Digit Polygons: Energy, Symmetry, Equidistribution, and Dedekind Sums
K. Fathi · Zenodo (CERN European Organization for Nuclear Research) · 2026
This paper develops a sequel theory for digit polygons arising from repeating base-b expansions of reciprocals of primes. Let p be a prime that does not divide b, and let k be the multiplicative order of b modulo p. The repeating digits of 1/p define a closed polygonal walk. Its signed area is exactly negative one quarter of the sum of the squared differences between digits two positions apart. The central question is whether the area divided by k approaches the universal constant negative (b squared minus 1) divided by 24 whenever k grows faster than log p. The paper converts this geometric problem into several exact arithmetic formulations. First, the area is shown to be a weighted distribution statistic for the canonical representatives of the multiplicative subgroup generated by b, reduced modulo the fixed integer b cubed. A finite Fourier formula identifies the particular carry-class modes that must remain biased in any counterexample. An exact coboundary decomposition also removes every continuous Fourier mode whose frequency is divisible by b. Known estimates for multiplicative subgroups yield an unconditional area theorem in the Bourgain-Shkredov subpolynomial range. The proposed superlogarithmic endpoint follows conditionally from the Montgomery-Vaughan-Wooley conjecture. In the reverse direction, a new entropy argument proves that persistent area bias would force polynomial overcrowding inside a single mesoscopic base-b interval. This produces a concrete nonconcentration problem that is weaker than full subgroup equidistribution but is not resolved by existing small-height estimates. At the logarithmic boundary, forced initial zero blocks and complementary digit plateaus create an unconditional obstruction. Exact repunit and cyclotomic constructions explain this boundary behavior. Bang-Zsigmondy theory also shows that the proposed superlogarithmic regime contains an unconditional infinite family. For general numerators a/p, the paper evaluates the exact mean over multiplicative numerator cosets and proves mean-square concentration. The exceptional numerator a equals 1 is reduced to twisted Dedekind sums, Bernoulli restriction fibers, and a positive projection norm for a filtered Stickelberger element. A separate base-change identity reconstructs the polygon area from raw square-digit totals in bases b, b squared, and b cubed. This gives an explicit half-order Dedekind formula. The accompanying reproducibility package contains the LaTeX source, exact census data, structured primitive-divisor and cyclotomic searches, identity verifiers, complete Lucas primality certificates for two large cyclotomic primes, and cryptographic checksums. The computations reveal substantial finite fluctuations but no asymptotic counterexample. The endpoint conjecture remains open.