An Effective-Response Framework for Galactic Rotation Curves: Frozen Kernel, Finite-Radius Response, and an Exponential-Disk Far-Field Result

This work presents an effective-response framework for galactic rotation curves, formulated as a two-dimensional static finite-radius response theory. The framework is based on a frozen momentum-space kernel and its corresponding coordinate-space Green function, together with an explicitly signed finite-radius response to a prescribed axisymmetric baryonic source. For an exponential-disk baryonic distribution, the resulting dimensionless response is analyzed in detail. A positivity theorem is established for the exponential-disk response, and a fixed-parameter far-field result is obtained under the stated asymptotic conditions. The analysis also demonstrates the non-commutativity of the limits of vanishing interaction mass scale and infinite radius. A local two-field effective-field-theory realization is constructed that reproduces the frozen static kernel. Its basic dynamical properties, including the absence of ghosts, tachyonic instabilities, and gradient instabilities within the stated effective framework, are examined. The phenomenological RAT_S rotation-curve ansatz and SPARC galaxy data are treated only as an external empirical baseline; RAT_S is not derived from the effective kernel. The framework does not establish a fundamental microscopic theory, a first-principles baryonic source map, Newtonian matching, galaxy formation or equilibrium, endogenous cross-galaxy parameter scaling, or a universal acceleration scale. In particular, the baryonic Tully–Fisher relation (BTFR) is not derived: the cross-galaxy scaling and the identification of a finite-radius observable with the asymptotic velocity remain open problems. This release corresponds to Version 1.0 of the public record and to Manuscript v4 FINAL.