A Certified Robustness Threshold for Constructible Cubic Angle Approximation

Exact angle trisection by straightedge and compass is impossible in general. This research establishes exact deterministic robustness thresholds for the publicly documented Baker D4 approximate angle-trisection construction. Necessary-and-sufficient chord-error and multistage measurement-error bounds are derived, together with convexity-based certification of a unique minimum admissible input angle. The analysis extends to random-effects calibration, statistical tolerance certification, readout-transfer safeguards, and continuous-angle Lipschitz envelopes. Four frozen numerical benchmark versions investigate Gaussian, heavy-tailed, and counterexample error distributions, demonstrating both the applicability and limitations of statistical certification. The archival package contains the complete manuscript, LaTeX source, Python scripts, generated tables, numerical results, and a SHA-256 manifest. The construction is approximate rather than an exact solution to classical angle trisection. Physical instrument calibration and independent full Monte Carlo replication remain outstanding. Copyright and licensing Copyright © 2026 C. Wayne Baker. Manuscript, PDF and explanatory documentation: Creative Commons Attribution 4.0 International (CC BY 4.0). Python source code: MIT License. Numerical datasets and generated tables: CC BY 4.0. The licenses apply to the respective materials as identified above. The author retains copyright. Reproducibility The release ZIP contains the verified manuscript, source code, benchmark outputs and SHA-256 manifest. The archival files are preserved unchanged.