We compute the homotopy type of the absolute order of the finite Coxeter group H3 after deleting its identity. The resulting complex is homotopy equivalent to a wedge of four circles and 872 two-spheres. Deleting the central involution as well gives a shellable wedge of 867 two-spheres. The ideals generated by classical and dual Coxeter elements are different: after deleting the identity, they are shellable wedges of 163 and 367 two-spheres, respectively. These assertions have explicit finite shelling certificates covering 4080 facets, checked by a standalone exact program that reconstructs the group and order. The link of the central involution is the disjoint union of five hexagons. It obstructs Cohen-Macaulayness and shellability for the full H3 complex, and the same interval gives an obstruction in H4.

This computer-assisted calculation addresses exceptional cases of AIM Problem 3.1 (dataset record AIM-PROBABILITY-0153), not the uniform finite-Coxeter-group classification or type-A/type-B EL-labelings. The supplementary source archive contains the complete certificates, a Python-standard-library checker, and the bounded source/literature audit. AI-assisted, self-audited, unrefereed preprint; no independent review, formal verification, or absolute priority certification is claimed.
