Spectral Kernel Dynamics for Planetary Surface Graphs: Distinction Dynamics and Topological Conservation
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908987689009152 |
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| author | Das, Jnaneshwar |
| author_facet | Das, Jnaneshwar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20887 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral Kernel Dynamics for Planetary Surface Graphs: Distinction Dynamics and Topological Conservation Das, Jnaneshwar Dynamical Systems Earth and Planetary Astrophysics Machine Learning Robotics 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. |
| title | Spectral Kernel Dynamics for Planetary Surface Graphs: Distinction Dynamics and Topological Conservation |
| topic | Dynamical Systems Earth and Planetary Astrophysics Machine Learning Robotics |
| url | https://arxiv.org/abs/2604.20887 |