Topological Photonic Sheaves and Bulk Gauge Continuity: A (4 +k)-Dimensional Geometric Extension of Maxwell Electrodynamics

This work contains some solutions for the category errors, degree mismatches, and differential geometry of the original U(1) toy model that was described in our earlier Maxwell work.

We formulate a geometric extension of abelian gauge theory by embedding four-dimensional spacetime X into a smooth, oriented fiber bundle pi : Y -> X with a compact, boundaryless k-dimensional Riemannian manifold K as fiber (dim_R Y = 4+k). To maintain categorical consistency, we model the bulk space Y as a complex analytic manifold with even fiber dimension k in 2N, letting E_P = O_Y(L) in Coh(Y) denote the coherent sheaf of holomorphic sections of a holomorphic line bundle L -> Y. The underlying smooth line bundle supports a U(1) gauge connection locally represented by potentials A with globally defined curvature F in Omega^2_cl(Y). Coupling E_P to a bulk Higgs sheaf H in Coh(Y), we formulate the interaction via the derived extension class eta in Ext_Y^{k+3}(pi* E_P (x)^L H, O_Y).

Through a postulated de Rham realization map c_eta, the class eta determines a closed differential form J in Omega^{k+3}_cl(Y) satisfying d_Y J = 0. Fiber integration along K induces a closed 3-form current on physical spacetime:
J_s := pi_* J = int_K J in Omega^3(X),    d_X J_s = pi_*(d_Y J) = 0.

The degree-consistent extended Maxwell equations are formulated as dF = 0 and d*F = J_s, where F in Omega^2(X) is the physical electromagnetic field strength. For any spatial 3-chain Sigma^3 subset X with boundary partial Sigma^3, this induces a localized Gauss-law topological charge:
Q_soul(Sigma^3) = int_{Sigma^3} J_s = int_{partial Sigma^3} *F.

We impose the explicit decoupling condition J_{k=0} = 0, guaranteeing that standard classical electrodynamics is recovered identically on uncompactified four-dimensional manifolds (k = 0).

Coupling J_s to the gauge potential via the effective action S_eff[A] = -(1/4) int_X F /\ *F + int_X A /\ J_s preserves 4D gauge invariance under A -> A + d lambda as a direct consequence of current closure d_X J_s = 0. Under compactification on an isotropic torus of characteristic scale Lambda, we formulate a phenomenological scaling hypothesis for multi-path quantum interference visibility:
1 - V approx (1/2)<(Delta phi_s)^2> ~ ||omega(eta)||^2 Lambda^{-2k}.

We structure the differential-cohomological lifting requirements with integral periods, formulate the anomaly-consistency conditions required of any complete bulk spectrum, and outline experimental detection bounds from precision optical interferometry.