TOPOLOGICAL OBSTRUCTION TO OFF-CRITICAL ZEROS ANNULAR COST, CARRIER ENERGY, AND THE RIEMANN ZETA FUNCTION

We introduce the annular J-cost, a convex-function energy functional defined on concentric-ring phase samples of a meromorphic function, and develop its theory using only standard complex analysis. Our first main result (Theorem 4.3) shows that a zero of the Riemann zeta function ζ(s) of multiplicity m ≥ 1 at a point ρ with 1/2 < Re(ρ) < 1 forces the annular cost to diverge as Θ(m² log N), where N is the mesh refinement depth. Our second main result (Theorem 6.1) shows that the Euler carrier—the regularized Fredholm determinant C(s) = det₂(I − A(s))², which is holomorphic and nonvanishing on Re(s) > 1/2—has uniformly bounded annular cost O(M²CR²), independent of N.

The excess decomposition (Theorem 5.2) shows that the annular cost of ζ⁻¹ separates cleanly into a divergent topological floor (from the winding number) and a bounded analytic excess (from the Euler carrier). We prove that any principle establishing a finite upper bound on the total annular cost of ζ⁻¹ in the critical strip immediately implies the Riemann Hypothesis (Theorem 8.1). We formulate such a bound as the Carrier Energy Hypothesis (Theorem 9.1), prove it is equivalent to RH (Theorem 9.2), and provide evidence from Carleson measure theory, the Lindelöf hypothesis, and the Recognition Science framework where the bound follows from the Law of Existence (T1).

All results in Sections 3–8 are machine-verified in Lean 4 (Mathlib), with no sorry and no custom axioms beyond propext, Classical.choice, and Quot.sound.