The four-dimensional scale space framework predicts five gravitational wave polarisation classes, including two vector modes hxs, hys coupling spatial and scale degrees of freedom. Paper 2 established the existence of these modes but could not evaluate their amplitude for lack of the rotating metric; Paper 6 derived the linearised scale-Kerr exterior rotating perturbation; Paper 8 performed the interior matching, fixing the mode spectrum and correcting the amplitude normalisation. We now combine these results to compute the vector-mode amplitude and LIGO detector response for a binary pulsar system, incorporating the Paper 8 correction factor 3GM/c3 ≈2 ×10−5. We derive the vector-mode polarisation tensors eVx ˆij= piˆ nj + ˆ niˆ pj and compute the antenna pattern functions FVx, FVy for an L-shaped interferometer. The sky- averaged RMS antenna pattern is ⟨F 2 Vx⟩1/2 ≈0.44, comparable to the tensor-mode value ⟨F 2 +⟩1/2 ≈0.45 — the vector modes are geometrically well-coupled to L-shaped detectors. However, the scale-space vector mode hxs couples spatial displacement dx to the scale coordinate s; a freely falling scale-stationary mirror produces no spatial strain at leading order. The effective coupling requires a mirror with nonzero scale velocity ˙smirror, introducing a suppression factor ˙ s/c∼1.4 ×10−9 for Earth-surface mirrors (derived from the linearised geodesic equation in Appendix D). For PSR J0737−3039 (PA = 22.7 ms, d = 1.15 kpc), the Paper 8-corrected radiated sideband amplitude is Asb xs ≈6.4 ×10−26 (Paper 8: ×5/7) at f ≈44.05 Hz. Combined with the scale-coupling suppression, the effective strain at a LIGO mirror is ≈6 ×10−40, giving SNR ≲10−13 — far below detection by any planned strain detector. For a hypothetical scale-velocity-sensitive detector with LIGO-comparable noise, the unsuppressed SNR for J0737 would be ≈5 ×10−7 (Paper 8 corrected); the nearby millisecond pulsar PSR J0437−4715 (d = 157 pc, P = 5.76 ms) gives unsuppressed SNR∼102 and is the most promising target for future scale-sensitive instrumentation.

The mathematical development in this paper was produced in dialogue with Claude.ai (Anthropic) in March 2026, directed by the author. Use of AI assistance is acknowledged in accordance with standard scholarly practice.