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. A completed native calculation now certifies exact rank 24 for the specified 39 × 768 original-active projection. Both invariance residual families vanish for all 48 supplied actions, whose induced matrices satisfy the complete multiplication table. We give the certificate argument and an exact reduction of the remaining full-active problem to discarded components and complete correction relations. The native full kernel and quotient remain unevaluated; practical feasibility is not established. From a conditional strain pullback K = J* H J, we derive cubic response and directional tensor-cost tests. Saint-Venant incompatibility connects strain to the spatial linearized Einstein tensor, while compatible displacement strain has no transverse-traceless component at nonzero wavevector. A four-dimensional treatment separates constraints from radiation, fixes the benchmark Newtonian and tidal responses, and shows why a static cost cannot determine dynamics. We distinguish local stress-tensor improvements from the universal tensor pole and derive scalar-pole, lensing and binary-radiation diagnostics. The result is a certified projected algebraic subsystem and an explicit conditional matching programme. It is not full C2 closure or a derivation of an Einstein limit, Newton's constant, a native matter source or compact-body scalar charges. Standard continuum results are attributed to their literature; project certificates, conditional proofs and open physical correspondences are kept distinct. Version 1.1. Updates the algebraic evidence with the reviewed native rank-24 projection, all 48 projected-invariance tests and all 2,304 induced multiplication identities. Adds the full-lift/kernel reduction, the complete stable-relation theorem and a conditional resource appendix. The full native kernel and correction quotient remain open; neither practical feasibility nor infeasibility is established. Scope and status. A 31-page graduate-level working preprint with 11 main sections, five mathematical appendices and 20 references. The revised synthesis has not undergone independent review as a complete manuscript. The native result reviews use separate arithmetic implementations, but producer and reviewer are Codex; no independent-author or third-party audit is claimed. This manuscript update performs no new native computation. 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.