C2 algebra and gravitational response: Algebraic structure, continuum obstructions, and observable tests

We develop a bounded route from the algebraic C2 construction to tests of a gravitational continuum interpretation. The starting point is a surviving symmetry sector and a conditional quadratic pullback K = J* H J from strain to an active module. We prove what sector survival does and does not imply for active invariance, derive the symmetry conditions under which the pullback reduces to three cubic response coefficients, and obtain the directional splitting of its transverse-traceless restriction. The Saint-Venant incompatibility operator supplies an exact connection to the three-dimensional linearized Einstein tensor, but also an obstruction: strain generated solely by compatible displacements has no transverse-traceless component at nonzero wavevector. A four-dimensional treatment separates the constraints and radiative modes, fixes the sourced Newtonian and tidal responses, and shows why a spatial cost cannot determine the dynamics. We then distinguish local stress-tensor improvement freedom from the universal massless spin-2 pole and derive scalar-pole, lensing, and binary-radiation diagnostics, with their coupling and source assumptions explicit. These results assemble a concrete matching problem for C2. They do not derive an Einstein limit, a value of Newton's constant, a native matter source, or a compact-body scalar charge. Standard continuum identities are identified as such; the contribution is the explicit chain of correspondences, obstructions, and discriminating tests that a C2 realization must satisfy. Scope and status. A 24-page graduate-level working preprint with 11 main sections, four mathematical appendices and 15 references. This synthesis has not undergone independent review as a complete manuscript. It distinguishes conditional results and established continuum benchmarks from the open native C2 construction. PDF-only release. This record contains only paper.pdf. The source notes, calculation scripts, replay records and supporting archive discussed in the manuscript are not included in this deposit and are retained separately by the author.