Spectral Geometry, Nodal Invariants, and a Transfer Principle from Selberg Zeta Functions to the Riemann Zeta Function

We develop a spectral–geometric framework comparing Selberg zeta functions on hyperbolic surfaces with the Riemann xi function via measurable invariants derived from nodal geometry, symmetry, and spectral fluctuations. Rather than constructing an operator associated with the Riemann zeta function, the approach formulates a transfer conjecture at the level of normalized valleys and trace-compatible invariants. The framework is fully compatible with the Selberg trace formula and avoids assumptions related to the Riemann Hypothesis. A numerical protocol for falsification is provided.