The Toroidal Impulse Mechanism in the Sigma–Schmidt Duality: How a 1D Extractive Axis Generates a 2D Expansion Surface

This paper introduces the Impulse–Toroid Mechanism that completes the foundational geometric structure of the Sigma–Schmidt Duality (SSD).The model proposes that the universe originates from a directed pre-spacetime 1-dimensional extractive axis SeS_eSe which, once crossing a critical energetic density, undergoes a curvature transition forming a 2-dimensional toroidal shock-front Sa(t)S_a(t)Sa(t).This expanding toroid acts as the generative surface from which the 3-dimensional σ-field emerges, defined by σ(t)=Se(t)Sa(t).σ(t) = S_e(t) S_a(t).σ(t)=Se(t)Sa(t). The work shows that: • the first curvature of SeS_eSe produces the toroidal heat-shell,• temperature emerges only after the 1D–2D interaction (“friction”),• energy is distributed anisotropically through σ-nodes,• stars act as local Sa-dominant expansions,• black holes act as Se-dominant contractions. This paper provides the missing geometric mechanism linking the Dual-Singularity Framework to observable cosmological structure, and serves as the precursor to the Sigma–Friedmann Equation v2.0.