From the cut to a numerical lower bound
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EXISTENCE, BALANCE, AND COMPLETE COVERAGE Write \(c=p\mathbin{++}q\). At this cut, Block 01 defines
and proves
Reversal does not change Toffoli count. The new coordinate degree theorem therefore gives
The least safe degree exists
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For a positive \(v\), take the point indicator
It is safe, detects \(v\), and has a degree-\(n\) certificate. Hence complete safe coverage exists at degree \(n\) for every function. Strong induction proves the existence of a least admissible natural degree. The definition beta chooses this bound and its specification and minimality are proved.
The balanced cut is constructed
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Because each gate has cost zero or one, a prefix can realize every Toffoli count from zero to \(T\). The source proves this fact and extracts a cut with
The complete safe cut cover then proves
Natural-number arithmetic retains the floor/ceiling and small-\(T\) cases. No balance assumption is added to the final theorem.