8 Computational Step
We define rational approximations of the 90 noperthedron vertices by \(\lfloor x \cdot 10^{16} \rfloor /10^{16}\).
nopertQ is a \(\kappa \)-rational approximation of the Noperthedron.
There exists a valid solution table whose zeroth row covers
By exhibiting the table and running the validity checking algorithm. Formalized as Noperthedron.NativeCaseAnalysis.solutionTable in the build-on-demand NativeCaseAnalysis library (checked with native_decide; a kernel-only KernelCaseAnalysis variant is in progress), which is kept out of the default build targets so that CI runs stay fast.
If a global node in the solution tree is valid, then there is no Rupert solution for its interval.
If a local node in the solution tree is valid, then there is no Rupert solution for its interval.
If we have a valid solution table, and in particular its \(i\)th row is valid, then there is no Rupert solution of the interval of its \(i\)th row.
By a mutual induction on the number of rows left in the table following the \(i\)th. This is because validity constrains each row to only refer to later entries. Appeal inductively to this same theorem if the row splits into other nodes in the tree, or appeal to Theorem 54 or Theorem 55) at the leaves.
If we have a valid solution table, then there is no Rupert solution of the interval of its zeroth row.
Immediate special case of theorem 56.