The Spacetime–Electromagnetic Framework: Mathematical Structures and Obstructions in Split-Signature Clifford Geometry

Can the mathematical structure of physics grow from a single algebraic seed? What if spacetime, electromagnetism, quantum structures, and matter representations were not independent ingredients of physical theory, but interconnected manifestations of one underlying algebra? The Spacetime–Electromagnetic Framework investigates this possibility through the real split-signature Clifford algebra [\mathrm{Cl}(3,3)\cong M_8(\mathbb{R}),] and its higher-dimensional hierarchy (\mathrm{Cl}(3n,3n)). The central ambition is generative rather than reconstructive: instead of imposing a universal dynamical action at the outset and arranging its ingredients to reproduce established physics, the programme begins with an algebraic seed and investigates the mathematical structures that can develop from it. Lorentz geometry, electromagnetic structures, quantum complex structures, spinorial representations, and internal gauge symmetries are explored as interconnected branches of this construction. The essential question is not simply whether the same algebra can represent these structures, but whether their mathematical relationships and mutual compatibility progressively restrict the possible physical theories. What emerges, and what is obstructed? The manuscript develops several constructive results, including: An algebraically distinguished complex structure and a real Clifford formulation related to conventional complex Dirac theory. A correspondence between spacetime-displacement and electromagnetic-field interpretations of vectors in (\mathbb{R}^{3,3}). Projector constructions supporting candidate chiral matter representations, including a Pati–Salam family with a right-handed neutrino. A hierarchy of higher-dimensional Clifford structures and associated symmetry reductions. Equally important are the mathematical obstructions encountered along these branches. The manuscript investigates restrictions on gauge-boson interpretations, ghost modes, low-rung gauge-algebra embeddings, induced connections, and representation selection. These obstructions are not discarded as unsuccessful attempts. They are investigated as possible selection mechanisms that may constrain the subsequent development of the framework. From a common seed toward a unique physical structure The long-term objective is to determine whether the compatibility of the different branches can progressively reduce the space of admissible physical structures until a unique, physically viable dynamical theory remains, up to appropriate equivalences. No universal parent action is imposed at the outset. The possibility of deriving or uniquely constraining such an action is itself an open problem. This distinction is fundamental: the programme does not assume that algebraic compatibility constitutes physical emergence, nor that mathematical unity necessarily guarantees a unique physical theory. The selection of a physical clock, chiral structures, vacuum dynamics, coupling parameters, and gravitational equations remains subject to further investigation. A research programme, not a completed unification This 412-page preprint documents the mathematical architecture, constructive results, explicit assumptions, and obstructions of an ongoing programme in mathematical physics. It does not claim experimental confirmation or a completed derivation of the Standard Model and general relativity. Rather, it establishes a foundation for investigating whether a common algebraic origin can impose sufficiently strong constraints to determine physical structure and dynamics. The work invites independent mathematical scrutiny, critical assessment, and further development. Substantial AI assistance was used during mathematical exploration, verification, manuscript development, and critical review, under the author's direction. The methodology is documented in Chapter 15, and scientific responsibility remains with the author. Author: Driss Bencheikh — Independent Researcher Version: 1.0.0 — Preprint, not peer reviewed DOI: https://doi.org/10.5281/zenodo.23900465 Source and verification materials:https://github.com/driss-bencheikh/spacetime-electromagnetic-framework