Stratum sweep equality: the level-u census of classes born at b equals the filtered diagram sum over bars with birth b and death below u

Let $M$ be a tame one-parameter persistence module: a filtered family of finite components indexed by the real line, governed by a filtration law, with a finite critical set $\mathrm{critSet}$ and a persistence diagram whose bar with birth $p$ and death $q$ carries multiplicity $\mu(p,q)$; the essential bar has death $q = \bot$ (the extended-real bottom). Assume the diagram has finite support: only finitely many points of the extended plane carry positive multiplicity. Fix a real birth value $b$ and a census level $u < b$ that is not a critical value ($u \notin \mathrm{critSet}$). Write $N_u(b)$ for the number of components present at level $u$ whose birth supremum is exactly $b$, and $\Sigma_u(b)$ for the sum of diagram multiplicities over all bars with birth exactly $b$ and death strictly below $u$:

where the essential bar $(b, \bot)$ is included (its death lies below every real level) and bars with death exactly $u$ are excluded (no bar dies at the crit-free level $u$). The sweep equality states that the level-$u$ census of the birth-$b$ stratum equals this filtered diagram mass. This is the level-by-level bridge between interleaving census counts and persistence-diagram multiplicities: it converts counting arguments over classes born at a fixed value into counting arguments over diagram bars, and is the engine behind the Hall-type matching arguments used in persistence-based clustering. Formalization note: components, structure maps and birth suprema are the platform's comps/structMap/birthSup structures; $\mu$ is mult of rankFn over the finite diagonal diagram t2_finDiag; deaths below $u$ are read in the extended-real order with $\bot$ minimal.