The Amplitude Law: the obstruction is the interpolation function, not the carrier -- and a monotone kernel escapes it with a0 untouched

v3 corrects v2 in the framework's favour. v2 identified the binding obstruction as an arm-level solar-system squeeze and left it undecided. It is now decided: a kernel clears it. On 175 SPARC rotation curves with Υ* refit per kernel and the conservative AQUAL quadrupole, the standard monotone family μn(x) = x/(1+xn)1/n passes all three tests at once. At n = 10 the Cassini quadrupole is 0.08–0.21× the ceiling across the full ±2σ of the measured external field on both footings, and the 1-AU monopole is 4×10-28 of the Mars budget — against 5.6–6.4× and 3.3×104 for the interpolation previously assumed. The family is monotone (dμ/dx > 0 proved symbolically), so AQUAL stays strictly convex and the uniqueness theorem survives the swap. a0 is untouched. The deep-MOND limit is identical across the family to 5×10-7, so a0 = κc√(GρΛ), the amplitude law, the baryonic Tully–Fisher relation and the fitted κ all survive unchanged. The solar system was never a statement about a0; it is a statement about the transition region, which is exactly where the interpolation is free. The cost is stated, not hidden: the previously assumed interpolation remains the better RAR fit (0.108 vs 0.127 dex; χ²/dof 21.1 vs 49.1). The kernel that survives Cassini fits rotation curves worse. That trade is the result. Two withdrawals, both of earlier claims of this work. A no-go asserting that every monotone kernel carries Q2 ≥ 3.5× the ceiling does not reproduce and is withdrawn; its error manufactured a deficit. A parallel-mode ghost attributed to the two-field mechanism does not exist on the stated kernel: the kinetic coefficient is 1 - 2x/√(1+4x²) > 0 for every real x with no root, and cT = 1 exactly. The binding obstruction moves. With the solar system cleared by a kernel choice, what remains is the double count between the phantom and any clustering dark sector — kernel-independent, untouched by every μn, and not removed by identifying the sector with the scalar (the cancelling cross term is capped at κ²/(192π) while cancellation needs a gradient 13.4× beyond a hard bound). The deciding question is now: what carries Ωdm cosmologically without depositing a clustered halo in galaxies? Two classes remain unevaluated — Vainshtein/k-mouflage screening, and multi-streaming support. κ = 1/2 remains fitted. All numbers on both footings. All load-bearing claims backed by committed scripts that exit zero, under adversarial verification in which manufacturing a deficit was penalised as heavily as manufacturing a win.