The Ananke (Gravitational Closure) Theorem
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| Format: | Recurso digital |
| Sprache: | Englisch |
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2026
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| _version_ | 1866901822122229760 |
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| author | Gates, Simon. F |
| author_facet | Gates, Simon. F |
| contents | <p>This preprint states and proves the <strong>Ananke (Gravitational Closure) Theorem</strong>, a foundational classification result for classical gravity.</p> <p>Rather than proposing a new model, modifying General Relativity, or fitting observational data, the theorem asks a prior structural question: <em>what gravitational field structures are admissible once gravity is required to close as a classical field?</em> Closure is defined as internal consistency under classical covariance, quadratic action structure with finite conserved energy, and exhaustion of functional freedom under symmetry.</p> <p>From three minimal axioms—classical covariance, quadratic closure, and orthogonal modes of response—the theorem derives a complete classification of admissible gravitational field structures:</p> <ul> <li> <p>In <strong>isolated vacuum regimes</strong>, closure is exact: the gravitational field admits zero residual degrees of freedom, exterior solutions are rigid and unique, inverse-square scaling is enforced in three spatial dimensions, and Newtonian gravity together with vacuum General Relativity emerge necessarily as structural consequences.</p> </li> <li> <p>In <strong>non-vacuum but symmetry-reduced regimes</strong>, closure admits exactly one residual degree of freedom, which is redistributive rather than sourcing, cannot propagate independently in vacuum, introduces no additional universal constants, and is uniquely constrained by conservation and orthogonality.</p> </li> <li> <p>These results fix the admissible covariant gravitational action uniquely up to equivalence, including field content, interaction structure, and coupling to matter. No phenomenological interpolation functions, screening mechanisms, or additional vacuum degrees of freedom are permitted.</p> </li> </ul> <p>The theorem is purely deductive. No observational input, phenomenological assumptions, or regime-specific tuning enters at any stage. Empirical consequences—such as galaxy scaling relations or cosmological behaviour—arise only as corollaries in subsequent work and are not required for the classification itself.</p> <p>The name <em>Ananke</em> is used in its classical Greek sense of necessity: given the axioms, it can be no other way.</p> <p>This document is released on Zenodo to establish intellectual priority for the classification result. It is not intended as a journal submission. Follow-up papers derive regime-specific phenomenological consequences of the theorem.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18335700 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Ananke (Gravitational Closure) Theorem Gates, Simon. F Classical gravity General Relativity Gravitational field theory Covariant field theory Quadratic action Vacuum solutions Gravitational rigidity Field closure Degrees of freedom Symmetry reduction Action principle Mathematical physics <p>This preprint states and proves the <strong>Ananke (Gravitational Closure) Theorem</strong>, a foundational classification result for classical gravity.</p> <p>Rather than proposing a new model, modifying General Relativity, or fitting observational data, the theorem asks a prior structural question: <em>what gravitational field structures are admissible once gravity is required to close as a classical field?</em> Closure is defined as internal consistency under classical covariance, quadratic action structure with finite conserved energy, and exhaustion of functional freedom under symmetry.</p> <p>From three minimal axioms—classical covariance, quadratic closure, and orthogonal modes of response—the theorem derives a complete classification of admissible gravitational field structures:</p> <ul> <li> <p>In <strong>isolated vacuum regimes</strong>, closure is exact: the gravitational field admits zero residual degrees of freedom, exterior solutions are rigid and unique, inverse-square scaling is enforced in three spatial dimensions, and Newtonian gravity together with vacuum General Relativity emerge necessarily as structural consequences.</p> </li> <li> <p>In <strong>non-vacuum but symmetry-reduced regimes</strong>, closure admits exactly one residual degree of freedom, which is redistributive rather than sourcing, cannot propagate independently in vacuum, introduces no additional universal constants, and is uniquely constrained by conservation and orthogonality.</p> </li> <li> <p>These results fix the admissible covariant gravitational action uniquely up to equivalence, including field content, interaction structure, and coupling to matter. No phenomenological interpolation functions, screening mechanisms, or additional vacuum degrees of freedom are permitted.</p> </li> </ul> <p>The theorem is purely deductive. No observational input, phenomenological assumptions, or regime-specific tuning enters at any stage. Empirical consequences—such as galaxy scaling relations or cosmological behaviour—arise only as corollaries in subsequent work and are not required for the classification itself.</p> <p>The name <em>Ananke</em> is used in its classical Greek sense of necessity: given the axioms, it can be no other way.</p> <p>This document is released on Zenodo to establish intellectual priority for the classification result. It is not intended as a journal submission. Follow-up papers derive regime-specific phenomenological consequences of the theorem.</p> |
| title | The Ananke (Gravitational Closure) Theorem |
| topic | Classical gravity General Relativity Gravitational field theory Covariant field theory Quadratic action Vacuum solutions Gravitational rigidity Field closure Degrees of freedom Symmetry reduction Action principle Mathematical physics |
| url | https://doi.org/10.5281/zenodo.18335700 |