Coherent Compression of Continuum Dirac Operators on LQG Graphs: A Conditional MTT Construction and the Doubling Boundary

Coupling chiral fermions to loop-quantum-gravity-type discretizations is often argued to produce fermion doubling in semiclassical regimes. We show that this conclusion depends on defining fermions by naive graph-local lattice operators. Within Modal Triplet Theory, fermions on graphs and spin foams arise instead by coherent compression of a continuum Dirac operator. The compression is implemented by heat-kernel smoothing followed by finite-element projection, yielding a discrete family of operators that converges in norm-resolvent sense under refinement. We treat the projector-variation (Berry) correction explicitly, establish uniform branch control for the associated matrix logarithm, and show that it does not spoil grading. Using resolvent convergence and spectral-projector stability, we prove infrared spectral stability and preservation of the chiral index in the coherent regime, excluding spurious low-energy mirror fermions. Refinement averaging then appears as a compatible corollary that further suppresses ultraviolet artifacts once operator-level control is established. These results provide a mathematically controlled route to chiral fermions in loop-quantum-gravity-type descriptions derived from Modal Triplet Theory.