We study whether the topological multiplicity of fermionic zero-mode crossings in the QGEFT surrogate is fundamentally tied to fluctuating gauge fields or to the deeper emergent geometry of the frozen graph background. Building on the previous Phase 4 result that established a stable Higgs-sensitive fermion mass response on a large N=1024 background, the present paper probes the chiral spectrum of the Wilson-Dirac operator through spectral flow and then subjects the gauge sector to explicit gradient-flow cooling while keeping the graph connectivity and spin connection fixed. Across the current benchmark, we find four linked results. First, the long-run N=1024 frozen background exhibits a stable topological skeleton of 111 spectral-flow crossings, corresponding to a net topological charge Q=5 in the standard convention. Second, thirty sweeps of gauge cooling reduce the bare gauge-action proxy from 56. 25 to 34. 57, indicating substantial smoothing of the gauge background. Third, despite nontrivial eigenvalue shifts as large as about 0. 70, the global topological content remains unchanged: both the original and cooled configurations preserve the same 111 crossings and the same net index, with a direct sign-preserving matching that yields a one-to-one correspondence for all crossings, median mass shift | m|0. 0365, and mean shift | m|0. 0867. Fourth, the large-volume fermion mass response remains intact on the cooled background: at y=2. 0 and m₀=1. 0, the Dirac gap remains D1. 16938, while the propagator gap remains ₑ₎2. 73193 with fit quality R^20. 98992. The narrow claim supported by these results is not that a continuum index theorem has already been derived on a random graph. It is that, in the present QGEFT surrogate, the fermionic topological sector is not an artifact of ultraviolet gauge noise. Instead, it is robust under substantial gauge smoothing and is best interpreted as a structurally stable property of the emergent discrete geometry. We refer to this regime as geometric dominance.
Yaniv Cohen (Thu,) studied this question.