T³/S³ Branch Dichotomy, Self-Dual Forcing, and a Topological Obstruction for the Riemann Hypothesis

The Riemann Hypothesis asserts that every non-trivial zero of the Riemann zeta function lies on the critical line where the real part equals one half. Despite its central role in analytic number theory, the hypothesis remains unproved, and a clear geometric or dynamical interpretation of the critical line is still lacking. This paper proposes a framework in which the critical line arises as the fixed-point set of a natural involutive symmetry, and its distinguished role is reflected in a branch-selection dynamics driven by the prime spectrum. The starting point is a two-nearest-neighbor contraction map defined on a finite metric space. Under iteration, the associated orbit spaces are modeled to converge, in the Gromov–Hausdorff sense, to a circle (see Theorem 1.1 for the precise formulation and assumptions). This contraction carries a canonical involution that exchanges the two nearest neighbors at each point. At the spectral level, this induces a symmetry that sends a spectral parameter to its reflection about one half, so that the fixed-point set of the involution coincides exactly with the critical line. In this way, the critical line is characterized intrinsically as a symmetry-fixed set, independently of analytic continuation. The connection to the zeta function is encoded through periodic orbit data on the limiting circle. Each prime is associated with an independent rotation mode, and the collection of these modes gives rise to a multiplicative spectral model that formally coincides with the Euler product for the zeta function. Under the involution, this structure reflects the functional symmetry relating values at a point and its reflection across the critical line. The dynamical model To describe branch selection, we introduce an order parameter taking values between zero and one. Values near zero correspond to an incoherent regime, modeled on a three-torus-type structure, while values near one correspond to a coherent regime modeled on a three-sphere-type structure. The evolution is described by a semilinear parabolic equation consisting of transport, diffusion, a double-well reaction term, and a forcing term encoding arithmetic information derived from the zeta function. At the critical value where the real part equals one half, the contributions from all prime modes are perfectly balanced between a mode and its dual counterpart. Away from this value, this balance is broken, and the resulting imbalance grows rapidly in higher modes. Within the model, this leads to the breakdown of stable coexistence between the incoherent and coherent regimes. In this sense, the critical line appears as a self-dual balance point of the dynamics. The topological obstruction At a higher geometric level, the involution extends to a natural action on a seven-dimensional sphere, whose fixed-point set is a five-dimensional sphere. We consider an equivariant section whose zero set corresponds to the zeros of the zeta function. If a zero lies off the critical line, equivariance forces the existence of a paired zero outside the fixed-point set, thereby extending the zero set beyond the symmetry-fixed region. This motivates Conjecture 7.5: that the Euler class of the quaternionic Hopf fibration acts as a topological obstruction to such extensions. If so, all zeros must be confined to the fixed-point set, which corresponds precisely to the critical line. Scope of the paper We do not prove the Riemann Hypothesis. The framework is presented as a self-contained mathematical model, without relying on external physical theories. The central conjecture is reduced to a collection of well-defined problems in algebraic topology and functional analysis, each of which may be studied independently. Organization Part I (Sections 1–3) develops the contraction framework, the involution, and the spectral model for the zeta function.Part II (Sections 4–5) introduces the branch dynamics and the dichotomy between incoherent and coherent regimes.Part III (Sections 6–7) analyzes imbalance mechanisms and formulates the topological obstruction conjecture.Section 8 summarizes the current status and outlines directions for further work.