The Auxetic Brane: Fano Plane Re-Entrant Geometry as the Structural Origin of Constant Brane Thickness, Negative Effective Poisson Ratio of the Brane Fabric, and Why the Observable Universe Maintains ℓP = const Throughout Cosmic Expansion

Paper XVI established that the brane thickness equals the Planck length ℓP =

ℏG/c3 and that this thickness is exactly constant throughout cosmic expansion,

protected by the cancellation ℏ(t)/c(t)3 ∝H3/H3 = const. This paper provides

the geometric explanation for that constancy: the brane fabric has an auxetic

microstructure. An auxetic material has a negative Poisson ratio ν < 0: when

stretched in one direction it expands rather than contracts in the perpendicular

directions, maintaining constant cross-sectional thickness against lateral strain.

We show that the seven imaginary units {e1,...,e7}of the octonions, connected

by the Fano plane incidence structure, define a re-entrant truss geometry on the

brane fabric with effective Poisson ratio νbrane =−1. Under cosmological lateral

expansion (growth of the brane’s spatial extent), this re-entrant Fano geometry

causes the brane to pull inward in the bulk direction, exactly compensating the

expansion and maintaining ℓP = const. The algebraic cancellation ℏ/c3 = const of

Paper XVI is thus not a numerical coincidence but a structural consequence of theoctonionic Fano-plane microstructure of the brane fabric. The auxetic geometry

also predicts a specific angular distribution of brane fabric fluctuations — the same

δfluc(t) ∝H(t)4 derived in Paper XVII — and a characteristic acoustic mode of

the brane at frequency ωFano ∼c/ℓP = ωP (the Planck frequency), above which

the auxetic response fails and the brane tears. A twenty-third falsifiable prediction

is added: the gravitational wave background should show a spectral peak at the

Planck frequency from residual Fano-mode oscillations of the brane fabric.