Scalar Impossibility in Multi-Pillar Complexity Measures: Why a Single Number Cannot Capture Algorithmic, Information-Theoretic, Dynamical, and Geometric Complexity
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2025
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| _version_ | 1866902058906419200 |
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| author | Sudoma, Oksana |
| author_facet | Sudoma, Oksana |
| contents | <div> <div>This paper proves a <strong>fundamental impossibility theorem for scalar universal complexity measures</strong>. We demonstrate that no single scalar quantity can consistently rank complex systems when they exhibit four independent types of complexity: algorithmic (circuit cost), information-theoretic (correlations and entropy), dynamical (chaos and scrambling), and geometric (topological structure).</div> <br> <div>The proof constructs an explicit 5-step cycle where each step increases complexity in one dimension while preserving the others. A closing isomorphism returns the system to its initial state. Any universal scalar attempting to track all four dimensions is forced into logical contradiction: it must assign higher values after accumulating increases, yet assign identical values to isomorphic states.</div> <br> <div>This result parallels Arrow's impossibility theorem in social choice theory, showing that certain aggregation problems have no solution satisfying natural axioms. The mathematical obstruction arises because the four complexity types impose incompatible partial orders on system states. The conclusion is that faithful complexity assessment requires vector reporting (C_alg, C_info, C_dyn, C_geom), not scalar reduction.</div> <div> </div> <div> <div> <div><strong>Significance: </strong>This result parallels Arrow's impossibility theorem in social choice theory, showing that certain aggregation problems have no solution satisfying natural axioms. The mathematical obstruction arises because the four complexity types impose incompatible partial orders on system states.</div> <div> </div> <div> <div> <div><strong>Computational Verification</strong>: Includes Python toy example demonstrating the 5-step cycle numerically (Σδ = 1.42 > 0), with all code and data available in the companion GitHub repository: <a href="https://github.com/boonespacedog/complexity-vector-1" target="_blank" rel="noopener">https://github.com/boonespacedog/complexity-vector-1</a></div> </div> </div> </div> </div> <br> <div>The paper includes computational verification via Python toy example demonstrating the 5-step cycle numerically, with all code and data available in the companion GitHub repository.</div> </div> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17562623 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Scalar Impossibility in Multi-Pillar Complexity Measures: Why a Single Number Cannot Capture Algorithmic, Information-Theoretic, Dynamical, and Geometric Complexity Sudoma, Oksana complexity theory impossibility theorem Arrow's theorem resource monotone algorithmic complexity information theory dynamical systems geometric complexity topological complexity quantum information category theory vector measures no-go theorem complexity science <div> <div>This paper proves a <strong>fundamental impossibility theorem for scalar universal complexity measures</strong>. We demonstrate that no single scalar quantity can consistently rank complex systems when they exhibit four independent types of complexity: algorithmic (circuit cost), information-theoretic (correlations and entropy), dynamical (chaos and scrambling), and geometric (topological structure).</div> <br> <div>The proof constructs an explicit 5-step cycle where each step increases complexity in one dimension while preserving the others. A closing isomorphism returns the system to its initial state. Any universal scalar attempting to track all four dimensions is forced into logical contradiction: it must assign higher values after accumulating increases, yet assign identical values to isomorphic states.</div> <br> <div>This result parallels Arrow's impossibility theorem in social choice theory, showing that certain aggregation problems have no solution satisfying natural axioms. The mathematical obstruction arises because the four complexity types impose incompatible partial orders on system states. The conclusion is that faithful complexity assessment requires vector reporting (C_alg, C_info, C_dyn, C_geom), not scalar reduction.</div> <div> </div> <div> <div> <div><strong>Significance: </strong>This result parallels Arrow's impossibility theorem in social choice theory, showing that certain aggregation problems have no solution satisfying natural axioms. The mathematical obstruction arises because the four complexity types impose incompatible partial orders on system states.</div> <div> </div> <div> <div> <div><strong>Computational Verification</strong>: Includes Python toy example demonstrating the 5-step cycle numerically (Σδ = 1.42 > 0), with all code and data available in the companion GitHub repository: <a href="https://github.com/boonespacedog/complexity-vector-1" target="_blank" rel="noopener">https://github.com/boonespacedog/complexity-vector-1</a></div> </div> </div> </div> </div> <br> <div>The paper includes computational verification via Python toy example demonstrating the 5-step cycle numerically, with all code and data available in the companion GitHub repository.</div> </div> |
| title | Scalar Impossibility in Multi-Pillar Complexity Measures: Why a Single Number Cannot Capture Algorithmic, Information-Theoretic, Dynamical, and Geometric Complexity |
| topic | complexity theory impossibility theorem Arrow's theorem resource monotone algorithmic complexity information theory dynamical systems geometric complexity topological complexity quantum information category theory vector measures no-go theorem complexity science |
| url | https://doi.org/10.5281/zenodo.17562623 |