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For a conserved binary partition a + b = 1, the incircle radius of the associated Thales right triangle satisfies the exact identity r_in = h/(1 + √(1+2h)), where h = √(ab) is the geometric mean. This expresses stability bandwidth as a nonlinear function of coupling alone. The derivative is everywhere positive and monotonically decreasing, establishing a diminishing-returns property: weakly coupled systems lose stability faster than strongly coupled ones under equal perturbation. The minimum Feuerbach gap at the square is exactly (√2−1)²/4, setting a hard geometric floor on the distance between coupling architecture and stability boundary. Validated empirically across 131 disk galaxies, 680 exoplanet pairs, early-type galaxies, and transient spectra, with the stability ratio converging to √2−1 within 1–2% in every system. Applications include post-Newtonian reliability bounds, neutron star equation-of-state discrimination, and adaptive mesh refinement criteria for numerical relativity.
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Title The Incircle–Mean Identity on the Thales Constraint Surface: Stability Bandwidth as a Function of the Geometric Mean
For a conserved binary partition a + b = 1, the incircle radius of the associated Thales right triangle satisfies the exact identity r_in = h/(1 + √(1+2h)), where h = √(ab) is the geometric mean. This expresses stability bandwidth as a nonlinear function of coupling alone. The derivative is everywhere positive and monotonically decreasing, establishing a diminishing-returns property: weakly coupled systems lose stability faster than strongly coupled ones under equal perturbation. The minimum Feuerbach gap at the square is exactly (√2−1)²/4, setting a hard geometric floor on the distance between coupling architecture and stability boundary. Validated empirically across 131 disk galaxies, 680 exoplanet pairs, early-type galaxies, and transient spectra, with the stability ratio converging to √2−1 within 1–2% in every system. Applications include post-Newtonian reliability bounds, neutron star equation-of-state discrimination, and adaptive mesh refinement criteria for numerical relativity.
Work type Technical Documentation
Tags general relativity, galaxy dynamics, thales theorem, partition geometry, feuerbach, geometric mean, stability, incircle, harmonic mean
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Identifier 2603154937024
Entry date Mar 15, 2026, 3:04 PM UTC
License Creative Commons Attribution 4.0
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Author. Holder Elias DeJesus. Date Mar 15, 2026.
Information available at https://www.safecreative.org/work/2603154937024-the-incircle-mean-identity-on-the-thales-constraint-surface-stability-bandwidth-as-a-function-of-the-geometric-mean
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