Residual Graph Theory: Source-Local Boundaries, Two Residual Algebras, and Residual Cohomology

We develop a residual theory for finite typed undirected, directed, and mixed graphs. For a visible vertex set, arrows whose sources remain visible and whose targets become hidden are replaced by anonymous outward residual arrows. Every finite typed mixed multigraph admits a canonical complete residual presentation with attachment and symmetric-pair data, and reconstruction is inverse to completion up to natural isomorphism. For undirected graphs over \(\mathbb F_2\), residual boundaries form the cut space, orthogonal to the cycle space. This yields coding, metric, spectral, enumerative, planar-duality, and Tutte-polynomial consequences. Directed source-cut transforms determine symmetric weights and vertex divergences modulo divergence-free skew-symmetric circulations. Two algebraic levels are distinguished. The residual graph algebra operates between graph objects through weighted addition, signed subtraction, interface composition, and factorization. The residual algebra of a fixed graph is instead a quotient of the path semiring of its propagation quiver. Weighted, fractional, and negative coefficients arise intrinsically from these operations. After quotienting by relabeling, graph addition forms a commutative hypergroup and composition forms a generally noncommutative hypersemigroup. Residual cohomology is defined on categories of graph comparisons. Residual assignments are one-cochains; their curvature measures local composition defects, while nonzero classes in \(H^1_{\mathrm{res}}\) obstruct globally compatible graphwise potentials. Relative and Mayer–Vietoris sequences describe extension and gluing, cup products provide a graded residual algebra, and endpoint realization relates graph-internal and graph-object obstruction classes. Residual Hodge theory gives unique minimum-energy representatives, while overlap-defect filtrations yield persistent residual cohomology. These constructions provide algebraic and computational invariants for residual structure, propagation, ambiguity, and obstruction. Keywords Residual graph theory; source boundary; mixed graphs; graph algebra; path semiring; weighted graphs; hyperrings; residual cohomology; obstruction classes; residual curvature; Hodge theory; persistent cohomology.