Motif-Aware Focus and Compilation

formal proposal. extends cyber/tri-kernel and CT-0 so that focus and the compiled transformer perceive non-linear subgraph structure (motifs) that the current pairwise pipeline is provably blind to. companion to the neural frontier survey (neural/docs/frontier.md, finding §5).

1. The problem — the compile is motif-blind

CT-0 derives the entire architecture from the raw pairwise adjacency $A$:

• φ* by power iteration of $P = A^\top D^{-1}$ (CT-0 §5.1; cyber/tri-kernel §1.1)
• the embedding by SVD of the φ*-weighted adjacency $M = \mathrm{diag}(\sqrt{\phi^*})\,A\,\mathrm{diag}(\sqrt{\phi^*})$ (CT-0 §5.2)
• the attention heads from per-dialect adjacency $A^{(s)}$ (CT-0 §7)

every one of these is a function of pairwise edges. message passing and diffusion of this form are bounded by the 1-Weisfeiler-Leman test (Xu et al. 2019; Morris et al. 2019): they provably cannot count triangles or cycles, and cannot distinguish a node closed in a triangle from a node on an open chain when their WL colours match. φ* assigns them identical focus; the embedding gives them identical roles. the compiled transformer inherits the blindness — corroboration, shape, analogy, and foresight are absent from the model, and no amount of downstream training recovers structure that was never in the weights.

1.1 The completeness theorem is consistent with the blindness

cyber/tri-kernel §4 proves that $\{M, L, H_\tau\}$ span every local LINEAR operator on a graph. this is true and it is exactly the boundary: the span of local linear operators on a node signal IS the 1-WL class. motif counts (subgraph-participation features, e.g. the diagonal of $A^3$ read as a feature, graphlet orbit counts) are not local linear operators on a signal — they live outside $\{M, L, H_\tau\}$. so the theorem does not contradict the blindness; it locates it. extending the basis is the work.

2. What this buys, in plain terms

take two concepts $A$ and $B$, each with two neighbours. $A$'s neighbours are linked to each other (a triangle — mutual corroboration); $B$'s are not (a chain — a lone bridge). to 1-WL they are identical, so today they compile to the same focus and the same embedding role; the model cannot tell a corroborated cluster from a lone bridge.

motif-awareness fixes this two ways, which are complementary:

• reweighting (a sharper formula). compute φ* on a motif-weighted graph, so a concept inside many triangles outranks an equally-connected concept that is not. the model attends to corroborated concepts, not merely popular ones.
• features (a motif vocabulary the model carries). give each token its motif-participation as part of its embedding ("in 5 triangles, hub of 1 star, on 2 cycles"). $A$ and $B$ now embed differently.

applied payoff at inference: confidence (corroborated vs lone), analogy (match concepts by structural role, not neighbour overlap), and foresight (predict the link that completes a near-finished motif — the Arrival engine).

3. The fit — CT-0 is already graded

CT-0 is not a scalar compile; it is geometric-algebra valued (CT-0 §2.5–2.6): a scalar grade (grade 0, stake-weighted sum) and a bivector grade (grade 2, $w_2 = \sum r\,a\,v\,(e_p \wedge e_q)$ — an oriented edge). a $k$-node motif is the next grade: a triad is a trivector $e_p \wedge e_q \wedge e_r$ (grade 3), a $k$-motif a $k$-vector. geometric algebra is the algebra of oriented simplices — the same higher-order / Hodge structure the literature points to. so motif-awareness is the grade-2 → grade-3+ extension of the algebra CT-0 already chose, and the degeneracy discipline already exists: CT-0 §13 / P-CLIFFORD-B already specify "when higher grades are zero, output is byte-identical to a scalar compile."

4. Integration points (by leverage)

1. φ* on a motif-weighted adjacency. run the cyber/tri-kernel update and CT-0 §5.1 power iteration on $W_M$ (the motif-weighted graph: edge weight = co-participation in a chosen motif; Benson, Gleich, Leskovec, Science 2016) instead of raw $A$. focus then concentrates on motif-rich rather than degree-rich nodes. foundational — φ* feeds everything below.
2. higher Clifford grades (CT-0 §2.5–2.6). extend the graded weight $w = w_0 + w_2$ to $w_0 + w_2 + w_3 + \dots$; the trivector grade is triadic consensus. wedge-augmented attention (§7.7) and the Clifford MLP (§8) then carry motif structure natively, making the compiled transformer super-WL.
3. motif-orbit features in the embedding (CT-0 §5.2 / Pass 4). concatenate per-node graphlet-orbit counts to the SVD input (Graph Substructure Networks, Bouritsas et al. 2022) — provably exceeds WL; each token carries its motif role.
4. higher-order Laplacian in the MLP context (CT-0 §8). the graph-Laplacian action $\mathcal{L}H$ becomes the Hodge Laplacian $L_1 / L_2$ on the simplicial lift, so the model's context sees triangles and cycles.
5. conformance predicate P-MOTIF (CT-0 §11). the compile must give distinct output on a pair of graphs that 1-WL cannot distinguish (a triangle-count or cycle-basis check). this is the formal guarantee that the blindness is removed.

minimal high-leverage set: (1) + (2). the rest are consequences.

5. The formal extension

generalize the operator basis. cyber/tri-kernel currently spans $\{M, L, H_\tau\}$ on grade-1 signals (1-WL). the extension adds the higher-order operators on the simplicial lift:

$$\{M_k, L_k, H_{\tau,k}\} \quad \text{for } k = 0, 1, 2, \dots$$

where $L_k$ is the $k$-th Hodge Laplacian on $k$-cells (motifs), and φ* becomes the fixed point of the composite operator over the lift. equivalently, at grade 1, replace $A$ with the motif-weighted $W_M$. the contraction/convergence guarantees of §2.2 must be re-derived on the lift (open question §8).

which motifs are first-class is discovered, not hand-set: the canonical motif set is the one that most compresses the cybergraph (minimum description length; Liu et al. 2024) and/or the motif-conductance communities (Benson 2016). a small bootloader motif set (triad, star, chain, diamond, cycle, feed-forward) may be seeded, mirroring the bootloader dialects.

6. Backward compatibility

when the input .graph carries no higher-grade or motif data, all $M_{k\geq 1}$, $w_{k\geq 3}$, and orbit features vanish and the output is byte-identical to the current CT-0 compile. same discipline as the existing bivector degeneracy (P-CLIFFORD-B). motif-awareness is additive.

7. Scope and touched specs

a CT-2 architecture feature, deliverable additively on CT-0. it touches:

• tru/specs/ct0.md — graded weights §2.5–2.6 (add trivector+), φ* §5.1 (motif-weighted), embedding §5.2 (orbit features), MLP §8 (Hodge Laplacian), conformance §11 (P-MOTIF)
• tru/specs/tri-kernel.md — the operator basis §4 (extend to higher-order); convergence §2.2 on the lift
• cybergraph / .graph — whether motif grades are stored in the snapshot or computed at compile time
• nox jets — shifted_wedge_product exists for grade 2; a grade-3 wedge jet is needed
• mc — the reference rust compiler

8. Resolutions

the four design questions, mostly closed.

8.1 the motif set — a pipeline, not a choice

the three criteria answer different questions; compose them. persistence GATES (real vs noise — a homology class born early and surviving scale; zigzag gives birth/death = the foresight signal). MDL SELECTS the vocabulary (the motif set that most compresses the graph — parameter-free, the same eigenvector as φ*). motif-conductance (Benson) PARTITIONS into dialects (downstream of selection). pipeline: candidates → persistence gate → MDL ranking → conductance clustering.

the $k$-vector blow-up is illusory. the graph is sparse, so the grade-$k$ structure is supported only on the $O(|\text{lexicon}_k|)$ discovered motifs, never $\binom{|P|}{k}$. carry grades up to 3 first-class — link (1), axon-bivector (2), triad-trivector (3); the motif zoo is triad-dominated (feed-forward loop, bi-fan). grade 4+ are compositions of grade-$\leq 3$ via the motif algebra (operadic substitution), not dense grades. assembly-index × copy-number bounds which higher motifs are worth keeping.

8.2 convergence — preserved, with a coupling bound

the §2.2 Banach proof is adjacency-agnostic: it holds for any non-negative ergodic operator. so φ* on the motif-weighted $W_M$ (PSD Laplacian, ergodic via teleport $\alpha$) contracts unchanged. cost: only the heat term's rate $e^{-\tau\lambda_2}$ degrades, since $W_M$ often has a smaller spectral gap (cleaner communities = slower intra-mixing) — compensate by raising $\lambda_d/\lambda_s$ or $\tau$, as diffusion and springs contract at $\alpha$ and $\|L\|/(\|L\|+\mu)$ independent of $\lambda_2$.

the Hodge lift, two-sided coupling — short proof. write the joint state $\phi = \bigoplus_k \phi_k$ over grades and the joint operator $R = D + \beta C$, where $D$ is block-diagonal (the per-grade tri-kernels, each contracting: $\|D\| \le \kappa_{\max} = \max_k \kappa_k < 1$) and $C$ is the off-diagonal boundary coupling ($\phi_k \leftarrow \partial_k\phi_{k-1}$ and $\phi_k \leftarrow \partial_{k+1}^\top\phi_{k+1}$) at strength $\beta$. the boundary maps are bounded: $\|\partial_k\|^2 = \|L_k^{\mathrm{down}}\| \le (k+1)\Delta$ for max coface degree $\Delta$, so $\|C\| \le C_\partial < \infty$. then

$$\|R\phi - R\psi\| \le (\kappa_{\max} + \beta\,C_\partial)\,\|\phi - \psi\|,$$

so $R$ is a contraction whenever

$$\beta < \frac{1 - \kappa_{\max}}{C_\partial}.$$

the per-grade margin $1-\kappa_{\max}$ is the budget; any two-sided coupling within it keeps the Banach fixed point, unique and converging linearly at rate $\kappa_{\max}+\beta C_\partial$. ∎ (one-sided read-out coupling is the $\beta\to 0$ limit and needs no condition.)

8.3 storage vs compute — compute, with an impulse cache

motifs are derived from links, so storing grades in .graph is redundant and staleness-prone. compute them at compile (triad counting is $O(|E|\cdot\deg)$, comparable to the existing power iteration + SVD). but extend CT-0's impulse reuse (§5.1): a signal carries its proven motif-delta, and by the §2.2 locality radius a new link changes only the motifs within $O(\log(1/\varepsilon))$ hops — so motif maintenance is local and cheap, carried as a proven delta like an impulse. snapshots stay light; the compiler skips the higher-order pass on proof-carrying snapshots.

8.4 proof — foresight as a verifiable query

yes, via the zheng verifiable-query compiler (CT-0 §12.8). claim: "link $(p,q)$ has predicted score $s$ under the current motif structure." witness: the local motif counts around $(p,q)$, the dialect's transition kernel $P(m'|m)$, the score computation. proof: the whole prediction is a nox program — look the local motifs (state-read + Brakedown opening), count them (a CCS-expressible polynomial in the local adjacency), look up the transition and compute $s$ (field ops) — so its trace IS the zheng witness (240-byte proof). honest scope: zheng proves the score is correctly computed from the committed graph, not that the future occurs — it certifies the present structure's implication, exactly the Arrival framing. this makes foresight unfakeable and stake-able (predict a link before it forms, provably), tying into the Shapley-of-Δφ* reward.

still open

the cross-grade soundness CONSTANT: a tight, graph-dependent bound on $C_\partial$ (hence the admissible $\beta$) for real cybergraph topologies, and whether the motif representation (hypergraph vs simplicial) shifts φ*-dynamics (Nat. Commun. 2023 says it can).

References

• Xu, Hu, Leskovec, Jegelka, "How Powerful are Graph Neural Networks?" (ICLR 2019); Morris et al., "Weisfeiler and Leman Go Neural" (AAAI 2019) — the 1-WL bound.
• Bouritsas, Frasca, Zafeiriou, Bronstein, "Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting" (GSN, TPAMI 2022) — motif counts exceed WL.
• Benson, Gleich, Leskovec, "Higher-order organization of complex networks" (Science 2016) — motif-weighted adjacency, motif conductance.
• Schaub, Barbarossa, Bianconi et al. — Hodge Laplacians / topological signal processing on simplicial complexes.
• Liu et al., "Compression-based inference of network motif sets" (PLOS Comp Bio 2024) — MDL motif discovery.

see ct0 for the compile contract, cyber/tri-kernel for the focus definition, and neural/docs/frontier.md §5 for the survey finding this proposal acts on.