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.
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 59 or Theorem 60) 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 61.