Structural Limits and Sensitivity Hierarchy of Energetic Continuous Computation
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2026
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| _version_ | 1866901068467666944 |
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| author | Fradkin, Yuval |
| author_facet | Fradkin, Yuval |
| contents | <p>This twelve-part series develops a resource-explicit structural theory of energetic continuous computation. The framework models computation as smooth dynamical evolution on finite-dimensional manifolds, parameterized by state dimension, descriptive complexity, evolution time, and accumulated dynamical sensitivity</p> <p><span><span><span>Σ(n)=∫0τ(n)∥DFφ(x(t))∥dt.\Sigma(n) = \int_0^{\tau(n)} \|DF_\varphi(x(t))\| dt .</span><span><span><span>Σ</span><span>(</span><span>n</span><span>)</span><span>=</span></span><span><span><span>∫</span><span><span><span><span><span><span>0</span></span><span><span><span>τ</span><span>(</span><span>n</span><span>)</span></span></span></span><span></span></span></span></span></span><span>∥</span><span>D</span><span><span>F</span><span><span><span><span><span><span>φ</span></span></span><span></span></span></span></span></span><span>(</span><span>x</span><span>(</span><span>t</span><span>))</span><span>∥</span><span>d</span><span>t</span><span>.</span></span></span></span></span></p> <p>The central result establishes the structural equivalence</p> <p><span><span><span>Epoly=P,E_{\text{poly}} = P,</span><span><span><span><span>E</span><span><span><span><span><span><span><span>poly</span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span>P</span><span>,</span></span></span></span></span></p> <p>showing that smooth energetic systems with polynomially bounded physical resources and logarithmic accumulated sensitivity are polynomial-time simulable by deterministic Turing machines, and conversely that every polynomial-time computation admits such an energetic realization.</p> <p>The series further introduces a sensitivity-based hierarchy in which computational amplification scales as <span><span>eΣ(n)e^{\Sigma(n)}</span><span><span><span><span>e</span><span><span><span><span><span><span>Σ<span>(</span><span>n</span><span>)</span></span></span></span></span></span></span></span></span></span></span>, yielding a unified resource conservation principle: super-polynomial computational power requires super-polynomial growth of accumulated dynamical sensitivity or other physical resources.</p> <p>The work does not resolve classical separation questions such as <span><span>P≠NPP \neq NP</span><span><span><span>P</span><span><span><span><span><span><span></span></span></span></span></span>=</span></span><span><span>NP</span></span></span></span>; rather, it provides a structural dynamical characterization of polynomial-time computation and identifies accumulated sensitivity as the fundamental bridge between continuous energetic evolution and discrete complexity theory.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18784428 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Structural Limits and Sensitivity Hierarchy of Energetic Continuous Computation Fradkin, Yuval <p>This twelve-part series develops a resource-explicit structural theory of energetic continuous computation. The framework models computation as smooth dynamical evolution on finite-dimensional manifolds, parameterized by state dimension, descriptive complexity, evolution time, and accumulated dynamical sensitivity</p> <p><span><span><span>Σ(n)=∫0τ(n)∥DFφ(x(t))∥dt.\Sigma(n) = \int_0^{\tau(n)} \|DF_\varphi(x(t))\| dt .</span><span><span><span>Σ</span><span>(</span><span>n</span><span>)</span><span>=</span></span><span><span><span>∫</span><span><span><span><span><span><span>0</span></span><span><span><span>τ</span><span>(</span><span>n</span><span>)</span></span></span></span><span></span></span></span></span></span><span>∥</span><span>D</span><span><span>F</span><span><span><span><span><span><span>φ</span></span></span><span></span></span></span></span></span><span>(</span><span>x</span><span>(</span><span>t</span><span>))</span><span>∥</span><span>d</span><span>t</span><span>.</span></span></span></span></span></p> <p>The central result establishes the structural equivalence</p> <p><span><span><span>Epoly=P,E_{\text{poly}} = P,</span><span><span><span><span>E</span><span><span><span><span><span><span><span>poly</span></span></span></span><span></span></span></span></span></span><span>=</span></span><span><span>P</span><span>,</span></span></span></span></span></p> <p>showing that smooth energetic systems with polynomially bounded physical resources and logarithmic accumulated sensitivity are polynomial-time simulable by deterministic Turing machines, and conversely that every polynomial-time computation admits such an energetic realization.</p> <p>The series further introduces a sensitivity-based hierarchy in which computational amplification scales as <span><span>eΣ(n)e^{\Sigma(n)}</span><span><span><span><span>e</span><span><span><span><span><span><span>Σ<span>(</span><span>n</span><span>)</span></span></span></span></span></span></span></span></span></span></span>, yielding a unified resource conservation principle: super-polynomial computational power requires super-polynomial growth of accumulated dynamical sensitivity or other physical resources.</p> <p>The work does not resolve classical separation questions such as <span><span>P≠NPP \neq NP</span><span><span><span>P</span><span><span><span><span><span><span></span></span></span></span></span>=</span></span><span><span>NP</span></span></span></span>; rather, it provides a structural dynamical characterization of polynomial-time computation and identifies accumulated sensitivity as the fundamental bridge between continuous energetic evolution and discrete complexity theory.</p> |
| title | Structural Limits and Sensitivity Hierarchy of Energetic Continuous Computation |
| url | https://doi.org/10.5281/zenodo.18784428 |