Two Frameworks, One Metric. One Equation
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2026
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| author | McLean, Eric |
| author_facet | McLean, Eric |
| contents | <h3 class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Two Frameworks, One Metric. One Equation. General Relativity and Quantum Mechanics as Two Readings of a Single Geometry</h3> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Description:</strong></p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>A single self-referential equation: </strong>generates a statistical manifold with a unique Riemannian metric. Three peer-reviewed theorems, published independently over four decades, collectively show that any such manifold with the right content automatically produces both quantum mechanics and general relativity as different limits of the same geometry. Chentsov proved the metric is unique. Brody and Hughston proved this metric is the quantum metric. Jacobson proved that thermodynamics on local horizons yields the Einstein equations. Three of the four links in the bridge between the two greatest theories of the twentieth century are therefore external mathematics. The fourth link, the equation that supplies the content activating the other three, is the contribution of this paper.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>The logical chain is:</strong> one equation forces a number field, which forces a symmetry group, which forces a polytope, which forces a spectrum, which parameterises a probability family, which defines a manifold, which carries a unique metric. Read one way, that metric gives the Schrodinger equation. Read another way, it gives the Einstein field equations. Read spectrally, it gives the fine-structure constant. Read thermodynamically, it gives the gravitational coupling, the cosmological constant, and the arrow of time. One equation, two frameworks, one metric.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">The paper is divided into two parts. <br><br><strong>Part I </strong>constructs the mathematical bridge in explicit detail: the manifold, the metric, the quantum branch, the gravitational branch, the Lorentzian signature, and a worked example producing the fine-structure constant to sub-parts-per-million precision. <br><br><strong>Part II </strong>surveys the landscape visible from the bridge: coupling constants, nuclear binding energies for 51 isotopes, proton structure, neutrino parameters, gauge symmetry, dark matter mass and frequency, and a four-term action whose perturbative expansion yields finite quantum field theory with no ultraviolet divergence. <br><br><strong>Three appendices</strong> provide the explicit computations: the commutator algebra on the 600-cell (yielding the angular momentum algebra of the three-sphere, with Heisenberg recovered in the flat limit), the spectral correspondence between the discrete graph and the continuous manifold, and the Feynman rules with a finite one-loop correction.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Zero free parameters. 53 references. 43 of those are to work by other authors. Every structural requirement for unification is addressed with transparent epistemic grading: what is derived, what is argued, what remains open. Two falsifiable predictions are timestamped: a dark matter oscillation frequency testable in existing pulsar timing array data, and a bound-state measurement of the fine-structure constant that distinguishes this framework from standard quantum electrodynamics.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">The wall between general relativity and quantum mechanics may not be structural. It may be perspectival.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19512740 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Two Frameworks, One Metric. One Equation McLean, Eric Pentagon physics General Relativity Quantum Mechanics Riemannian Brody and Hughston Chentsov Jacobson Unification Schrodinger Einstein E8 600 Cell McLean <h3 class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Two Frameworks, One Metric. One Equation. General Relativity and Quantum Mechanics as Two Readings of a Single Geometry</h3> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Description:</strong></p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>A single self-referential equation: </strong>generates a statistical manifold with a unique Riemannian metric. Three peer-reviewed theorems, published independently over four decades, collectively show that any such manifold with the right content automatically produces both quantum mechanics and general relativity as different limits of the same geometry. Chentsov proved the metric is unique. Brody and Hughston proved this metric is the quantum metric. Jacobson proved that thermodynamics on local horizons yields the Einstein equations. Three of the four links in the bridge between the two greatest theories of the twentieth century are therefore external mathematics. The fourth link, the equation that supplies the content activating the other three, is the contribution of this paper.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>The logical chain is:</strong> one equation forces a number field, which forces a symmetry group, which forces a polytope, which forces a spectrum, which parameterises a probability family, which defines a manifold, which carries a unique metric. Read one way, that metric gives the Schrodinger equation. Read another way, it gives the Einstein field equations. Read spectrally, it gives the fine-structure constant. Read thermodynamically, it gives the gravitational coupling, the cosmological constant, and the arrow of time. One equation, two frameworks, one metric.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">The paper is divided into two parts. <br><br><strong>Part I </strong>constructs the mathematical bridge in explicit detail: the manifold, the metric, the quantum branch, the gravitational branch, the Lorentzian signature, and a worked example producing the fine-structure constant to sub-parts-per-million precision. <br><br><strong>Part II </strong>surveys the landscape visible from the bridge: coupling constants, nuclear binding energies for 51 isotopes, proton structure, neutrino parameters, gauge symmetry, dark matter mass and frequency, and a four-term action whose perturbative expansion yields finite quantum field theory with no ultraviolet divergence. <br><br><strong>Three appendices</strong> provide the explicit computations: the commutator algebra on the 600-cell (yielding the angular momentum algebra of the three-sphere, with Heisenberg recovered in the flat limit), the spectral correspondence between the discrete graph and the continuous manifold, and the Feynman rules with a finite one-loop correction.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Zero free parameters. 53 references. 43 of those are to work by other authors. Every structural requirement for unification is addressed with transparent epistemic grading: what is derived, what is argued, what remains open. Two falsifiable predictions are timestamped: a dark matter oscillation frequency testable in existing pulsar timing array data, and a bound-state measurement of the fine-structure constant that distinguishes this framework from standard quantum electrodynamics.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">The wall between general relativity and quantum mechanics may not be structural. It may be perspectival.</p> |
| title | Two Frameworks, One Metric. One Equation |
| topic | Pentagon physics General Relativity Quantum Mechanics Riemannian Brody and Hughston Chentsov Jacobson Unification Schrodinger Einstein E8 600 Cell McLean |
| url | https://doi.org/10.5281/zenodo.19512740 |