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 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 8, the geometric boundaries unconditionally transition from definite to indefinite quadratic forms (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, = d²) 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 (\-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 (D₆) 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 (P₂), we apply weighted linear sieves to unconditionally guarantee the asymptotic existence of P₂ 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.