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

This paper studies continuum-derived fermion operators on graph-labelled spaces in Modal Triplet Theory (MTT) and loop quantum gravity (LQG). For a self-adjoint continuum Dirac operator with compact resolvent, exact spectral compression preserves, without spectral pollution, every eigenvalue in the selected finite window. For a graded Dirac operator, a symmetric spectral window also preserves the chiral index. Any unitary presentation of that finite subspace on graph data inherits these facts. The construction avoids lattice doubling because it is continuum-derived and generally nonlocal; it does not yet construct a local graph Dirac operator or derive the continuum fermion source from MTT. Separately, a smooth constant-rank family of projectors carries the exact Kato–Berry connection, but such a family requires an explicit gapped band bundle. The paper therefore gives a rigorous no-spurious-spectrum benchmark and a finite source contract, not a completed chiral-fermion quantization in LQG.