Higher-Order Geometry of the Abiotic Loom: Axis Alignment, Pore Tethering, and Side-Chain Packing of Secondary-Structure Elements in the Chrysene Tensor Space

Abstract

Paper 1 (Schaper, C. D. (2026). The Abiotic Loom: Discrete Morse Theory and the Geometric Resolution of Levinthal’s Paradox in the Chrysene Tensor Space. Annals of the Chrysene Formalism, 1(1), 273--282.  https://chrysene.com/index.php/acf/article/view/23) and Paper 2 (Schaper, C. D. (2026). Geometric Validation of the Abiotic Loom: Discrete Morse Predictions versus PDB Secondary-Structure Statistics and Ramachandran Densities. Annals of the Chrysene Formalism, 1(1), 283--291.  \url{https://chrysene.com/index.php/acf/article/view/24)  of this series established that the Chrysene Tensor Space functions as an abiotic loom whose interstitial voids select two families of local backbone geometries, and that the corresponding parameter-free angular windows are enriched in high-resolution public structures. The present work elevates the empirical test from local dihedral statistics to higher-order geometric compatibility.

Three independent lines of evidence are examined. First, the principal axes of native alpha-helices and beta-strands exhibit non-random alignment with the lattice frame. Second, direct rigid-body placement of high-resolution secondary-structure elements into the helical pores and rectangular channels of the lattice yields sterically admissible tethering configurations with positive backbone clearance; the same voids accommodate limited coiled-coil rotation. Third, side chains of tethered elements form systematic contacts with the lattice walls, display amphipathic residue-type preferences, and adopt sharpened rotamer distributions relative to unconstrained backgrounds.

Collectively these results demonstrate that the discrete geometry that enriches local dihedral angles also accommodates the three-dimensional shape, orientation, and side-chain packing of secondary-structure elements observed in native proteins. Continuous thermal fluctuations remain fully admissible inside the open cells; the lattice walls supply absolute steric bounds. The higher-order compatibility established here justifies the construction of functional voids by topological scission, the subject of Paper 4.