Interlevel Betti Token Transformer

1. Integration point: add a TopoScanEncoder beside a standard GNN in a graph-classification model. The encoder receives an input graph G and a scalar filtration h(v) for each node. Start with fixed choices such as node degree, normalized degree, atom type, or a random projection of node attributes; optionally add a learned scalar h(v)=MLP(x_v) but detach it initially so the homology computation is stable. Build a shared grid of N+1 quantiles of all node filtration values, choose m and s, and produce T windows. For each window I_t, construct the interlevel complex exactly according to the paper's filtration convention. In a graph-only MVP, include vertices and edges whose upper-star value lies in the interval, then enforce boundary closure; if the exact interlevel convention is unavailable, use the corresponding interval subcomplex as an explicit ablation. 2. Pseudocode: h = filtration(G, X); alpha = shared_quantile_grid(h, N+1); tokens=[]; for t in range(T): I=(alpha[t*s], alpha[t*s+m]); K=make_interlevel_complex(G,h,I); B0,B1=boundary_matrices(K, field=GF2); beta0=n0-rank_GF2(B0)-rank_GF2(B1); beta1=n1-rank_GF2(B1)-rank_GF2(B2); tokens.append([beta0,beta1,log1p(n0),log1p(n1)]); then project each token with a linear layer, add positional embeddings, apply a 2-4 layer Transformer, mean-pool, and concatenate with the baseline GNN embedding before classification. Use batched sparse GF(2) elimination or precompute incidence structure for fixed molecular graphs. 3. Computed from the mathematics: the grid/window definitions, upper-star values, boundary matrices, GF(2) ranks, and Betti counts. Estimated empirically: the best filtration function, grid density N, window width m, stride s, Transformer width, and whether learned filtrations improve over degree/attribute filtrations. Report the theorem's stability proxy directly by perturbing h to h+epsilon and measuring the empirical ratio ||beta(h+epsilon)-beta(h)||_1 / max(epsilon); bottleneck distances need not be computed for the first experiment. 4. First cheap experiment: use a 2-3 layer GIN baseline on MUTAG or PROTEINS, with graphs capped at a manageable size. Compare GIN, GIN plus raw degree histograms, and GIN plus the TopoScanEncoder at matched parameter count. Sweep N in {16,32}, m in {4,8}, and s=m/2. Evaluate validation accuracy, wall-clock preprocessing time, and robustness after randomly perturbing node attributes or filtration scores by 1%, 5%, and 10%. Success means a statistically significant accuracy gain over degree histograms at less than 2x preprocessing cost, plus a smaller increase in prediction error under filtration perturbations. A second ablation should compare exact Betti tokens against persistence-diagram features and against tokens containing only beta0 or only beta1.