Structural Bridges Across Millennium Problems: Invariants, Spectral Organization, and Dynamical Constraints

This work presents a methodological and mathematical research framework for constructing structural bridges between native mathematical formulations and unresolved global consequences in difficult mathematical problems, with the Riemann Hypothesis and the three-dimensional Navier–Stokes problem used as two test cases.The framework is developed within the broader IPCT — Invariant Phase Coherence Topology — research architecture. Its central objective is to investigate whether mathematical structures that are usually studied within separate domains can be connected through explicitly defined transformations, compatibility relations, structural functionals, invariants, monotone quantities, spectral representations, and dynamical constraints.A specific mathematical construction is developed in the Riemann Hypothesis case: a finite-height Gaussian spectral pairing functional based on the functional-equation symmetry of nontrivial zeta zeros. The construction yields an exact pairing identity and a rigorous local variational result showing that, for the selected kernel and fixed zero height, the critical-line position is a strict local maximum of the paired contribution under the stated conditions. This result is deliberately distinguished from an RH criterion or proof. The functional is therefore presented as a constructed mathematical object with a proved local property, while its possible global significance remains an open research question.The broader research hypothesis is that such mathematical constructions may serve as structural bridges between domains that are conventionally treated separately. In this sense, the framework explores a possible missing mathematical link between scientific descriptions: not a linear chain in which one discipline simply reduces to another, but a nonlinear architecture in which different domains may be related through shared structural constraints, transformations, state relations, and mathematical correspondences.The intended role of this mathematics is therefore not to claim that all sciences have the same physical mechanism. Rather, it investigates whether distinct sciences can be connected at the level of mathematical structure while preserving their domain-specific mechanisms. The guiding distinction is:Common Mathematical Architecture ≠ Common Physical Mechanism.Within this perspective, mathematics is treated as a possible cross-domain structural language through which relationships among physical, biological, cognitive, dynamical, and other scientific descriptions may be investigated. Such connections remain hypotheses unless they are supported by explicit mathematical constructions, proofs, estimates, equivalences, or empirical validation.The article maintains a strict separation between established mathematics, newly constructed mathematical objects, conjectural implications, and higher-level interpretation. In particular, no proof of the Riemann Hypothesis or global regularity of three-dimensional Navier–Stokes is claimed.The present version is intended as a research preprint and as a foundational version of an evolving research program. Future versions may extend, test, refine, or reject the proposed structural bridges as additional mathematical results become available.Date: 10 September 2026.Author: Farzaneh Seifi Fath Abad Independent Interdisciplinary Researcher Founder, IPCT–Wise Heart ORCID: 0009-0000-9930-3850 Framework: IPCT — Invariant Phase Coherence Topology Location: Shahr-e Babak, Kerman, IranEmail: [email protected]: ipctwiseheart.comResearch Website: seifi.ipctwiseheart.com