We study the closed polygon obtained by plotting consecutive digit pairs (dn, dn+1) from the decimal expansion of 1/p for a prime p coprime to 10, and develop a complete spectral theory connecting this elementary geometric object to Dirichlet L-functions, Artin's primitive root conjecture, class numbers of imaginary quadratic fields, and the factorization of cyclotomic polynomials evaluated at 10. Main results Spectral area identity. The signed area of the digit polygon admits an exact DFT decomposition whose Fourier coefficients factor through Gauss-type residue sums (Theorem 3. 3). Universal negative area. The signed area is strictly negative for every prime with period k ≥ 3, proved via two independent arguments—spectral negative-semidefiniteness and Cauchy–Schwarz—establishing a universal decorrelation property of the decimal digit map (Theorem 8. 1). Polygon discriminant and non-vanishing. For the binary split Φn (10) = pq, we define a polygon discriminant, prove its spectral factorization (Theorem 5. 5), and establish that its vanishing never occurs: distinct prime factors always produce geometrically inequivalent digit polygons (Theorem 9. 1). The proof uses a divisibility-versus-digit-range argument that appears to be new. Winding number resolution. Full-reptend primes always satisfy (10/p) = −1 (Theorem 8. 4) and the digit polygon always winds clockwise (Theorem 8. 8), resolving the Winding–QR conjecture. Finite-lag reduction. The signed area depends on exactly 4 autocorrelation values, and the cross-area on exactly 7 cross-correlation values, a consequence of the weight polynomial Pb (z) having degree 3 (Theorem 7. 1). Connection to Artin's conjecture. The polygon area is an explicit weighted second moment of |L (0, χ) |² over Dirichlet characters (Theorem 11. 7). We evaluate this moment unconditionally via Dedekind sum reciprocity, obtaining M (x) = (33/8) · x/log x + O (x/ (log x) ²) without GRH (Theorem 11. 13). We prove a character-orbit duality (Theorem 12. 1) and a coset decomposition theorem (Theorem 12. 3): the unrestricted moment decomposes exactly as Mp = |A (p) |/ (p−1) + CE (p), where CE (p) ≥ 0 is the coset energy, vanishing if and only if p is a full-reptend prime. This identifies the coset energy as the precise algebraic obstruction to the Artin conjecture for base 10. The Artin equivalence theorem (Theorem 12. 9) proves that the Artin conjecture (density form) for base 10 is equivalent to the evaluation of E1/fp². Spectral orthogonality of Artin's constant and class numbers. Girstmair's class number formula is recovered as the Nyquist-frequency Fourier coefficient (Theorem 13. 1). The polygon area is "class number silent"—the quadratic character's contribution vanishes identically, forced by the root z = −1 of the weight polynomial (Theorem 13. 2). This establishes a spectral partition: the polygon area detects primitive roots (Artin's constant), while the Nyquist coefficient detects class numbers, and these are complementary invariants. Base-independence. The entire framework holds for every integer base b ≥ 2. The weight polynomial factors universally as Pb (z) = (z² − 1) (bz − (b² + 1) ), and the leading coefficient of the unrestricted moment is Var (Ub) /2 = (b² − 1) /24 (Theorem 12. 5), establishing the digit polygon as a base-independent geometric invariant of the multiplicative action b ↷ (ℤ/pℤ) *. Large-sieve framework. We develop a Barban–Davenport–Halberstam-type framework for bounding the total coset energy unconditionally. The aggregate fluctuations of CE (p) around the equidistributed prediction are proved negligible (Theorem 15. 6), and the Artin density problem is reduced to a single sieve-theoretic input: any unconditional lower bound Σ 1/fp ≥ η · π (x) implies a positive density of full-reptend primes (Theorem 15. 8). Spectral bound on ω (Φn (10) ). We establish ω (Φn (10) ) ≤ n − 1 (Theorem 10. 3), prove this unconditionally for m ≤ 3 (Theorem 10. 4), and sharpen to ω (Φn (10) ) ≤ C φ (n) /log φ (n) via a Weil packing argument on the Gram matrix (Theorem 7. 3). All main theorems are verified computationally: the coset decomposition identity holds to relative error <10−12 for all 1, 227 primes below 10, 000 in base 10, and the exact identities (4-lag area formula, quartic factorization, discriminant non-vanishing) are verified via arbitrary-precision arithmetic across bases b ∈ 2, 3, 5, 7, 8, 10, 12, 16 and for primes up to 99, 990, 001.
Kevin Fathi (Thu,) studied this question.