Completing the Algebraic Framework: Non-Analytic Functions and Measure Theory in Rational Worlds

We complete the algebraic reformulation of mathematical analysis by resolving two fundamental limitations: treatment of non-analytic smooth functions (like e^(-1/x²)) and construction of Lebesgue measure theory without metric completeness. Through asymptotic algebras with infinitesimal scales, we incorporate flat functions that elude power series representations while maintaining algebraic computability. Definable Dedekind cuts over ℚ_alg provide rigorous foundations for Lebesgue measure, enabling construction of algebraic L^p spaces with functional completions via projective limits. The framework preserves all classical analysis results statable in definable structures while eliminating dependence on limits, epsilon-delta arguments, and real number completeness. Applications include exact treatment of heat equations with non-analytic initial data and quantum mechanics with singular potentials. This work unifies the Rational Worlds framework with o-minimal structure theory, providing complete limit-free foundations for analysis. Includes algorithmic implementations maintaining computational exactness throughout.