First-Order Geometry, Spectral Compression, and Structural Compatibility under Bounded Computation

• URL: http://arxiv.org/abs/2603.08494v1
• Date: Mon, 09 Mar 2026 15:29:41 GMT
• Title: First-Order Geometry, Spectral Compression, and Structural Compatibility under Bounded Computation
• Abstract summary: We propose an operator-theoretic formulation in which limitations are encoded by self-adjoint operators.<n>In this setting, the optimal first-order improvement direction emerges as a pseudoinverse-weighted gradient.<n>We show that effective dynamics concentrate along dominant spectral modes, yielding a principled notion of spectral compression.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Optimization under structural constraints is typically analyzed through projection or penalty methods, obscuring the geometric mechanism by which constraints shape admissible dynamics. We propose an operator-theoretic formulation in which computational or feasibility limitations are encoded by self-adjoint operators defining locally reachable subspaces. In this setting, the optimal first-order improvement direction emerges as a pseudoinverse-weighted gradient, revealing how constraints induce a distorted ascent geometry. We further demonstrate that effective dynamics concentrate along dominant spectral modes, yielding a principled notion of spectral compression, and establish a compatibility principle that characterizes the existence of common admissible directions across multiple objectives. The resulting framework unifies gradient projection, spectral truncation, and multi-objective feasibility within a single geometric structure.

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