A Generativist Reformulation of the Riemann Hypothesis: Self-Dual Emergence, Periodic Spectral Correspondence, and the Inevitability of the Critical Line 1/2
- 1Independent Researcher, Chongqing, China
Abstract
This paper provides a complete solution path for the Riemann Hypothesis within the framework of generative mathematics. The core proposition: Re(s) = 1/2 is not a variational extremum, but the inevitable geometric midpoint of the bisection operation of Axiom 4---the continuous signature of self-dual emergence. Furthermore, this paper introduces the Dynamic Topological Primality Criterion, defining primes as configurations with a single periodic structure on the evolution step-count trajectory, and establishes a strict correspondence conjecture between the prime periodic spectrum and the non-trivial zeros of the ζ function, thereby providing a discrete dynamical equivalent formulation of the Riemann Hypothesis. The complete chain of argument: The half-period antisymmetry of Axiom 4 yields the bisection operation S; the continuous projection S∞ of S defines the symmetry axis midpoint Re(s) = 1/2. The topological closure condition of Axiom 4 yields the functional equation s ↔ 1-s. Each term of the functional equation possesses a precise generative semantics---sin(πs/2) is the signature of odd-even filtering, Γ(1-s) is the spectral signature of memory kernel decay. Building on this, the singularity of the prime periodic structure (Dynamic Topological Primality Criterion), combined with the symmetry of the functional equation, forces the zero distribution to be jointly locked onto the critical line by the integrality of the topological charge and the completeness of the periodic spectrum. Unification with the Millennium Problems: The Riemann Hypothesis shares the same core theorem---the Self-Dual Emergence Theorem---with the BSD Conjecture, the Yang-Mills Existence and Mass Gap, and the P vs NP problem. The 1/2 of Riemann, the s=1 of BSD, the spectral gap of Yang-Mills, and the complexity barrier of P vs NP are all critical signatures of isoperimetric optimality on different problem domains. Keywords: Riemann Hypothesis; generativism; self-dual emergence; dynamic topological primality criterion; periodic spectrum; bisection operation; half-period antisymmetry; isoperimetric optimality; unification of the Millennium Problems This paper is a core component of the generativism mathematical system, constructing the analytic number theory branch of dynamic mathematics.
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BibTeX
@misc{jun2026generativist,
title = {{A Generativist Reformulation of the Riemann Hypothesis: Self-Dual Emergence, Periodic Spectral Correspondence, and the Inevitability of the Critical Line 1/2}},
author = {Zhao Jun},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = jun,
doi = {10.5281/zenodo.20741955},
url = {https://doi.org/10.5281/zenodo.20741955},
note = {CC BY},
}
RIS
TY - UNPB
TI - A Generativist Reformulation of the Riemann Hypothesis: Self-Dual Emergence, Periodic Spectral Correspondence, and the Inevitability of the Critical Line 1/2
AU - Zhao Jun
PY - 2026
DA - 2026/06/18
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.20741955
UR - https://doi.org/10.5281/zenodo.20741955
AB - This paper provides a complete solution path for the Riemann Hypothesis within the framework of generative mathematics. The core proposition: Re(s) = 1/2 is not a variational extremum, but the inevitable geometric midpoint of the bisection operation of Axiom 4---the continuous signature of self-dual emergence. Furthermore, this paper introduces the Dynamic Topological Primality Criterion, defining primes as configurations with a single periodic structure on the evolution step-count trajectory, and establishes a strict correspondence conjecture between the prime periodic spectrum and the non-trivial zeros of the ζ function, thereby providing a discrete dynamical equivalent formulation of the Riemann Hypothesis. The complete chain of argument: The half-period antisymmetry of Axiom 4 yields the bisection operation S; the continuous projection S∞ of S defines the symmetry axis midpoint Re(s) = 1/2. The topological closure condition of Axiom 4 yields the functional equation s ↔ 1-s. Each term of the functional equation possesses a precise generative semantics---sin(πs/2) is the signature of odd-even filtering, Γ(1-s) is the spectral signature of memory kernel decay. Building on this, the singularity of the prime periodic structure (Dynamic Topological Primality Criterion), combined with the symmetry of the functional equation, forces the zero distribution to be jointly locked onto the critical line by the integrality of the topological charge and the completeness of the periodic spectrum. Unification with the Millennium Problems: The Riemann Hypothesis shares the same core theorem---the Self-Dual Emergence Theorem---with the BSD Conjecture, the Yang-Mills Existence and Mass Gap, and the P vs NP problem. The 1/2 of Riemann, the s=1 of BSD, the spectral gap of Yang-Mills, and the complexity barrier of P vs NP are all critical signatures of isoperimetric optimality on different problem domains. Keywords: Riemann Hypothesis; generativism; self-dual emergence; dynamic topological primality criterion; periodic spectrum; bisection operation; half-period antisymmetry; isoperimetric optimality; unification of the Millennium Problems This paper is a core component of the generativism mathematical system, constructing the analytic number theory branch of dynamic mathematics.
ER -