Toward an Unobserved Photonic Law of Nature: A Gauge-Invariant Maxwell Extension from Higher-Dimensional Soul Currents

 We formulate and prove a mathematically precise, gauge-invariant extension of Maxwell’s
 equations in four dimensions that incorporates an invisible soul current arising from higher
dimensional structure. The extension takes the differential-form form dF = 0, d(⋆F −Js) = 0,
 where Js is a closed 3-form descending from a bulk coupling via pushforward. Our relative
 short work crosses mathematics with physics and avoids unnecessary expansions: we place the
 construction on a product bulk Y = X ×Σ (with compact Σ), use cohomological obstruction
 classes to pin the phenomenology of fringe visibility, establish well-posedness via deterministic
 hypocoercivity and stochastic regularity structures, and average over compactification moduli.
 At least one valid proof had to be provided to sustain interest in a Maxwell extension; here
 we furnish such a proof, together with falsifiable predictions. We emphasize from the outset
 that it is entirely possible to disprove this extension by experimental non-detection (e.g.
 in cavity interferometry) or by contradiction with precision constraints; such a disproof
 would still be productive for photonics. Nevertheless, following the program laid out in the
 five foundational works—Photon Soul Continuity: An Unobserved Extension of Maxwell’s
 Equations , A Unified Framework for Photon-Soul Continuity, On Higher
Dimensional Soul Perturbations via Higgs–Photon Couplings, The Photon Soul Theorem: A
 Cohomological Approach to Interference Visibility, and Photon Soul Resonance—we argue
 that a consistent extension is called for and provide a mathematically complete backbone
 supporting it