Truncation Levels and Spacetime D: Rigid Phase of Geometric Soft Truncation—An Explicit Anatomy of the Three-Dimensional Five-Configuration Deadlock

This paper provides an explicit dissection of the five-configuration in the threedimensional Kakeya direction-covering problem under the geometric soft-truncationframework. The five-configuration is a legal family of five δ-tubes that intersectpairwise while no three intersect; it is the smallest rigid unit in pure combinatorialbookkeeping. Four results. First, explicit existence: the tangent family of a strictlyconvex curve gives a closed-form construction of a general-position five-configuration,the intersection graph is complete, the Nerve complex contains no two-dimensionalsimplex, and the volume inclusion-exclusion truncates at the second order. Second,volume arithmetic: the leading volume of the transverse intersection of two tubes issixteen divided by three times the sine, multiplied by δ3; the union volume of the fiveconfiguration thereby obtains a closed second-order expansion. Third, linear structuretheorem: after fixing five general directions, the conditions for five lines to intersectpairwise form a homogeneous linear system; on a three-dimensional non-coplanar direction set this system has rank identically seven, null space identically three-dimensionaland exactly the concurrent family, and non-concurrent five-configurations exist onlyon coplanar direction sets; on a four-dimensional general direction set the rank is identically eleven, the null space is identically four-dimensional and exactly the concurrentfamily, and non-concurrent solutions are excluded by the rank structure. Fourth, dimensional ablation: in four dimensions non-concurrent five-configurations do not existfor general direction sets; this non-existence is carried by the exact value of the rank,while overdetermined counting only provides a criticality indication. The full threedimensional dimension count has been resolved by Wang and Zahl; this paper doesnot repeat their proof; its contribution is the explicit dissection of the rigid unit andthe dimensional comparison of the rank characterization.