The Evans Lattice Zeta and the Persistence of the s = 1 Pole

Research Note 37 in the "Geometry of the Critical Line" programme.

Paper 46 established that the leading-order Evans zeros in each winding sector are located at E_n = nπ/L − iD_m, with depth D_m growing linearly in m. This note assembles the cross-sector Evans zero lattice and studies its spectral zeta function. The associated modulus Epstein zeta has a simple pole at s = 1 with residue 2/π², and the lattice has aspect ratio 2b/a = 32/π — a pure geometric invariant of the SCT manifold. The phase-sensitive Evans zeta is numerically consistent with the same pole order. The Mellin transform identifies t⁻¹ short-time heat-trace asymptotics (two-dimensional Weyl law), whereas the Riemann side requires t⁻¹/². No mechanism internal to the Evans lattice reduces the pole from s = 1 to s = 1/2, confirming Kill #67 (dimensional saturation) at the spectral-zeta level and identifying the dimensional reduction as a genuine quotient problem.