Geometric Classification of Real Numbers via Relational Rank Geometry: An Empirical Recovery of the Lagrange Characterization
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| Langue: | anglais |
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2026
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| _version_ | 1866901481488121856 |
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| author | Calderas Cervantes, Jesus David |
| author_facet | Calderas Cervantes, Jesus David |
| contents | <p>This paper introduces the depth rate — a measurable observable computed from convergence sequences without requiring knowledge of the limit — and demonstrates that it empirically separates real numbers into their algebraic classes.</p> <p>Applied to admissible structural convergent sequences for 18 real numbers across three algebraic classes, the instrument recovers the Lagrange characterization with 18/18 accuracy. Rationals and quadratic irrationals exhibit logarithmic scaling O(log(1/ε)); tested transcendentals exhibit polynomial scaling Ω(1/ε), active beyond 10¹⁰ frames at ε=10⁻¹⁰. A companion experiment finds sustained relational coupling only among tested transcendental pairs, including cases sharing convergence rates that do not couple — an independent empirical pattern.</p> <p>The central result is proved: exponential increment decay implies logarithmic depth rate; polynomial increment decay implies polynomial depth rate. The period matrix eigenvalue of the CF expansion predicts the depth rate class for quadratic irrationals.</p> <p>Results apply to generator–sequence pairs within the admissible class. Stable under window variation W ∈ [20, 120] and adversarial tests (43/43 stress test, 25 quadratic irrationals across CF periods 1–11). Four open problems are stated connecting depth-rate scaling to normality theory; if all three normality-related problems are resolved, the argument constitutes an indirect geometric argument against the normality of algebraic irrationals.</p> <p>Part of the SFE research program applying Relational Rank Geometry across multiple scientific domains.</p> <p>Reproduction code: <a href="https://colab.research.google.com/drive/1r_6mYFafC6FYdrKyFDcGYVlHHMkROVFe">https://colab.research.google.com/drive/1r_6mYFafC6FYdrKyFDcGYVlHHMkROVFe</a></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19415231 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Geometric Classification of Real Numbers via Relational Rank Geometry: An Empirical Recovery of the Lagrange Characterization Calderas Cervantes, Jesus David continued fractions Lagrange theorem transcendental numbers algebraic irrationals convergence sequences relational rank geometry depth rate normality Khinchin compressibility experimental mathematics <p>This paper introduces the depth rate — a measurable observable computed from convergence sequences without requiring knowledge of the limit — and demonstrates that it empirically separates real numbers into their algebraic classes.</p> <p>Applied to admissible structural convergent sequences for 18 real numbers across three algebraic classes, the instrument recovers the Lagrange characterization with 18/18 accuracy. Rationals and quadratic irrationals exhibit logarithmic scaling O(log(1/ε)); tested transcendentals exhibit polynomial scaling Ω(1/ε), active beyond 10¹⁰ frames at ε=10⁻¹⁰. A companion experiment finds sustained relational coupling only among tested transcendental pairs, including cases sharing convergence rates that do not couple — an independent empirical pattern.</p> <p>The central result is proved: exponential increment decay implies logarithmic depth rate; polynomial increment decay implies polynomial depth rate. The period matrix eigenvalue of the CF expansion predicts the depth rate class for quadratic irrationals.</p> <p>Results apply to generator–sequence pairs within the admissible class. Stable under window variation W ∈ [20, 120] and adversarial tests (43/43 stress test, 25 quadratic irrationals across CF periods 1–11). Four open problems are stated connecting depth-rate scaling to normality theory; if all three normality-related problems are resolved, the argument constitutes an indirect geometric argument against the normality of algebraic irrationals.</p> <p>Part of the SFE research program applying Relational Rank Geometry across multiple scientific domains.</p> <p>Reproduction code: <a href="https://colab.research.google.com/drive/1r_6mYFafC6FYdrKyFDcGYVlHHMkROVFe">https://colab.research.google.com/drive/1r_6mYFafC6FYdrKyFDcGYVlHHMkROVFe</a></p> |
| title | Geometric Classification of Real Numbers via Relational Rank Geometry: An Empirical Recovery of the Lagrange Characterization |
| topic | continued fractions Lagrange theorem transcendental numbers algebraic irrationals convergence sequences relational rank geometry depth rate normality Khinchin compressibility experimental mathematics |
| url | https://doi.org/10.5281/zenodo.19415231 |