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

arXiv: https://arxiv.org/abs/2603.08494 · HTML: https://mlsanity.com/paper/2603.08494/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 5 references that are in the map.
- First posted on arXiv 9 Mar 2026.

## Builds on (most cited first, 1 of 1)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)

## Abstract (arXiv)

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.