Chern-Simons Boundary Completion of Degenerate Disformal Scalar-Tensor Theories

In degenerate disformal scalar-tensor theories, the degeneracy surface D: epsilon = 0 constitutes a hard boundary for classical metric evolution: hyperbolicity, finiteness of stress-energy, perturbative control, and curvature regularity all fail simultaneously (the No-Go Theorem of the companion paper, DOI 10. 5281/zenodo. 18866200). We propose a boundary Chern-Simons completion of the transition at D. In the epsilon -> 0+ limit, scalar freezing and destruction of the bulk metric leave a unique finite variational sector: the induced three-dimensional Einstein-Hilbert action on the transition hypersurface SigmaK with positive cosmological constant. By the Achucarro-Townsend-Witten correspondence, this is equivalent to a Chern-Simons theory with de Sitter gauge group SO+ (3, 1), working on the spin cover SL (2, C). Using the Atiyah-Bott-Goldman symplectic structure on the boundary character variety, we show that Dehn surgery along a framed knot K defines an exact correspondence whose associated filtration shift — the topological displacement Deltaₜop — is finite and determined entirely by the change in the real part of the complex Chern-Simons invariant. This finiteness guarantees that persistent homological data associated with defect lines on SigmaK survive the transition: right-infinite bars in the persistence barcode cannot be destroyed by a finite interleaving. The construction is purely (2+1) -dimensional and uses no four-dimensional bulk metric, consistent with the No-Go Theorem. The paper establishes consistency and finiteness of such a completion on a fixed framed-knot surgery sector; the selection of the surgery sector from four-dimensional dynamics is deferred to future work. Companion paper to "Degenerate Disformal Scalar-Tensor Gravity with Topological Interior Amplification" (DOI 10. 5281/zenodo. 18866200). This paper constructs the boundary Chern-Simons completion whose necessity was established by the No-Go Theorem of the companion paper. Developed through collaborative analysis with multiple AI systems within the TKWC Research Initiative.