Emergence of Riemann Zeta Zeros in Density Field Eigenmodes: Symmetry-Breaking Phase Transition and Super-Rigidity Isomorphism

This study verifies, within the FDNE density clustering network framework, whether the statistical properties of non-trivial zeros of the Riemann zeta function can naturally emerge in the eigenmode spectrum of a density field. The experiment underwent five iterative stages: (1) initial construction and establishment of unfolding methods; (2) discovery that the time-reversal symmetry of real-symmetric Laplacians leads to GOE statistics, creating a symmetry barrier against the GUE statistics of zeta zeros; (3) introduction of complex Hermitian Laplacian (Peierls substitution / random complex phases) to break time-reversal symmetry, achieving a GOE->GUE phase transition and confirming that both disorder and complex phases are indispensable; (4) deep comparison revealing that the SW complex Hermitian network not only matches the spacing distribution of zeta zeros (direct KS=0.058) but also spontaneously emerges the same spectral super-rigidity (Delta_3 deviation of only 0.009, far below the GUE theoretical value of 0.33). Core conclusion: the underlying structure of zeta function zeros is equivalent to the eigenmode spectrum of a density field with broken time-reversal symmetry, and the emergence of super-rigidity implies that the density field is a globally coherent field structure rather than a superposition of independent modes. This study provides experimental evidence for field mathematics as a foundational framework connecting number theory and physics.