\section{One problem only: the Hodge round}\label{sec:hodgeround}

The runs of Section~\ref{sec:experiments} spread the network over five problems. This section points the whole of it at one, the Hodge conjecture in the form of \cite{BB26c}: the conjecture is equivalent to two statements about abelian varieties, (F2), that every Hodge class on an abelian variety outside the part $R(Y)$ reached from the known constructions is algebraic, and (F3$'$), that on every smooth projective variety the Hodge classes are those coming from abelian varieties, $\mathrm{Hdg}^{p}=\mathrm{Ab}^{p}$. The paper proves that the locus on which either statement holds is dense in every family and is either everything or meagre, and it builds for each statement the object that would decide it next and proves at which step the construction stops. Those stopping points are the five target points of Table~\ref{tab:hodgetargets}. The question for the network is whether, with every result of \cite{BB26c}, of the companion paper on Weil classes \cite{BMB26b} and of the literature in its library, and with the fourteen external sources of Section~\ref{ssec:hodgesources}, it produces one object that passes one target point. The measured quantity is the gap of Section~\ref{ssec:gapnumbers}: for each target point the number of open steps between the best checked composite and the point, together with the counts of composites checked, rejected and stopped. Everything in this section was fixed before the runs except the numbers.

\begin{table*}[t]
\caption{The five target points of the Hodge round, from \cite{BB26c}. Each is the object that paper needs next and could not build; ``open steps'' is the depth of the deepest open step in the library's route to the point, the gap of Fig.~\ref{fig:hodgerounds}(c). None is reached.}
\label{tab:hodgetargets}
\centering
\footnotesize
\setlength{\tabcolsep}{4pt}
\begin{tabular}{@{}p{0.12\textwidth}p{0.42\textwidth}p{0.30\textwidth}p{0.08\textwidth}@{}}
\toprule
\rr target & \rr object that passes it & \rr where \cite{BB26c} stops & \rr open steps \\
\midrule
\rr F2.lift & \rr for one polarised abelian scheme $A\to S$ with a flat Hodge class $\xi$ algebraic at a CM point $c$, a semiregular object at $c$ with Chern character $\xi_{c}$; then $\xi$ is algebraic on all of $S$ & \rr the CM point exists in every family; the semiregular object at it does not & \rr 3 \\
\rr F2.E06 & \rr a convolution $E$ of the pieces $O_{Z_{\zeta}}\otimes L^{k}[s]$, $L^{l}[t]$ on $E_{0}^{6}$, $E_{0}=\mathbf{C}/\mathbf{Z}[i]$, with $\mathrm{ch}(E)$ of polarised shape and $\dim\mathrm{Ext}^{2}(E,E)=57$ & \rr the $35$ pieces of its Proposition on separation give $\dim\mathrm{Ext}^{2}\ge2605$; four configurations are left open & \rr 2 \\
\rr F2.link & \rr one algebraic class $\xi\in\mathrm{Hom}_{\mathrm{Hdg}}(T_{X}(-1),T_{Z})$ on $X\times X\times Z_{K}$ for a Mumford fourfold $X$ whose cubic field $F$ contains no imaginary quadratic field & \rr $\mathrm{Hom}(X,Z_{K})=0$; the class is a Hodge class on a CM sixfold and nothing produces it & \rr 1 \\
\rr F3.cubic & \rr an algebraic correspondence of $S_{\lambda}$ (Picard number $13$, transcendental rank $9$) acting on $T(S_{\lambda})$ through the cubic field $F$, with $(F,D)$ not $(\mathbf{Q}(\zeta_{7})^{+},D_{\infty})$ or $(\mathbf{Q}(\zeta_{9})^{+},D_{\infty})$ & \rr finite groups reach only $\mathbf{Q}(\zeta_{7})^{+}$ and $\mathbf{Q}(\zeta_{9})^{+}$; a self-map $f$ with $f^{2}=n$ in $F$ forces $f\in\mathbf{Q}$ & \rr 1 \\
\rr F3.sqrt & \rr a family of K3 surfaces with real multiplication by $\mathbf{Q}(\sqrt{10})$ or $\mathbf{Q}(\sqrt{14})$ on which a chain of symplectic quotients or an isogeny of elliptic surfaces starts & \rr the chain theorem gives $\sqrt6$ and the Fricke cases; no family starts the chain for $\sqrt{10}$, $\sqrt{14}$ & \rr 1 \\
\bottomrule
\end{tabular}
\end{table*}

\subsection{Library, agents and rounds}\label{ssec:hodgerun}

The library for this round has $76$ typed constructions: the $19$ results of \cite{BB26c} that bear on the target points; $9$ of \cite{BMB26b}, namely Markman's theorem for trivial discriminant, the Weil tori, the convolutions on $E_{0}^{6}$ and on $E_{0}^{8}$, the quartic rank, the abelian covers, Schoen's product, the Vandermonde intersections and the escape theorem; $34$ results of the literature from Deligne to Markman's recent work, among them one unrefereed claim, the family of preprints in \cite{OAI26b} that asserts the Hodge conjecture for abelian varieties of CM type and for products of K3 surfaces, which enters as a hypothesis and never as a result; two counts computed for this round (Section~\ref{ssec:hodgecount}); six moves that change the variety (blow-up at a CM point, degeneration to a CM fibre, Kuga--Satake passage, fibration, Hilbert scheme, isogeny cover); and six open conjectures from the OpenConjecture collection (Section~\ref{ssec:hodgesources}), which the refuter treats as hypotheses, so that a composite resting on one is recorded as conditional and stopped. One agent sits on each target point. In a \emph{steady} round each agent runs six generations of ten pairings, with the refuter and the memory of Section~\ref{ssec:memory}; in a \emph{burst} each agent runs ten generations of forty pairings and the round then checks three hundred random chains of three to six constructions, pruned by the same refuter. Every checked composite is recorded with its verdict and its reason, so that no later round checks it again.

Rounds $1$ to $11$ were steady and round $12$ was a burst. Fig.~\ref{fig:hodgerounds} shows the run. The memory records $78$ checked composites in the first round, $29$ in the second, and between $7$ and $16$ in each of the next eight, as the combinations of the library are used up; round $11$, run after the two counts and the moves had entered the library, checked $35$. In all $230$ composites were checked in the eleven steady rounds, $22$ survived, and every survivor is a known result: the reduction to (F2) and (F3$'$), the propagation theorem, the density of CM points, Markman's theorem and the cases it covers. No composite reaches a target point, and the gap of every target point is the same after round $11$ as after round $1$: three open steps for F2.lift, two for F2.E06 and one for each of the other three.

Round $12$ had a time limit of $7000$ seconds. It committed $29$ checked composites to the memory in its first $98$ minutes and was stopped at the limit before it wrote its summary and before the long chains began. None of the $29$ survived. Twenty-four were stopped because they rest on an open conjecture used as a hypothesis, the derived form of the Hodge conjecture or a statement on generic Hodge classes from the OpenConjecture entries; three stopped at the count on $E_{0}^{6}$, one at the semiregular object of F2.lift, and one was rejected. Most of the time of the round went into the channel, which brought $58{,}872$ records of the memory to the proposers, against $11{,}558$ in round $11$ and between $1{,}546$ and $4{,}108$ in each earlier round. The burst therefore multiplies the cost of a round and moves the search toward composites that assume what they need; it does not change the gap. A wider search of the same library does not move a flat line at the depth of the first open step, because that step is not a combination of what the library holds.

\begin{figure*}[t]
\centering
\includegraphics[width=0.98\textwidth]{figures/fig_hodgerounds}
\Description{Stacked bars of checked composites per round of the Hodge-only run: rejected, stopped at an open step, and survivors.}
\caption{The Hodge-only run. (a) Checked composites per round, from the memory: rejected by the refuter, stopped at an open step, and survivors, all of them known results; the green line, the survivors that are more than a known result, is zero in every round. Round $12$ (shaded) is the burst, stopped at its time limit; its hatched part rests on an open conjecture used as a hypothesis. (b) The composites aimed at each target point over rounds $1$ to $11$, with the depth of the gap after round $11$. (c) The gap of each target point by round: the number of open steps between the best checked composite and the point. It does not move.}
\label{fig:hodgerounds}
\end{figure*}

\subsection{Counting on \texorpdfstring{$E_{0}^{6}$}{E0\textasciicircum 6}}\label{ssec:hodgecount}

The target F2.E06 is the one that a finite computation can decide, and \cite{BB26c} leaves four configurations open after its $35$ pieces: (a) four or more line bundles, (b) one sheaf at two shifts, (c) one threefold at several twists, (d) two threefolds at the same shift meeting along an abelian surface. For a convolution $E$ of pieces $F_{a}$ the complex $\mathrm{Ext}^{\bullet}(E,E)$ is filtered by the blocks $\mathrm{Ext}^{k}(F_{a},F_{b})$ with total degree $k+s_{b}-s_{a}$, and $\dim\mathrm{Ext}^{2}(E,E)$ is at least the number of degree-two blocks that no differential reaches or leaves: a block $\mathrm{Ext}^{2}(F_{a},F_{b})$ is \emph{isolated} when no degree-one block maps to it along a chain of pieces and it maps to no degree-three block, and a bound that counts only isolated blocks is conservative, since a non-exact block can only add edges. The Ext groups of the pieces are the tables of \cite{BB26c} for $K=\mathbf{Q}(i)$, with $\chi_{Z_{\zeta}}=8$ and the groups between threefolds meeting along an abelian variety of dimension $d$ in $\prod_{j}\lvert\zeta_{j}-\zeta_{j}'\rvert^{2}$ components. We computed, for every polarised configuration of each family in a window of twists $1$ to $4$ and shifts $-3$ to $3$, the sums $C^{1},C^{2},C^{3}$ of the block dimensions by total degree, the Euler characteristic $C^{2}-C^{1}-C^{3}$, the isolated blocks, and a restricted Euler bound that removes the blocks that cannot interact. Fig.~\ref{fig:hodgepieces} shows the pieces of the least-bound configuration of family (a) and its block matrices; Fig.~\ref{fig:hodgebounds} shows the bounds over the families.

\begin{figure*}[p]
\centering
\includegraphics[width=0.98\textwidth,height=0.78\textheight,keepaspectratio]{figures/fig_hodgepieces}
\Description{Diagram of the thirty-two threefold pieces of candidate C1 on the sixfold and the blocks of their Ext groups.}
\caption{The pieces of candidate C1 on $E_{0}^{6}$ and their Ext blocks. (a) The $32$ threefold pieces $O_{Z_{\zeta}}[s]$ at the characters $\zeta\in\mu_{4}^{3}$ with $\prod\zeta=\pm1$, the sign fixing the shift, and the four line bundles $L^{1}[-2]$, $L^{2}[-1]$, $L^{3}[0]$, $L^{4}[1]$ of the least-bound configuration of family (a). (b) For each piece, the sum of the dimensions of the Ext groups to all pieces, by total degree. (c)--(e) The block matrices by total degree: the degree-one blocks carry the differential $d_{E}$, the degree-two blocks are the budget from which $\mathrm{Ext}^{2}(E,E)$ is cut, and the degree-three blocks are the targets of $d_{E}$ on degree two; the white lines separate the two parity classes of threefolds and the line bundles. Every degree-two block of this configuration is reached or leaves, so the counting bound is $0$ and the rank of the sixfold product decides.}
\label{fig:hodgepieces}
\end{figure*}

\begin{figure*}[t]
\centering
\includegraphics[width=0.98\textwidth]{figures/fig_hodgebounds}
\Description{Histograms of the isolated-block lower bound for the dimension of Ext squared over polarised configurations against the required rank 57, a scatter of Euler-characteristic terms, and the least bound of each family.}
\caption{Counting on $E_{0}^{6}$. (a) The isolated-block lower bound for $\dim\mathrm{Ext}^{2}(E,E)$ over the $2401$ polarised configurations of family (a) and the $432$ of family (c), against the required rank $r=57$. (b) $C^{2}$ against $C^{1}+C^{3}$ for the same configurations: the Euler characteristic is negative everywhere, so no configuration is excluded by it. (c) The least bound of each family against the rank it must meet: the $35$ pieces and family (d) are excluded, families (a) and (c) are not; the row for $E_{0}^{8}$ is the bound of \cite{BMB26b} within six shifts.}
\label{fig:hodgebounds}
\end{figure*}

The results are these. The $35$ pieces of \cite{BB26c} with the three line bundles at shifts $(0,0,0)$, $(2,3,4)$ or $(-3,-2,-1)$ have at least $7213$, $2605$ and $2605$ isolated degree-two blocks, so $\dim\mathrm{Ext}^{2}\ge2605>57$, in agreement with the bound $2560$ of \cite{BMB26b} for its line-bundle model. Family (d), two threefolds at the same shift meeting along a surface, has six polarised configurations in the window and the least bound is $5325$: it is excluded. Family (b) has no polarised configuration in the window. Family (a) has $2401$ polarised configurations, and for the four line bundles $L^{1}[-2],L^{2}[-1],L^{3}[0],L^{4}[1]$ the isolated blocks are $0$ and the Euler characteristic is $-17665$; family (c) has $432$, and for one threefold at the twists $1$ and $2$ at the shifts $-1,0,1,2$ the isolated blocks are again $0$. Counting decides nothing for (a) and (c). The closed chains of pieces along which a term of $d_{E}$ can return to a diagonal block decide which product acts: for (a) there is no closed chain with every step of total degree one up to length eight, and the one chain of \cite{BB26c}, through two threefolds and four line bundles with Ext degrees $0,3,3,0,0,0$, has length six, so the deciding product is the sixfold Massey product $m_{6}$; for (c) the chains have lengths two to five. The two open configurations are therefore a rank computation, not a search, and that computation is the step the network cannot take by combination.

\subsection{Candidates}\label{ssec:hodgecandidates}

Exclusion alone leaves a target point where it was, so the agents also construct. Table~\ref{tab:hodgecandidates} lists the seven explicit candidates written down in this round, each with its invariants, what it achieves, where it falls short and what decides it; Fig.~\ref{fig:hodgemap} places them under their target points. Three die here: the two threefolds at one shift (C3), by the count above; the link class as a construction (C5), since a Hodge class on the CM sixfold $X\times X\times Z_{K}$ is exactly the statement to prove and $\mathrm{Hom}(X,Z_{K})=0$ leaves no map to transport it, so C5 survives only as the reduction it already is in \cite{BB26c}; and the correspondences of $S_{\lambda}$ from Mukai vectors, Buskin's isometries and self-maps (C6), since a self-map $f$ of the transcendental lattice with $f^{2}=n$ in a cubic field is rational, and the moduli-of-sheaves correspondences act through the Hodge isometries that \cite{BB26c} classifies. Four are open and each names the computation that decides it: C1 and the Weil family through it (C4) by the rank of $m_{6}$ on the chain of length six, C2 by the ranks of $m_{3}$ to $m_{5}$ along its chains, in the theta-function model of \cite{BMB26b}; and C7, a chain for $\sqrt{10}$ through a symplectic involution and a symplectic automorphism of order five, multiplier $2/5\equiv10$ modulo squares, by one lattice question, whether a K3 surface with a symplectic automorphism of order five and transcendental rank six carries real multiplication by $\sqrt{10}$. Kummer surfaces of abelian surfaces with real multiplication are excluded from C7 because their Asai representation is irreducible. None of the seven is a proof of anything, and the table says so for each.

\begin{table*}[t]
\caption{The explicit candidates of the Hodge round. The status names the computation or the result that settles the candidate; ``dies'' means excluded with the reason given, ``open'' means that a stated computation decides it and has not been done. No candidate reaches its target point.}
\label{tab:hodgecandidates}
\centering
\footnotesize
\setlength{\tabcolsep}{4pt}
\begin{tabular}{@{}p{0.03\textwidth}p{0.07\textwidth}p{0.36\textwidth}p{0.30\textwidth}p{0.16\textwidth}@{}}
\toprule
\rr & \rr target & \rr object & \rr invariants and where it falls short & \rr status \\
\midrule
\rr C1 & \rr F2.E06 & \rr the $32$ threefolds $O_{Z_{\zeta}}[s]$ and $L^{1}[-2],L^{2}[-1],L^{3}[0],L^{4}[1]$ on $E_{0}^{6}$ & \rr $\mathrm{ch}$ polarised with $N=32$, Hankel rank $3$, $r=57$; $C^{1},C^{2},C^{3}=2523,20316,35458$; isolated $0$; one closed chain, length six & \rr open: rank of $m_{6}$ \\
\rr C2 & \rr F2.E06 & \rr one threefold $Z_{111}$ at the twists $1,1,2,2$ and shifts $-1,0,1,2$ with the $35$ pieces & \rr polarised, isolated $0$, Euler $-11423$; chains of lengths $2$ to $5$; the identity between the two copies contracts the pair, the zero map leaves the chains & \rr open: ranks of $m_{3}$ to $m_{5}$ \\
\rr C3 & \rr F2.E06 & \rr two threefolds meeting along an abelian surface at the same shift, with a cancelling pair & \rr six polarised configurations; isolated $\ge5325>57$ & \rr dies \\
\rr C4 & \rr F2.lift & \rr the Weil family of dimension $9$ through $A_{0}=E_{0}^{6}$ with the product object of \cite{BB26c} at the CM point & \rr the object at $A_{0}$ is C1; at trivial discriminant the result would be Markman's theorem again, and the non-trivial discriminant needs C1 first & \rr open: C1, then one discriminant \\
\rr C5 & \rr F2.link & \rr the class $\xi$ as a Hodge class on the CM sixfold $X\times X\times Z_{K}$ at a CM point & \rr it is the Hodge conjecture for that sixfold; $\mathrm{Hom}(X,Z_{K})=0$ gives no transport & \rr dies as a construction \\
\rr C6 & \rr F3.cubic & \rr correspondences of $S_{\lambda}$ from Mukai vectors, Buskin isometries and self-maps & \rr $f^{2}=n$ in a cubic field forces $f\in\mathbf{Q}$; the isometries are those classified in \cite{BB26c} & \rr dies \\
\rr C7 & \rr F3.sqrt & \rr $S\to Y\leftarrow S'$ through a symplectic involution and a symplectic automorphism of order $5$, multiplier $2/5\equiv10$ & \rr coinvariant rank of order five is $16$, transcendental rank $\le6$; Kummer surfaces of RM abelian surfaces excluded & \rr open: one lattice question \\
\bottomrule
\end{tabular}
\end{table*}

\begin{figure*}[p]
\centering
\includegraphics[width=0.98\textwidth,height=0.78\textheight,keepaspectratio]{figures/fig_hodgemap}
\Description{Map of the Hodge round: the two open statements at the top, the five target points with their gaps, and the seven candidates with their status.}
\caption{The map of the Hodge round. The two statements (F2) and (F3$'$) at the top, the five target points with their gap and their counts under them, the seven candidates under their targets (yellow open, red dead), and below them the three things every candidate is checked against before it is recorded: the library and the memory, the counting and chain results on $E_{0}^{6}$, and the fourteen sources. The bottom line is the state after the run.}
\label{fig:hodgemap}
\end{figure*}

\subsection{Sources}\label{ssec:hodgesources}

Fourteen sources were read during the round, and each is recorded in the memory with what was taken and what was left: a vault memory with embeddings and a knowledge graph, from which the community structure of the construction graph is kept and the server is not; a security reasoning stack with nothing for Hodge; an orchestration policy whose rule we adopt, that a problem is closed only after a kernel-checked Lean endpoint with an audited statement and the three standard axioms, never \texttt{native\_decide}, with evidence graded as lead, candidate, reproduced, formally verified and accepted; a Lean coding agent whose split of a theorem store from an axiom store the engine's Lean check follows; a hypothesis tournament whose Elo ranking orders the survivors of a burst; the output of a research agent on equivariant traces of the Hodge bundle over the moduli of curves, which enters as one construction; vault hygiene rules; four repositories with no mathematics; and the OpenConjecture collection of $3531$ open conjectures from the arXiv, in which a search finds $55$ on Hodge classes, abelian varieties, K3 surfaces and motives, six of which enter the library as conditional nodes. The memory shows that no source contains a construction for any target point; the ones that help change how the network keeps and ranks what it finds, not what it can find.

\subsection{What this round shows}\label{ssec:hodgeverdict}

Nothing closes. (F2) and (F3$'$) are where \cite{BB26c} left them, the gap of every target point is unchanged after the burst, and the only decisions of the round are negative: the configuration (d) and every configuration of the $35$ pieces on $E_{0}^{6}$ are excluded by counting, two candidates die by arguments of \cite{BB26c} applied to them, and the sources add method and no construction. What the round isolates is positive in a narrower sense: two of the four open configurations of \cite{BB26c} are reduced to the ranks of finitely many Massey products in an explicit model, and the companion paper \cite{BMB26b} has already carried out such a computation for the fourfold products at distance two or more and for $E_{0}^{8}$ within six shifts, with the answer that they fail. By the measured gap the search has reached its asymptote, and the next step is not a wider search but the rank of $m_{6}$ on the chain of candidate C1, which decides C1 and C4 either way. That rank is a finite computation in the theta-function model of \cite{BMB26b}; it is not carried out here.