Gravity as Shadow Price: Rigorous Derivation of Einstein Field Equations from Rank-Maintenance Constraints
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2026
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| _version_ | 1866901894993018880 |
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| author | Alcocer, Yuri |
| author_facet | Alcocer, Yuri |
| contents | <p><span>This technical supplement to the foundational work <em>Spacetime from Rank</em> provides the explicit mathematical derivation identifying the Einstein field equations not as fundamental dynamical laws assumed <em>a priori</em>, but as the equation of state arising from the marginal cost of maintaining a full-rank information-geometric constraint structure. We rigorously formalize the <strong>Full-Rank Principle</strong>, demonstrating that the existence of a non-degenerate metric manifold, characterized by locality, causality, and dimensionality, is conditional upon the invertibility of the underlying semantic Hessian. We introduce the <strong>Trace-Portal Mechanism</strong>, wherein the stress-energy trace </span><span></span><span><span> </span>acts as the exclusive coupling channel between pregeometric information content and the emergent geometric phase. By formulating the maintenance of the semantic invariant </span><span></span><span><span> </span>as a constrained optimization problem within the Canonical Semantic Framework (CSF), we identify the gravitational field with the Lagrange multipliers, specifically, the <strong>shadow prices</strong>, associated with the rank-maintenance constraint. We prove that the condition of stationarity under these constraints necessitates a geometric response indistinguishable from General Relativity (GR) in the semiclassical limit, effectively deriving </span><span></span><span><span> </span>as a condition of structural stability rather than a force law. Furthermore, we demonstrate that spacetime singularities represent "forbidden" rank-collapse events, which are dynamically averted through a "capacity injection" mechanism involving horizon entropy, effectively acting as a cosmic hover controller. This framework resolves the conflict between quantum mechanics and general relativity by reinterpreting gravity as an entropic restoring force that preserves the computational integrity (invertibility) of the vacuum.</span></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18521389 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Gravity as Shadow Price: Rigorous Derivation of Einstein Field Equations from Rank-Maintenance Constraints Alcocer, Yuri <p><span>This technical supplement to the foundational work <em>Spacetime from Rank</em> provides the explicit mathematical derivation identifying the Einstein field equations not as fundamental dynamical laws assumed <em>a priori</em>, but as the equation of state arising from the marginal cost of maintaining a full-rank information-geometric constraint structure. We rigorously formalize the <strong>Full-Rank Principle</strong>, demonstrating that the existence of a non-degenerate metric manifold, characterized by locality, causality, and dimensionality, is conditional upon the invertibility of the underlying semantic Hessian. We introduce the <strong>Trace-Portal Mechanism</strong>, wherein the stress-energy trace </span><span></span><span><span> </span>acts as the exclusive coupling channel between pregeometric information content and the emergent geometric phase. By formulating the maintenance of the semantic invariant </span><span></span><span><span> </span>as a constrained optimization problem within the Canonical Semantic Framework (CSF), we identify the gravitational field with the Lagrange multipliers, specifically, the <strong>shadow prices</strong>, associated with the rank-maintenance constraint. We prove that the condition of stationarity under these constraints necessitates a geometric response indistinguishable from General Relativity (GR) in the semiclassical limit, effectively deriving </span><span></span><span><span> </span>as a condition of structural stability rather than a force law. Furthermore, we demonstrate that spacetime singularities represent "forbidden" rank-collapse events, which are dynamically averted through a "capacity injection" mechanism involving horizon entropy, effectively acting as a cosmic hover controller. This framework resolves the conflict between quantum mechanics and general relativity by reinterpreting gravity as an entropic restoring force that preserves the computational integrity (invertibility) of the vacuum.</span></p> |
| title | Gravity as Shadow Price: Rigorous Derivation of Einstein Field Equations from Rank-Maintenance Constraints |
| url | https://doi.org/10.5281/zenodo.18521389 |