Riemann Hypothesis via Self-Dual Trace Confinement: A Conditional Closure

Abstract

For any positive integer weights mρ, the Riemann hypothesis is equivalent to the vanishing of AZ = Σρ mρ(Re ρ − ½)2 over the nontrivial zeros. A companion paper proves a duality-confinement theorem: for a strongly measurable observable with values in a real Hilbert space that intertwines an involution with a linear isometric involution and has square-integrable odd part, the Gram operator of the odd part vanishes whenever it is dominated by positive trace-class operators of vanishing trace. This paper applies the theorem to the reflection s ↦ 1 − s̄ in the critical line and measures what the application is worth. A named hypothesis ΓSDTCSelberg on a shell of represented zeros, supplying domination records and a bridge to the zeros' coordinates, implies that every represented zero has coordinate ½; together with a map from coordinates to the reals that sends ½ to ½ and carries the represented zeros onto the real parts of all nontrivial zeros, it implies the Riemann hypothesis. Conversely, the hypothesis holds exactly when every represented zero has coordinate ½, so on the bare shell of all nontrivial zeros with their real parts it is equivalent to the Riemann hypothesis; so is the concrete version on finite windows of actual zeros that are closed under s ↦ 1 − s̄, carry unit weights and eventually contain every zero. The conditional theorem is therefore a reformulation, not evidence. By elementary linear algebra we also show that, for windows closed under s ↦ 1 − s with symmetric positive weights that eventually contain every zero, several natural bounds for the window sums are again equivalent to the Riemann hypothesis, and that on a nonempty window operator-level transport bounds for the identity target cannot tend to zero. No arithmetic estimate is supplied, and no unconditional result is claimed. The mathematical statements are formalized in Lean 4 with Mathlib, using its statement of the Riemann hypothesis; Appendix A gives the scope.