The Sigma–Schmidt Extension: A Geometric and Dualistic Completion of the Standard Model Version 2.0

This paper introduces the Sigma–Schmidt Extension as a geometric and dualistic completion of the Standard Model of particle physics. Without altering the core Lagrangian, the model embeds all known interactions within a higher-dimensional framework defined by a dimensionless dual field, $\sigma(t) = S_e \times S_a$, coupling an extractive axis ($S_e$) and an expansive surface ($S_a$). The framework resolves long-standing conceptual gaps such as the origin of vacuum energy, dimensional emergence, and cosmological tensions (e.g., $H_0$ and $\sigma_8$) through energetic duality and geometric constraints. By defining space as a dynamic interplay of extraction and expansion, the Sigma–Schmidt Extension preserves all Standard Model predictions while offering deeper explanatory power. The model is mathematically consistent, testable with cosmological data (DESI, Pantheon+), and compatible with quantum field theory and general relativity. It represents an adaptable geometric layer that connects the microscopic and cosmic domains through the same dualistic structure. Previous works outlining the geometric foundation of the Sigma–Schmidt Duality are archived on Zenodo: Sigma–Schmidt Duality Series I: Local Dimensional Genesis — https://zenodo.org/records/17469826 Sigma–Schmidt Duality Series II: Farwell–Type Geometric Constraints — https://zenodo.org/records/17497957 This publication can be read independently or as a continuation of those earlier studies.