Green–Poisson Reconstruction and Hypersurface Recursion for Dirac Laplacians

We study reconstruction of sections of Hermitian Dirac bundles through finite chains of hypersurfaces. For the invertible full Dirichlet realization of the Dirac Laplacian, the Green–Poisson formula separates boundary data from an interior source. We establish a composition formula that retains the sources at every level and specifies the Banach spaces and assembly maps required for its use. The induced Clifford connection, the multiplicity of restricted modules, and the interface conditions are treated explicitly. A transmission formula explains the difference between partitioning one global Green integral and using independent Dirichlet inverses on the pieces. A cylinder and a product neighborhood of a complex hypersurface provide exact two-step examples with nonzero intermediate sources. We also give a global source-corrected lift on a normal disk bundle and explain the obstruction to choosing a real diameter in a nontrivial normal bundle. The general composition theorem is conditional on a realized chain; the product example has bounded transfers on its specified finite-dimensional spaces.