Towards Geometry-Grounded Layer-Wise Adaptivity for Graph Foundation Models

As the receptive field of a graph neural network expands across hops, the distribution of structural bottlenecks changes, raising the question whether different message-passing layers might favour different operators. Heterogeneous per-layer architectures emerge consistently across the NAS literature yet are treated as a means to performance rather than investigated. The broader graph neural network and graph transformer literature defaults to homogeneous backbones repeating a single operator at every depth — an assumption central to graph foundation models (GFMs) that fix a backbone across graphs of vastly different geometric profiles. We use NAS as a diagnostic instrument and ask whether the per-layer signal is real, robust across architectural envelopes and operator search spaces, and geometrically grounded. Across 31 configurations spanning three envelopes and two operator pools, layer-wise heterogeneity is statistically validated against optimisation noise via an ensemble-consensus protocol, brute-force corroborated against random baselines, and matches or approaches SOTA on six of eight benchmarks. Under controlled synthetic conditions varying only graph geometry, for a given operator pool the separation between operator MSEs grows from negligible on positively curved cliques, to mild on grids, to sharp on negatively curved trees, with within-graph per-node variation responding both to bottleneck position and to which task each node carries. Two propositions and a structural remark ground the observations combinatorially. Together, this elevates layer-wise heterogeneity from a recurring search output to a statistically validated, geometrically responsive phenomenon and makes tractable the foundational question of whether graph geometry admits a principled mapping to operator preference.