Geometric Lattices and Prime Distribution: Short-Interval Conjectures on the Cartesian and Eisenstein Spirals

The geometry of integer lattices provides a structural framework for studying the local distribution of prime numbers. This paper analyzes the algebraic properties of two geometric mappings---the Cartesian square lattice (the Ulam spiral) and the optimally packed Eisenstein integer lattice Z---to formulate explicit short-interval conjectures regarding prime distribution. On the Cartesian plane, we isolate the arithmetic sequence governing the left boundary of the Ulam spiral. We show that this geometric boundary algebraically bisects Oppermann’s interval for even perfect squares, restricting primes within intervals of width 0. 5X^1/2. Assuming the proposed Ulam Left-Boundary Conjecture, we establish structural lower bounds for prime distribution within Legendre's and Brocard's intervals, and demonstrate via polynomial discriminants and Quadratic Reciprocity that this boundary systematically avoids divisibility by early odd primes (3, 5, 7, 11). To address orthogonal symmetries and parity biases inherent to the Cartesian metric, we subsequently transition to a discrete centered hexagonal spiral on the Eisenstein lattice. We show that the algebraic discriminants of all six geometric boundaries evaluate to the Stark-Heegner set \-3, -8, -11, -12\, corresponding to imaginary quadratic fields of class number h=1. By mapping the associated binary quadratic forms onto the Poincar\'e upper half-plane H, we observe that all Heegner points generated by the Eisenstein boundaries exhibit exact geodesic alignment, lying uniquely on the semi-circular geodesic of radius R = 1/3. We explicitly calculate the hyperbolic metric symmetry of these points, confirming they preserve the discrete D₆ symmetry of the lattice. Finally, under the action of the modular group = PSL (2, Z), we show that these boundary invariants map naturally to the fundamental domain F. Supported by these algebraic regularities, analytic variance heuristics over discrete coordinates via the explicit formula, and unconditional empirical verification up to 2 10⁹ using computational prime gap records, we propose two geometrically derived short-interval open problems.