OG-II-04 — T — Overlap Fermions and the Dirac Continuum Limit on an Aperiodic D₆/H₃ Carrier - Uniform Wilson coercivity, quantitative continuum convergence, and selected Weyl bands

We construct a free Wilson–overlap fermion regulator on a specified aperiodic D₆/H₃ cut-and-project carrier. The Pauli operator and Wilson Laplacian use the full Voronoi geometry and its volume-weighted Hilbert space. Exact separation and covering bounds give a distributional consistency estimate in the homogeneous H⁻¹ norm. A continuum Dirac lift then proves uniform Wilson coercivity with constant r²/(8+25r)², and hence a strong Wilson-kernel gap for 0<μ≤r/[2(8+25r)²]. The gapped overlap operator obeys the Ginsparg–Wilson identities and is exponentially local. A comparison of noncommuting polar factors yields O(h) generalized norm-resolvent convergence to one four-component continuum Dirac operator, uniformly over the permitted carrier phases. Reconstructed states with bounded physical energy are locally strongly precompact, excluding additional channels whose norm remains in a fixed bounded region while their local weak limit vanishes. For the specified β-Hermitianized overlap Hamiltonian, a conserved overlap charge defines invariant interior-band sectors converging to the two continuum Weyl summands. Selecting one sector is additional model data; the complete regulator remains paired. The proof uses real-space geometry rather than periodic momentum counting and extends to uniformly Delone Voronoi families satisfying the same quantitative hypotheses. The results concern a free regulator and its continuum content; an interacting chiral quantum theory requires additional constructions.