Phase-transition thresholds for curriculum scheduling

Structural Skeleton

The source paper studies a system whose macroscopic behavior changes when local magnetic polarons become connected enough to support a new transport regime. The reusable skeleton is a thresholded connectivity process with a measurable order parameter.

Physics Concept / Mathematical Object

The transferable object is percolation near a regime boundary: sparse local clusters become a system-wide pathway once connectivity crosses a critical point.

AI Target Problem

Target a curriculum scheduler for sparse or modular models. Instead of advancing phases by wall-clock time, advance when a representation-level connectivity statistic crosses a threshold.

Mapping of Variables / Operators / Objective

• Local polaron cluster -> locally useful feature islands or specialist subnetworks
• Connectivity/percolation threshold -> criterion for switching from isolated skill learning to coordinated optimization
• Macroscopic transport change -> measurable improvement in transfer, routing efficiency, or cross-task generalization

Why this might work

Curricula often fail because they change phases too early or too late. A percolation view suggests waiting until the learned substructures become sufficiently connected, which is closer to a state-based control policy than a schedule heuristic.

Why it may fail

The analogy breaks if the monitored connectivity statistic is not causally related to the downstream regime change. It also fails when training dynamics are smooth enough that there is no meaningful threshold to exploit.

Smallest falsifiable experiment

Train a sparse MoE or modular sequence model on a staged task suite. Compare fixed-step curriculum switches against switches triggered by a graph connectivity statistic measured over router co-activation. Reject the hypothesis if threshold-triggered switching shows no benefit in stability or transfer under matched compute.