Cut-by-cut ledgers

Let $c$ range over all semantic cuts of all branches. For each declared qubit register $A$, let $q(A)$ be its qubit count and $1_c(A)$ indicate whether it is live and laboratory-owned at that cut. Define

Reserved accessible blanks count when they are allocated and held. Preinstalled supplier factors count from their actual supply time, not only from the moment of use.

Returned registers and remote supplier holdings have their own storage function $Q_{ret/sup}(c)$. The untouched reference ledger is

when the reference consists of $q(E)$ qubits. A total convention, if desired, is

These are different ledgers, not interchangeable bounds.

A useful conservative specialization is

where the summands refer to disjoint currently allocated registers. If two roles use the same physical allocation at different times, the actual indicator ledger, not simultaneous double counting, determines the peak.

The cumulative clock supply count is separate:

It can be much larger than the maximum number of simultaneously live clock qubits.