Paper XXV: The Auxetic Brane: Fano Geometry, Planck Stability, and the Sump Resonance Con-

The brane fabric is proposed to have Poisson ratio ν = -1 (the auxetic condition): when stretched in one direction, it expands laterally rather than contracting. This property, uniquely xed by the requirement that the sump mechanism reduce exactly to Newtonian gravity at high accelerations, is geometrically encoded in the Fano plane structure of the G2 automorphism group. The Fano truss architecture of the brane provides a mechanical explanation for why the brane thickness is exactly the Planck length and remains constant throughout cosmic expansion. The resonance condition a0 = cH0/2π emerges from the requirement that the sump mode completes exactly one full oscillation per Hubble time. The T1/T2/T2* dephasing hierarchy is derived from the auxetic response and the three-level structure of sump coupling: uniform background (T1 = a0), coherent BH focal zone (T2* = JBH e-r/rT2*), and incoherent stellar cluster dephasing (T2 = JT2 e-r2/λ2T2). Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10. 5281/zenodo. 19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M. D. , University of Minnesota. ORCID: 0000-0002-1143-941X.