GEOMETRIC PERIODICITY IN SYSTEMS FROM GALAXIES TO MOLECULAR CLUSTERS

10.5281/zenodo.18622453 We present observational evidence for a universal discrete periodicity across systemsof vastly different scales. Analysis of 175 spiral galaxies from the SPARC catalog reveals adimensionless parameter αG = V 2R/GM = 1.332 ± 0.012 (8.2σ). An empirical sequenceΠ(a), a = 1..6, characterizing characteristic scales in galactic systems, is derived: Π(1) =32/27, Π(2) = 4/3, Π(3) = 3/2, Π(4) = 5/3, Π(5) = 15/8, Π(6) = 2187/1024. Five of sixvalues are confirmed by observations with significances ranging from 2.1σ to 8.2σ. Thesixth, Π(5), constitutes a testable prediction for Euclid DR1 (2027).In BOSS+eBOSS DR18 data, a second peak in the correlation function is detected at224.0 ± 5.2 Mpc (4.0σ), with a ratio to the BAO scale of 2.131 ± 0.050, matching Π(6) towithin 0.2%.We demonstrate that water clusters H+(H2O)n obey the law (0, −m, −n) = 3m · 7n,where (m, n) are integers. Values n = 7, 21, 9 are experimentally confirmed; predictedvalues include 49, 63, 147, 441.A criterion for planetary magnetic field generation is formulated: β = (MFe/M⊕) ·(R⊕/R)3 ≥ 0.453. For Earth, β = 0.453; for Venus, β = 0.449. This prediction is testableby the VERITAS/EnVision missions (20282031). Keywords: large-scale structure, BAO, galaxies, water clusters, dynamo mechanism,Pythagorean tuning