Two-Dimensional Lattice Distribution of Primes via Vertical Scanning along the Main Chord Axis A(k)

This update presents a geometric and algorithmic enhancement to the Chord and Angular Network framework by transitioning from traditional scanning to Vertical Chord Scanning originating from the Main Chord Axis nodes, A(k) = 2k^2 + 2k. We demonstrate that the composite odd space is entirely mapped by vertical vectors launched from A(k) with a linear step size of (2k + 1). This formulation optimizes memory access locality and computational throughput, enabling the exact evaluation of the prime-counting function \pi(10^9) = 50,847,534 in 8.06 seconds using a parallel segmented sieve architecture.