Deterministic filtering: from a modular counting oracle to All-Edges Exact Triangle

For every program $P$ whose procedure $p_o$ solves the counting oracle task oracleTask within $T_o$ steps with need $\nu_o$, there are procedures $R$ and an index $p'$ such that procedure $p'$ of $P\mathbin{+\!\!+}R$ solves All-Edges Exact Triangle (aeTask: for each pair $(a,b)$ of an $n$-vertex instance with weights bounded by $U$, the flag whether some $c$ has $w(a,b)+w(b,c)+w(a,c)=0$) within filterTime To n U steps and with need filterNeed νo n U; and filterNeed νo is polynomially bounded whenever $\nu_o$ is.

filterTime is the time of the host: computing the parameters, a sieve for the prime pool in $[R,2R)$, the marks, and roundsOf n rounds of rOf n steps, each step calling the oracle once per pool prime and each round twice more, every call within the largest oracle time over the moduli up to $P_{\max}=(2R)^r$; then the final answer (the certified flags under the guard, or a scan of the remaining candidates).