This paper introduces the Charlton Basis (denoted “@”) and Charlton Operator (denoted “@ () ”), which when applied to δ produces the Lorentzian metric @ (δ), ” establishing the framework of Charlton Geometry as a new foundation for spacetime and emergent physics. The approach provides a basis shift that makes much of Lorentzian mathematics more intuitive and accessible to students at all levels, while remaining rigorous enough for advanced research. Within this framework, this first paper establishes the axiomatic foundation and demonstrates a full derivation of first post-Newtonian (1PN) recovery, with falsifiable beyond-1PN corrections for orbital predictions and a proof that Maxwell’s equations emerge naturally from @, @ (). In Minkowski spacetime, Maxwell’s equations are treated as axioms, written in Lorentz-covariant form. In @, @ (δ), Maxwell’s equations arise as identities of the reflected calculus; the current J is not inserted but generated by geometry. Maxwell is no longer an axiom, but an emergent fact. Even if specific predictions are challenged or refined, the enduring breakthrough is the mathematical structure itself, which reframes how relativity, electromagnetism, and geometry can be taught, learned, and extended. As the first publication in the Charlton Geometry Series, it invites both researchers and students to explore a new language for time, space, and physical law. Licensed under CC BY 4. 0 for open dissemination to researchers, educators, and learners in general relativity, mathematical physics, quantum foundations, and geometric field theory.
J. P. Charlton Jr. (Fri,) studied this question.