Cubic-residue quadruplets: elliptic formulas and arithmetic fluctuations

Let \(A_p\) count the monochromatic consecutive four-windows of a nontrivial cubic character modulo a prime \(p≡13\), with zero excluded and overlapping windows counted separately. We give an exact formula for \(A_p\) in terms of fixed elliptic traces and a finite boundary correction. Its complex-multiplication terms are related cubic twists. This produces two arithmetic fluctuation regimes: the first four moments of \((27A_p-p)/√ p\) tend to \((0,44,0,4224)\) when three is a cube and to \((0,17,0,738)\) otherwise. Over all admissible primes they tend to \((0,26,0,1900)\). The proof combines a four-position identity following from Greene's hypergeometric formalism and Lennon's trace formula with symmetric-power automorphy, cyclic base change and a finite collection of Rankin-Selberg prime number theorems; it uses no GRH. A direct geometric proof exposes the phases and exceptional points, with the genus-two and isogeny constructions of Freeman-Satoh and Guillevic-Vergnaud explicitly credited. Point-order and Hasse certificates verify numerical examples without enumerating their billion-position character words. The conclusions concern fixed pattern length and four moments; no complete limiting distribution, constant-order centring of the raw count, or law for logarithmic-length maxima is proved.