Fix an integer base b ≥ 2 and a prime p coprime to b. Write the purely periodic base-b expansion of 1/p with period k equal to the multiplicative order of b modulo p, plot consecutive digit pairs as planar points, and join them in order to form a closed polygon — the digit polygon of 1/p. The foundation of the paper is a single algebraic identity: the signed shoelace area equals minus one quarter of the lag-2 Dirichlet energy of the digit stream, from which negativity, clockwise orientation, and half-integrality of the area are immediate. The extremal area, negative k times (b−1) squared over 4, is attained if and only if 4 divides k; among prime reciprocals the unique attainer is p = b squared plus 1 whenever that number is prime, whose digit polygon is the full digit square traversed clockwise. Admissible closed polygons on the digit grid are exactly the closed walks in the order-2 de Bruijn graph on b symbols, and a polygon represents a prime reciprocal precisely when its numerator divides b to the k minus 1 with prime quotient, separating polygon geometry from prime selection cleanly. The digit stream is the output of a one-tap causal linear filter driven by the residue orbit, and cascading this with the lag-2 difference operator yields a four-tap filter with symbol q(z) = (z² − 1)(b − z) and a perfect-square identity expressing negative four times p squared times the area as the squared norm of the filtered residue orbit. Diagonalizing via the discrete Fourier transform yields spectral area formulas, and expanding the perfect square gives an exact four-lag autocorrelation formula whose coefficient polynomial satisfies a spectral factorization identifying the area as the output power of a factored filter. A step-function transfer principle then converts the digit statistics appearing in the normalized area into Riemann-sum approximations controlled by the interval discrepancy of the residue orbit, yielding a master bound that ties the normalized area to the negative of half the variance of a uniform digit, which is negative (b squared minus 1) over 24. Gauss sums over the multiplicative subgroup generated by b modulo p and the Erdős–Turán inequality bound the discrepancy, and combining this with the Erdős–Murty lower bound on the multiplicative order yields the unconditional headline: for every fixed base b, the area divided by the period converges to negative (b squared minus 1) over 24 with an error of order 1 over log p, for all primes outside a set of relative density zero. For full-reptend primes, where the period equals p minus 1, the four-lag formula over the complete multiplicative group converts each lag-autocorrelation into a Dedekind-type sum, and Dedekind reciprocity collapses the result to an exact closed form: the area equals negative (b squared minus 1) over 24 times p, plus a correction term that is a explicit linear combination of three classical Dedekind sums evaluated at p modulo b cubed. The deviation in the variance law is therefore exactly periodic in p, takes finitely many rational values, and exhibits no fluctuations at any growing scale; the evaluation extends to prime powers, to every cyclic digit autocorrelation of 1/p, and to half-reptend primes with p congruent to 3 modulo 4 via a character decomposition. All evaluations unify over the character group through a spectral identity expressing the area as a weighted quadratic form in generalized Bernoulli numbers over odd Dirichlet characters, with the filter symbol providing the weights and strict negativity following from the non-vanishing of Dirichlet L-functions at 1. Every identity in the paper has been verified in exact arithmetic by a companion script, and all numerical tables were produced by the same script.
K. Fathi (Thu,) studied this question.