Investigating an Unobserved Photonic Law (Paper III of III): Torsional Anomaly Inflow: Riemann-Cartan Holography and the Protected Photon

We extend the Holographic Photon Continuity framework into Riemann-Cartan geometry
(U4) to analyze the dynamical interaction between extra-dimensional topological currents
and gravitational torsion. Standard general relativity assumes a torsion-free Levi-Civita
connection; however, coupling the bulk Higgs-photon sheaf to a non-symmetric affine con
nection fundamentally alters the spectral geometry of the system. We demonstrate that
mapping the holographic boundary coupling to a Riemann-Cartan background naturally
leads to an APS-type index formulation including torsion-dependent corrections, allowing
the Nieh-Yan invariant together with torsion-dependent characteristic classes to contribute
to the bulk obstruction class.
Consequently, the Holographic Photon Law (Lholo), which protects the integrity of the
4D Bianchi identity (dF = 0), becomes strictly torsion-dependent. This implies that macro
scopic gravitational torsion acts as a direct catalyst for anomaly inflow, dynamically shifting
the quantized index charge (Qindex) flowing from the extra dimensions. This formulation
predicts profound cosmological and astrophysical signatures: photons traversing torsion
rich environments—such as primordial defect backgrounds or the vicinities of ultra-dense
fermionic spin matter—must exhibit anomalous phase shifts and topological vacuum birefrin
gence. Ultimately, this framework establishes a mathematically consistent, testable bridge
between Einstein-Cartan gravity, derived category topology, and observable quantum op
tics, providing a microscopic derivation, proving interpretation uniqueness, and identifying
testable observables.