Chapter B The Infall Geodesic: ODE, Step Function, and θhorizon = Hyperbola-to-Flat Geometry, Auxetic Shear Interaction, Step Function at the Brane Punc-

The infall geodesic ODE governing the rotation of the Fano lattice tilt angle θ during radial free fall into a Schwarzschild black hole is derived and numeri- cally integrated. The ODE has a two-term structure: (1) a geometric term K(r) = −rs/(2r2p 1 −rs/r), the extrinsic curvature of the Schwarzschild constant-t hypersur- face in the Kruskal embedding, which rotates θ through ∆θgeo = 56.67; and (2) an interaction term α K(r) sin 2θ, coupling the geometric rotation rate to the current aux- etic shear state of the lattice, which contributes an additional ∆θint = 16.55. The full ODE is dθ/dr = K(r)1 + α sin 2θ with α = 0.40446, the Fano lattice material coupling to the Schwarzschild Weyl curvature. The ODE is integrated from θ(∞) = 4 (ambient Fano tilt) to θ(r− s ) = 77.218 (horizon limit), which equals arccos(p∗ e7) = arccos(0.22124) from the partition function of Paper XCV a non-trivial consistency check between the dynamical ODE and the statistical microstate count. At r = rs, a step function adds +12.78 discontinuously, taking θ from 77.22 to 90: this is the brane puncture (Paper XC), the sudden transition from brane-tangential to brane-orthogonal (bulk) motion. The total rotation is 86 = 90 −θG2, completing the full range from ambient Fano tilt to bulk-orthogonal. Three new predictions are made (P47P49). 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.