Digit Polygons, Gauss Sums, and the Geometry of Repunit Factorization

This paper introduces and systematically studies the digit polygon P(p), the closed planar polygon formed by plotting consecutive digit pairs from the decimal expansion of 1/p for a prime p coprime to 10. The paper establishes a spectral area identity decomposing the signed polygon area via the discrete Fourier transform, with Fourier coefficients factoring through Gauss-type residue sums. A universal negative area theorem is proved by two independent methods -- spectral negative-semidefiniteness and Cauchy-Schwarz -- showing that the signed area is strictly negative for every prime with period at least 3, a consequence of a universal decorrelation property of the decimal digit map. The paper proves an entropy separation theorem distinguishing unique primes of the cyclotomic value Phi_n(10) from the split case, introduces the polygon discriminant for binary splits Phi_n(10) = pq, and establishes that its vanishing forces a rigid Chinese Remainder Theorem constraint -- then proves that this vanishing never occurs, so that distinct prime factors always produce geometrically inequivalent polygons. The winding number and quadratic residue conjecture for full-reptend primes is resolved unconditionally. A finite-lag reduction theorem shows that the signed area depends on exactly four autocorrelation values of the residue orbit and the cross-area on exactly seven cross-correlation values, a consequence of the spectral weight polynomial having degree three. A character resonance condition is identified controlling which Dirichlet character pairs contribute to cross-correlations. The paper establishes a spectral bound on the number of prime factors of Phi_n(10), proved unconditionally for small cases via an algebraic determinant identity and sharpened via a Weil packing argument. For full-reptend primes, the polygon area is expressed as an exact weighted second moment of Dirichlet L-functions at s = 0, evaluated unconditionally via character orthogonality and Dedekind sum reciprocity, yielding an asymptotic formula without recourse to the Generalized Riemann Hypothesis. Under GRH the average polygon area recovers Artin's constant as its leading coefficient, demonstrating that polygon geometry detects primitive roots. Three new structural results are established in this version. The character-orbit duality theorem proves that the character-averaged Bernoulli sum equals the subgroup index times the conjugate DFT coefficient of the residue orbit, with contributions from residues outside the orbit of 10 killed by kernel orthogonality. This yields the coset decomposition theorem: the unrestricted per-prime moment decomposes exactly into the normalized polygon area plus a nonnegative coset energy term that vanishes if and only if p is a full-reptend prime. The resulting identity identifies the total coset energy as the precise algebraic obstruction to Artin's conjecture for base 10, reducing the primitive root problem to bounding this energy sum. The quartic factorization theorem proves that the squared weighted second moment of the L-function values factors as a perfect square, forced by the vanishing of the weight polynomial at 1, and the variance detection theorem shows that the gap between the unrestricted and actual squared moments directly measures the count of non-full-reptend primes with explicit bounds. All three results are proved to be base-independent, holding for every integer base at least 2 via the universal root structure of the weight polynomial. The entire framework is proved to be base-independent more broadly: for every integer base b, the weight polynomial factors universally, three structural cancellations in the Dedekind evaluation hold identically in b, and the leading asymptotic coefficient equals the variance of the uniform distribution on digits, establishing the digit polygon as a base-independent geometric invariant of the multiplicative orbit of b modulo p. The paper concludes by proving that Girstmair's class number formula is the Nyquist-frequency Fourier coefficient in the spectral decomposition and that the polygon area is class-number silent, partitioning the spectral energy into complementary invariants: the polygon area detects primitive roots while the Nyquist coefficient detects class numbers.