Homotopy Types and Shelling Certificates for the H3 Absolute Order

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.