Direct Shot: Bypassing the Computational Wall via Constant-Time Coordinate Reaching

Description This work presents Direct Shot, a constant‑time execution model that bypasses dense matrix multiplication through direct coordinate reaching. The study is organized around three complementary pillars: Appendix A – Reproducibility: Jupyter notebook benchmarks and raw latency measurements, ensuring transparency and independent verification. Appendix B – Theoretical Framework: A number‑theoretic foundation involving elliptic curve isogenies, j‑invariants, Hecke operators, and spectral properties of the Riemann zeta function. Appendix C – Practical Applications: Dimensional interpretation and exponential sparsity, including experimental validation on the Rio Botanical Garden LiDAR dataset. Together, these components demonstrate the empirical reliability, the mathematical inevitability, and the broader technological significance of Direct Shot. The work establishes Direct Shot as both a reproducible research contribution and a defensible innovation for real‑world applications in high‑dimensional sparse data processing.