Thermodynamic Consistency and Holographic Obstructions in Lorentz-Violating BTZ Gravity

We examine whether the established thermodynamics of the exact rotating radial BTZ-like branch of Einstein-bumblebee gravity closes within a single action-derived fixed-normal framework linking asymptotic charges, horizon entropy, Euclidean thermodynamics, and the standard Cardy dictionary. The full action fixes distinct radial-kinetic and curvature-boundary coefficients, and the corresponding Hamilton-Jacobi counterterms yield Brown-York charges equal to the known solution-phase-space mass and angular momentum after normalization matching. On the real nonextremal exterior branch, a direct finite-radius evaluation of the complete adjusted covariant surface charge reproduces the first-law entropy, whereas the local curvature-derivative term accounts for only part of it. Their finite difference is independent of the stretched-horizon radial regulator within the specified action, boundary prescription, and presymplectic representative; the extremal value follows by continuity. For the complex stationary Euclidean saddle, the symmetry-preserving radial fixed-norm sector has no smooth thermal-disk continuation, and its inherited Lorentz-violating tube increment has the opposite sign from the finite completion required by Hamiltonian thermodynamics. At infinity, the reduced logarithmic counterterm reproduces the known Virasoro central coefficient, but the displayed standard Cardy temperature, zero-mode, and Hamiltonian entropy assignments do not form one undeformed dictionary in the controlled two-sector Cardy regime.