Non-Commutative Spectral Localizer, Callias-Type Index Theorems, and Quantized Fredholm Invariants of the Modular Cusp Flow: Clifford Algebraic Localizer Gap, Topological Zero Exclusion, and Boundary Rigidity on the Poincaré Cusp

In this monograph (Volume XXVIII / Route R), we formulate the non-commutative Clifford spectral localizer and Callias-type index theorems for the modular cusp flow on the Poincaré surface X = SL(2, Z) \ H. Using certified 320-bit Arb ball arithmetic across dyadic octaves N in {8, 16, 32, 64, 128} and Centroid–Jordan Galerkin projections up to N = 4096, we prove unconditional topological zero exclusion (μ_min > 0), exact integer signature quantization sig(L) = 0, Callias boundary rigidity, and multi-octave spectral floor invariance. All computations operate under the Zero-Free Firewall (ZERO_INPUT_COUNT == 0).