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

2026-04-17

AI Summary

This work addresses the absence of conservation-law analogues in spectral kernel field equations, which leads to violations of topological integrity and conservation deficits during planetary surface map compression. To remedy this, we propose a discriminative dynamical equation \( \mathrm{d}c/\mathrm{d}t = G[c, h_t] \) that compensates for the conservation deficit and, for the first time, establishes an exact relationship between this deficit and the Hessian stability margin. Drawing on spectral graph theory, differential geometry, Maximum Caliber (MaxCal) optimization, and algebraic topology, we derive a conditional topological-preserving compression theorem guaranteeing invariance of Betti numbers when sufficient spectral modes are retained. We further introduce MaxCal-optimal discriminative dynamics and a triple-spectral diagnostic criterion for planetary drainage networks. The method achieves efficient spectral diagnostics at \( O(N) \) complexity, demonstrates empirical consistency on real-world data, and delineates its theoretical limits via short-cycle counterexamples.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Computational social science

Abstract

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.

Problem

• spectral kernel
• topological conservation
• Betti numbers
• planetary surface graphs
• conservation deficit

Innovation

• spectral kernel dynamics
• topological conservation
• distinction dynamics
• Betti-number preservation
• MaxCal optimization