Doubling and Jamming: Universal Chern Hamiltonians on Triangulated Surfaces with Boundary

Higson and Prodan introduced a universal tight-binding Chern-insulator model on triangulated closed oriented surfaces, defined from the simplicial boundary operator $B$ and a Hermitian Poincaré-duality operator $S$ obtained from cap product data [1, 2]. The closed-surface Hamiltonians $H_\pm=(B+B^\dagger)\pm S$ are local, self-adjoint, and carry Chern numbers $\pm1$ in the refinement limit. For triangulated surfaces with boundary, the closed-case cap-product construction of $S$ is not directly available. We treat the boundary case by a canonical doubling-and-jamming procedure: glue an orientation-reversed copy along $\partial K$ to form the closed double $D(K)$, build $H_\pm(D(K))$, and add a large on-site potential $MP_{\mathrm{copy}}$ on the copy degrees of freedom. The unconditional part of the paper is finite-dimensional and purely linear-algebraic. We prove explicit simplex-count identities for doubling, norm-resolvent convergence with rate $O(1/M)$, a first-order Schur-complement effective operator, low-energy eigenvalue localization, and eigenvector leakage bounds. We also correct the spectral-projection statement from the earlier draft: for an isolated spectral cluster of the original-side block $A$, the full spectral projections satisfy $\chi_I(H_{\pm,M})\to\chi_I(A)\oplus0$ in operator norm at rate $O(1/M)$, with eventual rank stability. For bulk–interface physics we separate what is proved here from what belongs to existing infinite-volume frameworks. Under an explicit refinement/admissibility hypothesis, doubling-and-jamming realizes a topological/trivial class-A interface whose boundary class is the image of the bulk class of the doubled universal model under the standard bulk–interface boundary map. Consequently the interface index equals the Chern jump $\pm1$ and changes sign under $H_+\leftrightarrow H_-$. This is a rigorous reduction to established bulk–interface theorems rather than a new stand-alone bulk–boundary theorem for arbitrary triangulation refinements. Numerically, we study three examples with one, two, and three boundary components, and observe bulk gaps on the doubles, in-gap states created by jamming, boundary localization, chirality reversal under $H_+\leftrightarrow H_-$, and the predicted $1/M$ and $1/M^2$ decoupling scalings. AI-use disclosure from the manuscript: Declaration of generative AI and AI-assisted technologies in the manuscript preparation process During the preparation of this work the author used generative AI tools to accelerate drafting and revision. All mathematical claims, proofs, numerical interpretations, and bibliographic information were subsequently reviewed and edited by the author, who takes full responsibility for the content of the manuscript. Source version: This deposit preserves the manuscript dated 6 March 2026 from the author's GitHub website. The related ResearchGate record also lists revisions dated 27 April 2026. Source commit: 9ffb0271ef824d08373f26ec3b8543503108f727. Repository snapshot: 2026-10-04T01:06:07Z.