arithmetic descent and factor graphs. Opposite-root descent terminates at prime inputs and has logarithmic depth, but its image has limited capacity. The prime–semiprime shared-factor graph has explicitly classified components, including prime-free components; short paths reach little rough semiprime mass. Allowing four prime factors gives a connected graph of diameter two without supplying the needed weighted prime bound. Fixed rational norm maps are classified, including the square factors forced by nontrivial maps. Restricted digits and reciprocity. A decimal family with digits drawn from 0 through 8 supports squarefree values with at most four prime factors, with mass bounded below by a positive constant times 9ᵏ/k for all sufficiently large k. A different, weaker restriction forbidding one aligned eight-digit block gives squarefree three-almost-prime mass in base B = 10⁸, bounded below by a positive constant times (B − 1)ᵏ/k for all sufficiently large k. Exact odd-shift reciprocity cancellation, a marked-prime incidence estimate, and weighted quadratic Fourier identities are also established. These estimates do not control the full Möbius-weighted prime-counting sum. Methods and remaining questions The arguments combine the shoelace formula, the Chinese remainder theorem, finite Fourier analysis, Pell equations, Chebyshev extremality, linear and weighted sieves, Vaughan’s identity, Gaussian integers, affine-sieve counting, and spectral and transport inequalities. Explicit rational certificates and reproducible programs accompany the proofs. Finite computations are distinguished from asymptotic theorems. The principal unresolved question is a one-sided estimate for the inherited Möbius-weighted sum, or the weaker strict positive gap sufficient for prime infinitude. Related open questions concern the signed digital bilinear remainder, arithmetic comparisons between different inputs, and the contribution of odd composite values. The stated limitations apply to the specified constructions and estimates; they are not general impossibility results for other methods. Contents and related work The deposit contains the manuscript, editable LaTeX sources, figures, verification programs, data, and build instructions. The exposition has a main geometric and prime-counting argument followed by eleven technical appendices.