Exact finite reset-ledger balance

The theorem is a conjunction of two telescoping identities. First, for every small edge type $E$, every ledger consisting of a natural number $n$ and integer-valued edge cochains $T_i:E\to\mathbb Z$ for all natural $i$, and every edge $e\in E$, one has

Second, for every small vertex type $V$, every finite small edge type $E$, every arbitrary coefficient function $b:E\times V\to\mathbb Z$, every such ledger, and every integer chain $c:E\to\mathbb Z$ equipped with the certificate $\sum_e c(e)b(e,v)=0$ for every $v$, one has

The reset orientation is next state minus current state, and the left side is final winding minus initial winding. When $n=0$, both sides of each identity are zero; values $T_i$ for $i>n$ are unrestricted and unused. If $E$ is empty, the first universal claim is vacuous because it quantifies an edge, while the second identity consists of zero sums. If $V$ is empty, the certified closedness condition is vacuous. No incidence or orientation laws are required of $b$, and neither $V$ nor the first assertion’s $E$ is required to be finite. The second identity is algebraic and does not use the supplied closedness certificate, so it would state the same equality for an arbitrary chain.