The Kepler Conjecture 2025, 52 - Branch and Bound Strategies

Bounds can be improved by branching and bounding. There are strategies applicable to specific situations. An upright diagonal that is a loop, R a standard region containing it and surrounding simplicies, then the contexts (n, k) are the only ones possible. There are constant upper and lower bounds σ_R(D), τ_R(D) when R contains a loop of that context. If R contains an upright diagonal that isn't a loop and its surrounding quarters than only (4, 1) and (5, 1) is possible. A contravening decomposition star doesn't contain any upright diagonals that are 3-crowded. A large gap along a 4-crowded upright diagonal with the dihedral angle α, may have a union of the four upright quarters along the upright diagonal F. The anchors of U(D) along the large gap v₁, v₂, if |v₁| + |v₂| < 4.6, α > 1.78 and σ_F(D) < -0.31547. A contravening decomposition star doesn't contain 4-crowded upright diagonals. A contravening decomposition star D doesn't have loops with context (5, 1). A single region can be broken into smaller regions by taking the part that meets the cone over a quarter, anchored simplex, creating subregions. An anchored simplex overlapping a flat quarter is "masking" it. If an upright diagonal {0, 3} is 3-unconfined, it can be erased with penalty 0.008. If it's 3-unconfined, masking a flat quarter, then it can be erased with penalty 0, and if a flat quarter is masked generally, its diagonal has length of at least 2.6.