ALGEBRAIC CLASSIFICATION OF GEOMETRIC LATTICES: SHORT-INTERVAL PRIME DISTRIBUTION ON k-GONAL SPIRALS

The geometry of integer lattices provides a structural framework for modeling quadratic polynomials associated with prime distribution. This paper proposes a unified algebraic classification of continuous centered $k$-gonal integer spirals. We initially analyze the Cartesian mapping (the Ulam spiral), isolating a bounding sequence that algebraically bisects Oppermann's interval. While this geometry yields localized prime constraints via quadratic reciprocity, elementary algebraic analysis reveals inherent structural flaws, restricting contiguous prime-admissible bands to micro-segmentations of $\mathcal{O}(X^{1/4})$ due to intersecting rational nodes and bipartite parity fault lines. To contextualize these artifacts, we expand our framework to generalized $k$-gonal geometries by mapping their polynomial boundaries to binary quadratic forms. We demonstrate an algebraic phase transition: for any $k \ge 8$, the geometric boundaries unconditionally transition from definite to indefinite quadratic forms ($\Delta \ge 0$), causing their roots to degenerate onto the real axis and transitioning into real quadratic fields. Crucially, by solving the underlying Diophantine equations, we prove that exact rational factorization (reducible collapse, $\Delta = d^2$) occurs uniquely for the terminal rays at $k=8$ and $k=9$, geometrically forcing these boundaries to consist entirely of composite numbers. Conversely, tight geometries like the triangular metric ($k=3$) maintain definite forms but suffer from class group ramification ($h>1$). This rigorous elimination mathematically isolates the orthogonally evaluated diamond lattice ($k=4$) and the optimally packed Eisenstein hexagonal lattice ($k=6$) as the unique planar configurations capable of strictly binding their boundary discriminants to the Stark-Heegner set of Class Number One imaginary quadratic fields ($\Delta \in \{-3, -4, -7, -8, -11, -12\}$). Utilizing this optimal $k=6$ lattice, we demonstrate that its non-bipartite geometry structurally sweeps fixed divisors into interstitial valleys, preserving its principal boundaries as continuous prime ridges. We then map its binary forms to the Poincaré upper half-plane to reveal exact hyperbolic metric symmetries ($D_6$) natively aligned on a universal geodesic. Furthermore, we leverage the explicit formula to demonstrate that the discrete geometry of these lattices induces a powerful spectral phase cancellation over the non-trivial zeta zeros, conditionally bounding the discrete prime variance. By relaxing the analytic target to almost-primes ($P_2$), we apply weighted linear sieves to unconditionally guarantee the asymptotic existence of $P_2$ within these hexagonal intervals, effectively bypassing the parity barrier. Concluding with these theoretical guarantees, spectral density heuristics, and computational sanity checks, we formulate unified short-interval open problems.