A Fiber-Integration Extension of Maxwell's Equations on the Boundary of a Holographic Bulk inspired by Hertog

Standard quantum field theory typically models the electromagnetic vacuum on a fixed
four-manifold with no additional global topological sectors. Hertog and Hawking in recent
years made efforts to develop alternative theory. We consider a relaxation motivated by
holographic/bulk-boundary ideas: observed 4D spacetime X is the base of a fibration π :
Y → X with compact k-dimensional fiber Σ. Instead of invoking a derived pushforward
directly at the level of differential forms, we define a concrete transgression current on X
by integration along the fiber : a closed bulk (3 + k)-form eJ induces a boundary 3-form
Js := π!( eJ) ∈ Ω3(X ),
which is automatically conserved (dJs = 0) under mild geometric hypotheses. The effective
Maxwell system becomes
dF = 0, d ⋆ F = Jmatter + Js,
equivalently d(⋆F − Js) = Jmatter. We give a gauge-invariant action principle, clarify when
a Proca mass term is excluded, and outline (as clearly labelled speculation) how such a
cohomological source could be constrained experimentally via phase-sensitive cavity or in-
terferometric measurements