# Spectral Kernel Dynamics for Planetary Surface Graphs: Distinction Dynamics and Topological Conservation

arXiv: https://arxiv.org/abs/2604.20887 · HTML: https://mlsanity.com/paper/2604.20887/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 37 references that are in the map.
- First posted on arXiv 17 Apr 2026.

## Builds on (most cited first, 5 of 5)

- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Path Homotopy Invariants and their Application to Optimal Trajectory Planning](https://mlsanity.com/paper/1710.02871/) (2017)
- [Kernel Dynamics under Path Entropy Maximization](https://mlsanity.com/paper/2603.27880/) (2026)
- [Distributed Optimization of Bivariate Polynomial Graph Spectral Functions via Subgraph Optimization](https://mlsanity.com/paper/2511.11517/) (2025)
- [Spectral Kernel Dynamics via Maximum Caliber: Fixed Points, Geodesics, and Phase Transitions](https://mlsanity.com/paper/2604.09745/) (2026)

## Abstract (arXiv)

The spectral kernel field equation R[k] = T[k] lacks a conservation-law analog. We prove (i) the fixed-point flow is strictly volume-expanding (tr DF > 0), precluding automatic conservation, and (ii) the conservation deficit per mode equals the Hessian stability margin exactly: D_m = -Delta'. Closing the deficit requires a scene-side compensating contribution, which we formalise as the distinction dynamics equation dc/dt = G[c, h_t], with MaxCal-optimal realisation G_opt. On fixed-topology 3D surface graphs we derive a conditional topology-preserving compression theorem: retaining k >= beta_0 + beta_1 modes (under a spectral-ordering assumption) preserves all Betti-number charges; we include a worked short-cycle counterexample (figure-eight) calibrating when the assumption fails. A triple necessary spectral diagnostic -- Fiedler-mode concentration, elevated curl energy, anomalous beta_1 -- is derived for planetary drainage networks at O(N) cost. Two internal real-data sequences serve as preliminary consistency checks; full benchmarks and adaptive-topology extensions are deferred.