A Persistence Equivalence for the Riemann Hypothesis

We prove that the Riemann Hypothesis is equivalent to a topological statement: all nontrivial zeta zeros lie on the critical line if and only if the maximum persistence lifetime of the cumulative prime gravitational potential W (N) = Σ V (k) satisfies Lₘax (N) = O (√N polylog (N) ). We establish an unconditional packing limit: the local potential is bounded by O (log²N). The forward direction chains von Koch through Abel summation to the Stability Theorem. The reverse direction shows the log (n) prefactor amplifies the Ingham wave, with construction noise bounded at O (log²N) by the packing limit, and the 1D Elder Rule traps the resulting basin into a persistence feature that violates the assumed bound. Computational verification using true 1D sublevel set persistence confirms LₘaxW ~ N^0. 49 to N = 10⁹ (50. 8 million primes) with monotonically decreasing ratio to the theoretical bound across six orders of magnitude. The paper concludes with a formal statement of the remaining open problem — proving the O (√N polylog) bound unconditionally from the algebraic and dynamical structure of the integers alone — as an invitation to the ergodic theory and discrepancy theory communities. The companion paper (Paper 6, DOI: 10. 5281/zenodo. 19430529) establishes the spectral disjointness framework and empirical evidence that closing this bound requires the infinite Euler product structure of Halász's theorem. Fifth paper in the Prime Gravity series. Version 5. 0.