Investigating an Unobserved Photonic Law (Paper II of III): Holographic Photon Continuity — A Topological Extension of Maxwell’s Equations

We propose a minimal, topologically driven extension of Maxwell’s equations designed to encode a hidden topological current, Jtop, arising from a higher-dimensional Higgs-photon coupling. In standard classical electromagnetism within four-dimensional spacetime, pho- tons are strictly massless and decoupled from the Higgs sector, governed by the homogenous and inhomogeneous equations dF = 0 and d∗F = 0. We hypothesize that our observable 4D spacetime is a boundary of a higher-dimensional bulk Y of real dimension 4 + k. While the additional current Jtop vanishes identically in a strict 4D limit, in the (4 + k)-dimensional bulk, a nontrivial morphism in the derived category between the photon and Higgs sheaves induces a conserved topological obstruction class. By mapping this sheaf-theoretic obstruction into differential cohomology, we perform a fiber integration over the compactified extra dimensions to yield a non-vanishing 3-form current on the 4D spacetime manifold. We extend the standard electromagnetic Lagrangian by a topological coupling term, A ∧Jtop. Crucially, by leveraging the Atiyah-Singer Index Theorem and anomaly inflow mechanisms, we prove that the resulting conserved topological charge is strictly quantized. Finally, we elevate this framework to a foundational paradigm shift by treating 4D space- time as a boundary manifold governed by the Atiyah-Patodi-Singer theorem. We introduce the “Holographic Photon Law,” proving that the structural perfection of the classical photon (dF = 0) is not a standalone axiom, but is dynamically protected by the spectral asymmetry of the bulk. This transforms classical electrodynamics into a holographic boundary theory where the photon acts as a measurable probe for hidden bulk geometry, providing a mi- croscopic derivation, proving the uniqueness of this interpretation, and identifying testable experimental observables.

@article{deceuster2026_81-investigating-an-,
  title = {Investigating an Unobserved Photonic Law (Paper II of III): Holographic Photon Continuity — A Topological Extension of Maxwell’s Equations},
  author = {De Ceuster, Peter},
  year = {2026},
  month = {02},
  institution = {SIG Labs},
  doi = {10.5281/zenodo.21557719},
  url = {https://peterdeceuster.uk/papers/81-investigating-an-unobserved-photonic-law-paper-ii-of},
  note = {Full text available at https://peterdeceuster.uk/articenter/work/light2.pdf}
}