The Geometry of Prime Constellations in Quasimodular Space: Algebraic Detectors and the Cubic Mimic Phenomenon

Building on the recent discovery of prime power detectors via MacMahon partition functions, we generalize the framework to establish a broad structural principle: any number-theoretic set parameterized by polynomials lies on a one-dimensional algebraic variety in the quasimodular plane $(n, M_1)$. We prove the Quasimodular Parameterization Theorem and use it to derive explicit polynomial detectors for prime constellations (twins, triplets, quadruplets), Sophie Germain chains, and a parameterized subclass of Carmichael numbers. We establish the General Gap Theorem, proving that the product of any pair of primes $p(p+k)$ satisfies a quadratic identity where the gap $k$ is encoded strictly in the constant term $-(k^2-1)$. Furthermore, we characterize the geometric boundary of this theory by identifying the phenomenon of "Algebraic Aliasing". We demonstrate that the varieties defined by these detectors contain a sparse exceptional set of "Cubic Mimics" (e.g., $n=8, 135, 1485$) where the geometric divisor sum of a prime cube $\sigma_1(p^3)$ numerically mimics the linear product structure of a constellation. Outside this sparse exceptional locus, the detectors are shown to be robust.