The Irrational Geometry Engine – Definitive Scientific Monograph v26.0: Dimensional Shadows, Bekenstein Relaxation, and the Geometric Origin of Physical Laws

This is the complete scientific monograph (Version 26.0, March 13, 2026) presenting the Irrational Geometry Engine (IGE) framework. IGE posits that irrational numbers are generative causal seeds whose non-repeating expansions project higher-dimensional geometry into observable 3D forms, producing physical laws, molecular properties, and the arrow of time via Bekenstein-like relaxation. Core contributions include: • Dimensional Shadow Theory and Closure Impossibility Theorem • Five independent geometric derivations of the fine-structure constant α ≈ 1/137.036 (notably 4π³ + π² + π) • Geometric Lagrangian with mandatory radial expansion (ℛ̈ > 0) • Bekenstein Relaxation applied to chemistry and biology (conductivity, magnetism, catalysis as stress-response modes) • Three falsifiable laboratory predictions (ribosome pressure threshold 0.3 GPa, Au cluster magic numbers, 10⁻¹² chiral bias) Accompanied by open-source Python code repository: https://codeberg.org/ishrikantbhosale/IRRATIONAL-GEOMETRY-ENGINE Related Zenodo records: • ISL Paper 1: https://zenodo.org/records/18766730 • ISL Paper 2: https://zenodo.org/records/18733645 • Screaming Numbers essay: https://zenodo.org/records/18940193