The Chladni Universe: Feigenbaum Sub-Harmonic Structure of Einstein's Spacetime Metric and the Discrete Spectrum of Astrophysical Objects

Description / Abstract: Version 2 — Revised with expanded statistical validation, formal hypothesis framing, and two-population discovery. We apply the Lucian Method (Mono-Variable Extreme Scale Analysis) to the interior Schwarzschild solution across 46 orders of magnitude in energy density. The metric reveals a five-cascade harmonic structure with phase transitions at compactness values η = 0.001, 0.01, 0.1, 0.5, and 8/9 (Buchdahl limit). We hypothesize that the cascades generate a Feigenbaum sub-harmonic spectrum spaced by δ = 4.669201609, motivated by fractal classification of Einstein's equations in prior work. Testing against eight astrophysical objects: the full catalog does not cluster more tightly than random placement (p = 0.64), partially validating the coverage argument. However, the Sun and Earth sit at nearly identical ratios (0.53×) to their respective sub-harmonics (p = 0.013). The expanded catalog reveals a two-population structure: objects with active core energy generation (Sun, Earth, PSR J0348+0432) cluster at 0.53–0.66×, while passive objects (Jupiter, Saturn, Mars, Moon, Sirius B) cluster at 1.05–1.66×. We propose this as a testable prediction for Gaia DR3 stellar surveys. Paper XXI — Resonance Theory. Changes from v1: Self-similarity language sharpened: scale-collapse acknowledged as consequence of dimensionless parameterization, not presented as empirical discovery Feigenbaum application explicitly framed as hypothesis motivated by prior fractal classification, not derivation from static metric Expanded astrophysical catalog from 5 to 8 objects with peer-reviewed reference values Full statistical validation: Monte Carlo null hypothesis test (10⁶ trials), pairwise offset test, coverage argument analysis Discovery of two-population structure correlated with internal energy generation mechanism Three new figures (ratio distribution, Monte Carlo results, expanded Feigenbaum map) Acknowledgment of reviewer critique that motivated the revision Keywords: general relativity, Feigenbaum constant, fractal geometry, astrophysical density distribution, interior Schwarzschild solution, Chladni patterns, sub-harmonic spectrum, stellar structure, Lucian Method, nonlinear dynamics, self-similarity, harmonic structure, astrophysics, spacetime metric, compactness parameter, Buchdahl limit Additional Notes: All computational code available at github.com/lucian-png/resonance-theory-code. The Lucian Method was calibrated against Mandelbrot's equation z → z² + c before application to Einstein's equations. No equations were modified, linearized, or approximated. All results are reproducible.