Fundamental Barriers in Geometric Primality Testing: A Synthesis of Galois Theory, Transcendence, and Computational Limits

We prove that any deterministic primality test based on finite geometric constructions or bounded modular sampling must fail for density-one sets of integers. Our main theorems establish: (1) an Ω( √ n) lower bound on numerical instability arising from the transcendence of π; (2) a Galois-theoretic barrier showing that tests examining O(log k n) samples cannot decide primality for all sufficiently large n; (3) constructibility barriers via Gauss-Wantzel, making exact geometric realizations impossible for almost all primes; and (4) model-theoretic impossibility results via o-minimality. These results unify geometric and modular approaches under a single framework, revealing that primality's continuous nature resists complete discrete characterization. The work synthesizes techniques from transcendence theory, Galois theory, model theory, and computational complexity.