Reset Dynamics, Extra 4: A Cohomological Framework for Structural Reconfiguration

This paper establishes a cohomological framework for structural reset dynamics, building on the residual accumulation mechanisms of Papers 16–18 and the resource-deficit obstruction of Extra 2. We formalize structural incompatibility as a residual obstruction class R_t ∈ H¹(X, O_X) — a non-trivial Čech 1-cocycle indicating that local operational rules are consistent on every patch but cannot be glued into a global section. When the residual norm exceeds a critical threshold under resource deficit (Γ < n), the obstruction condenses into a structured zero Z_Σ = (0, Σ), where Σ ∈ H¹(X, O_X) encodes the topological invariant of the incompatibility. A modulo-bounded reset functor R_{Z_Σ} performs constrained structural reconfiguration: F_new ∼ F ⊘ Z_Σ (mod B), where ⊘ is the projection-based division from Paper 18 and B = (X_parent, d, C_max) is a boundary modulus that limits reconfiguration complexity. An evolutionary kernel K ⊆ Struct_X, invariant under reset, preserves meta-rules across structural collapse. We prove three principles (Reset Existence, Kernel Preservation, Complexity Boundedness) supported by a worked example on the circle S¹. This framework provides the mathematical closure for Extra 5's computational toy model. This is a working paper, not peer-reviewed. Version 1.1 — Final (abelian sheaf, residual norm construction, principle downgrade).