Investigations into an unobserved photonic law of nature: Torsional Boundary States in Warped Compactifications: An Effective APS-Type Index Formula with Nieh-Yan Contributions (Paper IV)

This is the first of a new set of three, where we continue to expand upon our initial work.
Hercule Poirot had his suspects, we have our hidden variables. Poirot, however, had the
distinct advantage of a closed room; our unobserved laws operate in arbitrary dimension
without a smoking gun in sight. Following the Topological Emergence Principle, we de
rive an effective APS-type index formula including torsion-dependent contributions for a
classical gauge field confined to the boundary of a warped extra dimension. While previ
ous work in this program established the general Riemann-Cartan holographic framework,
this manuscript moves to a specific, realistic compactification geometry: a slice of AdS5
(a Randall-Sundrum throat) permeated by a background torsional flux. By evaluating the
Nieh-Yan invariant explicitly within this warped background, we derive the corresponding
torsion-dependent contribution to the bulk spectral asymmetry within the effective frame
work adopted here. Crucially, we formulate and analyze the generalized equations governing
the gauge-field zero mode coupled to this torsional background. Within the effective model,
we show that the torsion-dependent contribution can compensate the effective anomaly
inflow contribution, leading to the massless gauge zero-mode localizing on the infrared (IR)
boundary brane. This provides an effective geometric mechanism for the Holographic Pho
ton Law: the four-dimensional photon remains structurally intact (dF = 0) because its
wavefunction localization is topologically protected by the Nieh-Yan invariant of the bulk
geometry