Degenerate Disformal Scalar-Tensor Gravity with Topological Interior Amplification: Ghost-Freedom, Interior Cosmological Constant, and a No-Go Theorem for Classical Completion

We present the action S_STKWC for a degenerate disformal Brans-Dicke scalar-tensor theory within the TKWC (Topological Knot-Web Cosmology) framework. The theory features non-minimal coupling f(φ)R, canonical scalar kinetics, and disformal matter coupling to the Jordan-frame metric g̃ = A g_μν + B ∂_μφ ∂_νφ. The paper establishes ghost-freedom via Dirac-Bergmann Hamiltonian analysis (N_dof: 3 → 2 at the degeneracy surface), spectral stability via Sturm-Liouville/Poincaré-Friedrichs analysis, GW170817 compliance (α_T = 0 from Bellini-Sawicki parametrization), and an Israel junction derivation of the interior de Sitter geometry with Λ = k̃²/3 in the classical thin-wall regime. The central result is a No-Go Theorem (Theorem C.2): no classical metric description can propagate matter through the degeneracy surface D: ε = 0. Four independent mechanisms fail simultaneously: (H) Loss of Lorentz hyperbolicity in the matter sector (c_matter → ∞). (F) The effective stress-energy T^(eff) ~ ε^(-1/2) diverges but does not generate a distributional source on the boundary (thin-wall integral vanishes). (P) The EFT strong-coupling scale Λ_EFT ~ M_Pl ε^(1/2) → 0, destroying perturbative control. (C) The Einstein-frame curvature R_μν ~ ε^(-1/2) diverges, invalidating standard Israel junction conditions in the bulk. This establishes a hard boundary for classical general relativity at D: the continuous Riemannian description must yield to boundary data (induced metric, extrinsic curvature, topological surgery) that determine the post-transition geometry without bulk extrapolation. The theorem provides a precise criterion for any proposed non-perturbative completion: it must operate natively in (2+1) dimensions on the degeneracy surface without relying on bulk matter propagation. Volume IV in the STKWC series. Developed within the TKWC Research Initiative through collaborative analysis with multiple AI systems (Claude Opus, ChatGPT, Gemini, DeepSeek, Grok). Previous version (v3) published as DOI 10.5281/zenodo.18812729. Supplementary Python code for numerical convergence test included.