The core procedure of the modular counting oracle writes all counts of the candidates (Lemma 4.1)

Let $P_0\mathbin{+\!\!+}R$ be a program containing the procedures of the counting oracle (OCoreCtx: the procedures of the host of Theorem 17 that it reuses, the loop over the instances, and a #Lop-AE-SparseTri solver with time $T_n$ and need $\nu$). Let $x$ be an oracle instance: an Exact Triangle instance with $n$ vertices per part and weight bound $U$ (three $n\times n$ matrices $AB$, $BC$, $AC$ in memory), a block size $1\le s\le n$, a modulus $p\ge1$ and a place cnt for the $\lceil n/s\rceil n^2$ counts, with the precondition x.Pre at free pointer $f$, the host ready at depth $d$ for the data of the instances (oDataOf x), and the limits HostLim allowing the host and the solver.

Then the core body oCoreBody, run in the frame oCoreArgs x f, ends within $\mathrm{blockCost}+t_{\mathrm{coreRun}}(T_n,\mathrm{oDataOf}\,x,U)$ steps and leaves the memory with the list cntList n s p AB BC AC at cnt (for the candidate $i=jn^2+an+b$, the number of $c$ in the block $[js,\min((j+1)s,n))$ with $p\mid w(a,b)+w(b,c)+w(a,c)$), changing no other cell below $f$.

Lemma 4.1: with $\rho=w(a,b)\bmod p$, the count is the number of common neighbours of $(a,\rho)$ and $(b)$ in the lopsided graph with $X_{a,(c,r)}=[w(a,c)\equiv r]$, $Y_{(c,r),b}=[w(b,c)+\rho\equiv -r]$; the queries are grouped by $\rho$ and cut into chunks, so each block costs at most $2p+1$ solver calls at inner dimension $sp$.