arXiv:2607.27535 2026 Geometry 1 ideas extracted · analyzed Aug 31, 2026
The paper gives a constructive geometric-transfer mechanism: repeated Freudenthal-Kuhn subdivision followed by rectification produces a piecewise-linear complex whose intrinsic metric approximates the original triangulated space with explicit multiplicative distortion. The transferable asset is not persistent homology itself, but a resolution-controlled, nearly isometric simplicial representation that can remain meaningful across smooth pieces and singular strata. A practical neural-network adaptation is to use the refined piecewise-linear metric as a graph or attention positional bias. The explicit distortion constant provides a principled refinement target and a falsifiable test of geometric stability under remeshing and nonuniform sampling.
Build a refined Freudenthal-Kuhn simplicial graph over mesh vertices or geometric samples and inject its piecewise-linear metric into attention logits. Refinement is increased until the explicit distortion target is met, making attention scores less sensitive to arbitrary mesh resolution, triangulation, or local curvature.