Part 10_Geometric Correlation Between Prime Statistics, Riemann Zeta Zeros, and Nuclear Spectra

This paper investigates a geometric correlation between three seemingly unrelated domains: prime number statistics, nontrivial zeros of the Riemann zeta function, and nuclear energy level spectra. A recurring limitation in earlier comparative studies is the direct use of raw, unitless numerical sequences, which obscures the distinction between genuine physical structure and accidental statistical similarity. To address this, the present work adopts a ratio-coordinate framework motivated by physical density variables, ensuring that all inputs are mapped through the same scale-invariant, saturating transformation before comparison. Within this unified framework, prime gaps, zeta zero spacings, and heavy-nucleus level spacings are projected onto a common geometric coordinate and subjected to a self-falsifiable restoration test derived from an elastic manifold interpretation. This procedure involves no parameter fitting and admits direct counterexamples when the test fails. The results show that small primes do not satisfy the restoration criterion, while sufficiently large prime statistics become statistically indistinguishable from both Riemann zeta zeros and heavy-nucleus spectra within the saturation regime. This behavior identifies a well-defined scale threshold beyond which prime statistics exhibit the same geometric response as physical spectral data. The paper does not claim a proof of the Riemann Hypothesis or an identity between primes and quantum spectra. Instead, it provides a reproducible, physically motivated diagnostic that clarifies when and why number-theoretic data can meaningfully align with physical spectral statistics. All mappings, definitions, and statistical tests are explicitly specified, and the analysis is fully reproducible from the provided methodology.