<b>Paper 2: Mathematical Formalism of the D1 Field Theory</b>

This paper presents a rigorous mathematical formalisation of the D1 Field Theory, a novel framework positing a single, fundamental scalar field, ΦD1​, as the origin of spacetime, gravity, and all physical reality. Building upon previous conceptual work, this document establishes the action principle governing the ΦD1​ field's dynamics, detailing its kinetic, self-interaction (quartic potential), and matter/energy interaction terms, which collectively yield a non-linear Klein-Gordon-like equation of motion. A cornerstone of this formalism is the explicit mathematical description of emergent spacetime, introducing a non-minimal coupling term that demonstrates how this interaction leads to an effective gravitational constant inversely proportional to ΦD12​. This mechanism fundamentally links spacetime geometry and gravitational strength to the D1 field's amplitude, outlining how specific amplitude thresholds of ΦD1​ dictate the very existence and effective dimensionality of spacetime, including a "no-space" regime. Applying this framework to cosmology, the paper derives the D1 field equation in an FLRW background and establishes the coupled effective Friedmann equations, showing how the D1 field's self-interaction potential naturally drives an early universe inflationary phase. The theory proposes a unique multi-stage cosmological evolution, including an initial D1-driven inflation, a hot Big Bang, subsequent deceleration, and a secondary expansion phase (driven by matter-antimatter annihilation and D2-4 dynamics) that aims to explain the observed late-time accelerated expansion. Finally, it initiates a preliminary discussion on the quantum aspects of the D1 Field Theory, proposing a promising pathway to resolve the long-standing problem of quantum gravity by quantizing the fundamental ΦD1​ field rather than the emergent spacetime metric. This paper sets the mathematical foundation for future investigations into the D1 field's profound implications for the universe's fundamental nature, with ongoing work focused on quantitatively aligning its cosmological predictions with observational data.