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This technical note proves an exact identity on the conserved binary partition manifold: the ratio of adjacent mean gaps in the AM–GM–HM hierarchy is equal to the manifold Lorentz factor,
\frac{\mathrm{AM}-\mathrm{GM}}{\mathrm{GM}-\mathrm{HM}}=\gamma=\frac{1}{2h}.
For binary partitions with a+b=1, the classical means reduce to simple functions of the Thales altitude h=\sqrt{ab}, allowing the full hierarchy of adjacent and total mean gaps to be written in closed form. The result shows that each deeper layer of the AM–GM–HM inequality is contracted by one additional factor of \gamma, giving the hierarchy a precise Lorentz-type structure on the partition manifold. The note also derives the complete decomposition of the three mean gaps and interprets them as a contraction hierarchy governed by asymmetry.
The identity is verified to machine precision across 4,149 binary black hole simulations from the SXS catalog. Beyond the exact algebra, the paper reports that systems lying close to rational Pythagorean closure points on the partition manifold exhibit lower remnant-spin scatter than systems farther from that lattice, suggesting that the rational geometric skeleton carries empirical structure beyond smooth coupling alone. A primitive Pythagorean triple inside the coherence corridor reproduces the corridor constants to three significant figures, and the radiative fraction shows a monotonic regime structure with increasing asymmetry. The work is presented as an exact geometric and diagnostic result on conserved binary partitions, without proposing any modification to gravitational dynamics.
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Title Adjacent Mean Gaps on the Thales Partition Manifold: The Exact Lorentz Identity
This technical note proves an exact identity on the conserved binary partition manifold: the ratio of adjacent mean gaps in the AM–GM–HM hierarchy is equal to the manifold Lorentz factor,
\frac{\mathrm{AM}-\mathrm{GM}}{\mathrm{GM}-\mathrm{HM}}=\gamma=\frac{1}{2h}.
For binary partitions with a+b=1, the classical means reduce to simple functions of the Thales altitude h=\sqrt{ab}, allowing the full hierarchy of adjacent and total mean gaps to be written in closed form. The result shows that each deeper layer of the AM–GM–HM inequality is contracted by one additional factor of \gamma, giving the hierarchy a precise Lorentz-type structure on the partition manifold. The note also derives the complete decomposition of the three mean gaps and interprets them as a contraction hierarchy governed by asymmetry.
The identity is verified to machine precision across 4,149 binary black hole simulations from the SXS catalog. Beyond the exact algebra, the paper reports that systems lying close to rational Pythagorean closure points on the partition manifold exhibit lower remnant-spin scatter than systems farther from that lattice, suggesting that the rational geometric skeleton carries empirical structure beyond smooth coupling alone. A primitive Pythagorean triple inside the coherence corridor reproduces the corridor constants to three significant figures, and the radiative fraction shows a monotonic regime structure with increasing asymmetry. The work is presented as an exact geometric and diagnostic result on conserved binary partitions, without proposing any modification to gravitational dynamics.
Work type Technical Documentation
Tags remnant spin, binary black holes, adjacent mean gaps, harmonic mean, conserved binary partition, sxs catalog, rational lattice, gravitational-wave diagnostics, asymmetry parameter, thales altitude, lorentz factor, contraction hierarchy, mean gap ratio, coherence corridor, pythagorean triples, am–gm–hm inequality, geometric mean, thales partition manifold
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Identifier 2604035153564
Entry date Apr 3, 2026, 3:16 AM UTC
License Creative Commons Attribution 4.0
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Author. Holder Elias DeJesus. Date Apr 3, 2026.
Information available at https://www.safecreative.org/work/2604035153564-adjacent-mean-gaps-on-the-thales-partition-manifold-the-exact-lorentz-identity
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