A Spectral Rank-Loss Criterion for Intrinsic Structural Order in Hierarchical Correlative Structures
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| Format: | Recurso digital |
| Langue: | anglais |
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2026
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| _version_ | 1866901815766810624 |
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| author | Bédrouni, Smaïn |
| author_facet | Bédrouni, Smaïn |
| contents | <p>This manuscript establishes a necessary and sufficient spectral condition for intrinsic structural order in hierarchical sequences of finite positive semi-definite operators.</p> <p>Given a sequence obtained through admissible projections, we prove that strict rank loss is equivalent to a strict reduction of accessible spectral dimension and induces a non-degenerate partial order across levels. The result is purely spectral and does not rely on any metric, probabilistic structure, dynamical law, or external time parameter.</p> <p>We further show that, under natural invariance and monotonicity requirements, rank is the unique scalar spectral invariant compatible with admissible compressions, up to monotone reparametrization.</p> <p>The framework is representation-independent and invariant under permutation, orthogonal transformations, admissible block coarse-graining, and Laplacian normalization. A minimal computational illustration is provided using a two-block Laplacian model.</p> <p>The manuscript is self-contained and focuses exclusively on the structural properties of finite-dimensional PSD operators.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18764782 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Spectral Rank-Loss Criterion for Intrinsic Structural Order in Hierarchical Correlative Structures Bédrouni, Smaïn Mathematics Discrete mathematics Linear algebra Algebra Graph theory <p>This manuscript establishes a necessary and sufficient spectral condition for intrinsic structural order in hierarchical sequences of finite positive semi-definite operators.</p> <p>Given a sequence obtained through admissible projections, we prove that strict rank loss is equivalent to a strict reduction of accessible spectral dimension and induces a non-degenerate partial order across levels. The result is purely spectral and does not rely on any metric, probabilistic structure, dynamical law, or external time parameter.</p> <p>We further show that, under natural invariance and monotonicity requirements, rank is the unique scalar spectral invariant compatible with admissible compressions, up to monotone reparametrization.</p> <p>The framework is representation-independent and invariant under permutation, orthogonal transformations, admissible block coarse-graining, and Laplacian normalization. A minimal computational illustration is provided using a two-block Laplacian model.</p> <p>The manuscript is self-contained and focuses exclusively on the structural properties of finite-dimensional PSD operators.</p> |
| title | A Spectral Rank-Loss Criterion for Intrinsic Structural Order in Hierarchical Correlative Structures |
| topic | Mathematics Discrete mathematics Linear algebra Algebra Graph theory |
| url | https://doi.org/10.5281/zenodo.18764782 |