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"We establish a correspondence between iterated geometric similarity composition and the structural content of the Einstein field equations. A discrete similarity operator defined by the Thales right-triangle construction yields, in its continuous limit, conformal metric rescaling. Three properties of the construction — scaling invariance, chain closure, and second-order locality — produce equivalent constraints to Lovelock's uniqueness theorem, identifying the Einstein tensor G_μν + Λg_μν as the unique similarity-invariant curvature tensor. The binary partition a + b = 1 emerges directly from the operator structure of the Einstein equation: the exact decomposition into linear and nonlinear parts defines a conserved two-channel system with a = C/(1+C), b = 1/(1+C), where C is the Schwarzschild compactness, with no free parameters. The Thales altitude decomposition identifies cubic order as the first level at which a self-consistency condition constrains the partition, yielding the compactness threshold C_crit = 0.4656 as the unique positive root of C³ + 2C² + C − 1 = 0. This threshold coincides with the compactness of the most massive observed neutron stars and the known onset of post-Newtonian degradation. The prediction is falsifiable through neutron star observations, numerical relativity convergence studies, and gravitational wave spectroscopy. No modification of general relativity is proposed. The contribution is a structural re-reading that identifies why the field equations take their specific form from the perspective of self-similar geometric composition
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Title terated Similarity Composition and the Structure of the Einstein Field Equations: A Reverse Thales Construction
Excerpt/Description:
"We establish a correspondence between iterated geometric similarity composition and the structural content of the Einstein field equations. A discrete similarity operator defined by the Thales right-triangle construction yields, in its continuous limit, conformal metric rescaling. Three properties of the construction — scaling invariance, chain closure, and second-order locality — produce equivalent constraints to Lovelock's uniqueness theorem, identifying the Einstein tensor G_μν + Λg_μν as the unique similarity-invariant curvature tensor. The binary partition a + b = 1 emerges directly from the operator structure of the Einstein equation: the exact decomposition into linear and nonlinear parts defines a conserved two-channel system with a = C/(1+C), b = 1/(1+C), where C is the Schwarzschild compactness, with no free parameters. The Thales altitude decomposition identifies cubic order as the first level at which a self-consistency condition constrains the partition, yielding the compactness threshold C_crit = 0.4656 as the unique positive root of C³ + 2C² + C − 1 = 0. This threshold coincides with the compactness of the most massive observed neutron stars and the known onset of post-Newtonian degradation. The prediction is falsifiable through neutron star observations, numerical relativity convergence studies, and gravitational wave spectroscopy. No modification of general relativity is proposed. The contribution is a structural re-reading that identifies why the field equations take their specific form from the perspective of self-similar geometric composition
Work type Technical Documentation
Tags weyl geometry, schwarzschild, perturbation theory, gravitational self-energy, picard iteration, similarity operator, einstein field equations, compactness threshold, lovelock theorem, binary partition, partition geometry, conformal geometry, thales theorem, post-newtonian, neutron stars
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Identifier 2603154937093
Entry date Mar 15, 2026, 3:15 PM UTC
License Creative Commons Attribution 4.0
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Author. Holder Elias DeJesus. Date Mar 15, 2026.
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