Topological void nucleation in stochastic Regge calculus: oblate geometry produces toroidal boundaries from first principles

We demonstrate the formation of toroidal voids (χ = 0) from oblate gravitational po-tential wells in Regge calculus - the first physical mechanism producing the geometricdark matter structures established in companion papers 1, 2, without periodic boundaryconditions, rotation, or imposed dynamics.The formation pathway has four steps, each independently established: (i) Action-driventopology selection: the Regge action preferentially produces χ = 0 at 5.3× the random rate,with a sharp phase transition at Tc ≈ 3 × 10−6 (9 experiments, >70 runs). (ii) Gravita-tional tear closure: a dense spherical blob produces closed tear boundaries (χ = 2, 24/25clusters, 66× enrichment at the density gradient) without periodic boundary conditions (Ex-periment 13). (iii) Equatorial concentration : oblate blobs channel tears into the equatorialplane, with cluster aspect ratio scaling from 1.7 (spherical) to 8.9 (aspect ratio 10:1) (Ex-periment 14, 5 aspect ratios × 5 seeds). (iv) Necklace-to-ring merger : at N = 40,000 with2000 evolution steps, equatorial χ = 2 beads coalesce into toroidal handles, producing χ = 0in 3/3 seeds with peaks of 48, 49, and 38 toroidal clusters (Experiment 14b). This is theheadline result.The mechanism requires only an oblate density concentration - which every disk galaxyprovides - and the Regge action. A control experiment replacing the oblate blob with aprecession-swept toroidal initial geometry (ϕmax = 360◦) produces statistically identicalχ = 0 counts (mean 42.0 vs. 42.3 for oblate, 3 seeds each at N = 40,000), confirming thatthe formation is driven by the dynamical tearing mechanism, not by the initial blob topology.The logical chain is complete: galaxy shape (oblate) → equatorial tearing → bead merger→ toroidal boundary (χ = 0) → Gauss-Bonnet curvature balance → rotation curves andlensing.Three negative results provide honest controls: homogeneous χ = 0 voids are transient(Experiment 5), collision χ = 0 requires periodic boundaries (Experiment 7), and rotationproduces isotropic damage due to solver limitations (Experiment 12) - resolved by Ex-periment 14b, which achieves equatorial concentration through initial geometry rather thanimposed dynamics. A 4D pentachoral engine with baryonic coupling confirms the equato-rial tearing in 15/15 runs across 5 rotation rates (Experiment 12b), establishing that themechanism is robust to kinematic perturbations. The formation problem for geometric darkmatter is solved: every disk galaxy's oblate shape is sufficient to produce the toroidal voidswhose curvature replaces dark matter.