Modular Congruence Structures, Parallel Lattice Paths, and Quadratic Polynomial Orbits on the Ulam Spiral

This paper examines algebraic relationships between linear arithmetic progressions and a symmetric family of quadratic-polynomial lattice paths mapped onto the two-dimensional integer lattice Z^2 via the Ulam spiral. By decomposing lattice points into four quadrant-aligned polynomial orbits attached to the Heegner discriminant Δ = -163, we analyze modular-congruence constraints for parallel linear paths whose common differences differ by square values exemplified by the value Δd = 12^2. We further derive modular boundary conditions that characterize candidate intersections of these arithmetic progressions with the principal coordinate axes and concentric square shells of the lattice. Numerical computations include prime-orbit statistics up to magnitude 10^10 and exact divisibility verification for constructed composite integers at the 10^17 scale. These computations illustrate exact divisibility identities, segments of elevated prime density, and occurrences of twin primes along these paths, providing structured empirical data for the study of prime-generating quadratic polynomials.