Spectral Kernel Dynamics for Planetary Surface Graphs: Distinction Dynamics and Topological Conservation
For researchers in topological data analysis and planetary science, this work provides theoretical foundations and diagnostics for graph-based surface analysis, but results are preliminary and lack full benchmarks.
The paper addresses the lack of a conservation-law analog in the spectral kernel field equation for planetary surface graphs, proving that fixed-point flow is volume-expanding and deriving a distinction dynamics equation to close the conservation deficit. It presents a topology-preserving compression theorem and a spectral diagnostic for drainage networks, with preliminary consistency checks on internal data.
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.