Digit Polygons from Repeating Expansions: Exact Area Identities, Finite-Lag Reduction, and Base-b Asymptotics

Let b >= 2 be an integer and p a prime with gcd (b, p) =1. Write the base‑b expansion of 1/p as 0. d0 d1. . . d₊-₁ (repeating), where k = ordₚ (b) is the multiplicative order of b modulo p. The digit polygon is the closed polygon with vertices (dᵢ, d₈+₁) for i=0,. . . , k-1 (indices mod k). This work isolates the mathematically robust core of the digit‑polygon construction and adds a reverse construction. The reverse construction starts from a pointed directed polygon on the digit‑pair grid 0, 1, …, b-1². Such a polygon determines a repeating base‑b expansion exactly when consecutive vertices overlap, i. e. , when Vᵢ = (xᵢ, yᵢ) satisfies yᵢ = x₈+₁ cyclically. The resulting digit block d0…d₊-₁ yields the repeating expansion 0. (d0 d1 … d₊-₁) = N/ (bᵏ - 1) with N = d0 b^k-1 + d1 b^k-2 + … + d₊-₁. Prime reciprocals are an arithmetic subfamily: the reverse‑generated expansion equals 1/p for a prime p exactly when N divides bᵏ - 1 and (bᵏ - 1) /N is prime. The signed area Ab (p) satisfies an exact Dirichlet‑energy identity: Ab (p) = -1/4 * sumᵢ (d₈+₂ - dᵢ) ². This immediately implies Ab (p) = 3. The Fourier representation is Ab (p) = -1/ (k p²) * sumⱼ |b - ωʲ|² sin² (2π j/k) |r̂ⱼ|², where rᵢ = bⁱ mod p and ω = e^2π i/k. Expanding in residues yields a finite‑lag reduction: the area depends on exactly four orbit correlations, with coefficient polynomial Pb (z) = - (b²+1) + b z + (b²+1) z² - b z³ = - (z²-1) (b z - (b²+1) ). For two digit sequences of the same period we prove an exact discriminant theorem: the Gram determinant of the area form equals a Cauchy‑Schwarz determinant for the lag‑2 difference operator; it is always nonnegative, and equality is characterized completely by the kernel of that operator. Finally, the reverse construction gives a random‑digit baseline. For independent uniform base‑b digits, the expected normalized area is -Var (Ub) /2 = - (b² - 1) /24, where Ub is uniform on 0, …, b-1. For full‑reptend primes (ordₚ (b) = p-1) we prove the matching asymptotic law: Ab (p) / (p-1) = - (b²-1) /24 + Ob (1/p).