Mechanism of convergence-induced reshaping of the within-channel graph

Explain why convergence reshapes the local-neighborhood graph underlying the within-class persistent-homology channel, beyond the quantified observation that neighborhood relations are reshuffled during convergence.

Background

The paper finds that the within-class topological quantity varies with training position and can decline during convergence, even when the relevant trajectory changes are measured. The authors quantify the neighborhood reshuffling associated with this change but do not provide a mechanistic explanation for why convergence reorganizes the local-neighborhood graph.

References

What remains open: (i) the residual image term in $C$ ($\approx\pm0.02$ at the $\eta$50 / $K{=}12500$ caliber, beyond the feature-displacement route; Sec.~\ref{sec:C-mechanism}); (ii) determining factor of the co-evolution advantage (SVHN resnet); (iii) breadth --- more architectures/datasets/noise schedules; (iv) the dynamics of within --- why convergence reshapes the local-neighborhood graph (the reshuffle is quantified but not mechanistically explained); (v) the ``chaos region'' ($n{=}3$ cannot distinguish deterministic-effect-plus-s42-anomaly from true chaos).

— Measuring Memory and Generalization as Separable Geometric Channels: The Topo^2 Framework  (2608.30487 - Zhang et al., 31 Aug 2026) in Section L1 — within is a training-position function; Discussion and Outlook