Recent numerical constructions of constant-velocity, subluminal warp-shell spacetimes that satisfy all pointwise energy conditions have shifted the central open problem of warp-drive research away from existence and onto the acceleration phase: how such a configuration can change its velocity. This work analyzes the geometric structure of that phase for a broad class of warp metrics with a rigid shift profile — an arbitrary static lapse α(r), an arbitrary static spherically symmetric spatial metric with radial factor h(r), and a shift vector β = v(t)W(r)∂ᵤ — which contains the Alcubierre metric as a special case and is the analytic counterpart of the numerically constructed positive-energy shell class. The central result is an exact linearity theorem: for any foliation with static lapse, static spatial metric, and rigid shift β = v(t)X, the Einstein tensor is exactly affine in the acceleration v̇ and independent of all higher time derivatives of the velocity. The acceleration-linear part of any null contraction is captured by a single ADM identity, ∂_v̇(G_μν kᵘkᵛ) = −(K̂_ij sⁱsʲ − tr K̂)/α, with K̂_ij = K_ij/v the velocity-stripped extrinsic curvature. Closed-form evaluation of this identity yields three structural results: (i) the acceleration cost vanishes identically for longitudinal null rays if and only if the spatial slices have constant curvature, so that longitudinal coupling is controlled purely by the spatial-curvature inhomogeneity of the bubble wall; (ii) the worst-case acceleration-linear degradation of the null energy condition is localized on the rear wall and admits a closed form through the eigenvalues of K̂; and (iii) the entire acceleration cost scales as α⁻², so that a lapse hill in the wall suppresses it quadratically. Combining (i) with the vanishing ADM energy of flat-slice warp models gives a structural trade-off: a positive-ADM-energy shell necessarily acquires a long-range longitudinal acceleration-coupling tail of order E_ADM/r² per unit acceleration. Two further results complete the acceleration accounting. In the momentum sector, the Eulerian momentum density is shown to be independent of the acceleration (a "constraint blindness" corollary), while the total ADM axial momentum admits a closed form whose exterior slope is fixed entirely by the boundary data, 8πG dP_z/dv|_ext = (4π/3)(2b₁ − w₁) with h ≃ 1 + b₁/r and W ≃ 1 + w₁/r. This slope vanishes for the zero-ADM-mass (Alcubierre) falloff but is nonzero for a positive-mass shell, so that Ṗ_ADM ≠ 0 forces a null-dust exhaust: a positive-mass warp shell cannot accelerate without radiating momentum to infinity — the momentum-sector counterpart of the energy-sector tail. Finally, the analysis is extended to a nonzero cosmological constant: because Λg_μν is static and vanishes on the null cone, the acceleration identity carries over verbatim for Λ ≠ 0, with the shift confined to the static Hamiltonian constraint (8πG E = G_nn + Λ). With the Schwarzschild–de Sitter static lapse α² = 1 − 2M/r − Λr²/3 this connects the same machinery to the frozen-horizon (Nariai) sector. All statements are geometric identities of the metric class, obtained in an agonistic (Synge G-method) sense: the geometry is fixed first and the required matter content is read off afterward, with no claim of physicality. Every result is labeled by epistemic status (derived / borrowed / candidate-new / conditional) and all symbolic computations are reproducible from the self-contained SymPy scripts bundled with this record. As part of the preparation, the full source of the Barzegar–Buchert–Vigneron classification (arXiv:2602.16495) was cross-checked; the general Hamiltonian constraint of the present class was verified against their second-form constraint with identically zero residual, and a minor cross-invariant omission in one printed energy-density formula (vacuous for all their flat-slice applications) is noted in a footnote.