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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.00349 |
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Table of Contents:
- We study the evolution of a positive operator under weighted residual maps determined by a finite family of orthogonal projections. Iterating these maps along the rooted tree of multi-indices produces a "weighted residual energy tree", together with natural path measures obtained by normalizing the dissipated energy or trace at each step. Under a quantitative coverage condition on the projections, we show that along almost every branch the residuals converge strongly to zero and the dissipated pieces admit a rank-one decomposition that reconstructs the initial operator. In the special case where the initial operator is the identity on a subspace, this yields almost surely a random Parseval frame generated intrinsically by the weighted residual dynamics.