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This technical note introduces a diagnostic tool for testing whether population-level statistics faithfully preserve the geometry of individual two-channel systems on the Thales constraint surface. For partitions satisfying p+q=1, each system is described by the Thales altitude h=\sqrt{pq} and the associated incircle radius r_{\mathrm{in}}=h/(1+\sqrt{1+2h}). The central idea is a commutativity test: compare the harmonic expectation of a diagnostic across a population with the diagnostic evaluated at the harmonic expectation of the underlying partition coordinate. The resulting commutativity error measures how much geometric distortion is introduced by aggregation.
The note develops two complementary tests. The first uses the harmonic mean diagnostic HM=2pq and proves that, under the Thales conservation law p+q=1, harmonic expectation commutes exactly with the diagnostic. This exactness is observed empirically at machine precision in all datasets analyzed, and is interpreted as an instance of contextual finiteness: the conservation law annihilates the commutativity error identically. The second uses the nonlinear incircle diagnostic r_{\mathrm{in}}(h), for which commutativity is only approximate. Its error is shown to be controlled by the Jensen gap of the concave function r_{\mathrm{in}}, and therefore by the variance of the population in altitude space.
The tool is applied to three independent physical populations: 110 SPARC disk galaxies, 16 early-type galaxies from the Lelli et al. sample, and 52 Planck CMB TT–EE multipole bins. Across all three systems, the harmonic mean diagnostic commutes exactly, while the incircle diagnostic commutes to better than 0.05%. The smallest errors occur in populations with narrow Feuerbach-gap distributions, especially inside the corridor, where variance suppression drives the commutativity error down by factors of up to 4{,}600\times. This shows that corridor populations are exceptionally robust under aggregation, while intermediate populations are more vulnerable to distortion.
The note further shows that the early-type galaxy sample independently confirms the predicted stability ratio r_{\mathrm{in}}/h \approx \sqrt{2}-1, while the SPARC sample reveals systematic drift away from this target in high-altitude systems. This drift is explained by the reflection coefficient \Gamma from the series/parallel rulers framework, which orders the mass discrepancy almost perfectly and resolves residual structure not captured by the coarse commutativity tests. Two distinct mechanisms for low commutativity error are identified: variance suppression in the corridor, and curvature suppression in the deep cascade. The resulting tool serves as a pre-diagnostic for new datasets: if commutativity error is small, aggregation is faithful; if not, cross-scale comparisons may be contaminated by population-structure artifacts.
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Title Harmonic Commutativity on the Thales Constraint Surface: A Diagnostic Tool for Population-Level Partition Aggregation
This technical note introduces a diagnostic tool for testing whether population-level statistics faithfully preserve the geometry of individual two-channel systems on the Thales constraint surface. For partitions satisfying p+q=1, each system is described by the Thales altitude h=\sqrt{pq} and the associated incircle radius r_{\mathrm{in}}=h/(1+\sqrt{1+2h}). The central idea is a commutativity test: compare the harmonic expectation of a diagnostic across a population with the diagnostic evaluated at the harmonic expectation of the underlying partition coordinate. The resulting commutativity error measures how much geometric distortion is introduced by aggregation.
The note develops two complementary tests. The first uses the harmonic mean diagnostic HM=2pq and proves that, under the Thales conservation law p+q=1, harmonic expectation commutes exactly with the diagnostic. This exactness is observed empirically at machine precision in all datasets analyzed, and is interpreted as an instance of contextual finiteness: the conservation law annihilates the commutativity error identically. The second uses the nonlinear incircle diagnostic r_{\mathrm{in}}(h), for which commutativity is only approximate. Its error is shown to be controlled by the Jensen gap of the concave function r_{\mathrm{in}}, and therefore by the variance of the population in altitude space.
The tool is applied to three independent physical populations: 110 SPARC disk galaxies, 16 early-type galaxies from the Lelli et al. sample, and 52 Planck CMB TT–EE multipole bins. Across all three systems, the harmonic mean diagnostic commutes exactly, while the incircle diagnostic commutes to better than 0.05%. The smallest errors occur in populations with narrow Feuerbach-gap distributions, especially inside the corridor, where variance suppression drives the commutativity error down by factors of up to 4{,}600\times. This shows that corridor populations are exceptionally robust under aggregation, while intermediate populations are more vulnerable to distortion.
The note further shows that the early-type galaxy sample independently confirms the predicted stability ratio r_{\mathrm{in}}/h \approx \sqrt{2}-1, while the SPARC sample reveals systematic drift away from this target in high-altitude systems. This drift is explained by the reflection coefficient \Gamma from the series/parallel rulers framework, which orders the mass discrepancy almost perfectly and resolves residual structure not captured by the coarse commutativity tests. Two distinct mechanisms for low commutativity error are identified: variance suppression in the corridor, and curvature suppression in the deep cascade. The resulting tool serves as a pre-diagnostic for new datasets: if commutativity error is small, aggregation is faithful; if not, cross-scale comparisons may be contaminated by population-structure artifacts.
Work type Technical Documentation
Tags harmonic commutativity; thales constraint surface
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Identifier 2603195028378
Entry date Mar 19, 2026, 7:56 PM UTC
License Creative Commons Attribution 4.0
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Author. Holder Elias DeJesus. Date Mar 19, 2026.
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