A Spectral Subcriticality Framework for Precoloring Extension in Graph-Minor Decompositions

We introduce a spectral subcriticality principle for precoloring extension (PE) across graph-minor-style decompositions. The method associates to each local component (surface part, apex set, and vortices) a transfer operator modeling boundary-constraint propagation. If the induced operators are contractive and their aggregated spectral budget satisfies gammag + gammaₐ + gammaw < 1, then boundary restrictions dissipate rather than amplify, implying PE and hence global t-colorability. The framework yields verifiable certificates based on operator norms and local structural parameters, independent of global boundary size. For quasi-embeddable torsos in Kₜ-minor-free graphs, we establish explicit bounds yielding chi (G) <= eta (G) in the regime covered by these certificates. The approach includes a complete L2 toolbox connecting vortex pathwidth to circular min-cut and matching-linkedness, plus a calibrated trichotomy resolving high-pathwidth cases via grid, cylinder, or hub structures.