Prime-Impurity Wells and an Eulerised Relative-Determinant Program for the Riemann Zeta Function

We study a “container + impurities” viewpoint in which the critical strip 0 < σ < 1 is modelled as a one-dimensional confinement x ∈ (0, 1) with a singular cosecant wall,while primes enter as localised impurities. Two numerical diagnostics are recorded: (i) unfolded level-spacing statistics for a discretised strip Hamiltonian exhibit clear level repulsion (GOE-like for the present real-symmetric implementation), and (ii) a “Riemann wave”superposition ΨN (X) = −2∑n≤N cos(γn log X) (with X > 1) produces sharp constructiveinterference peaks near primes, illustrating the explicit-formula mechanism linking primesand zeros.On the analytic side we formulate an Eulerised scattering construction of a prime-definedrelative (perturbation) determinant, designed so that it matches the Euler product on 1 and—after completion—satisfies a functional symmetry s ↔ 1 − s. If these analytic properties are achieved, equality with the completed Riemann ξ-function follows by the identity theorem. Within this blueprint, a key remaining obstruction is an endpoint regularity estimate (IR1†) controlling the infinite-prime cutoff limit uniformly as the smoothing is removed.Status. This manuscript is not a proof of RH; it is a programmatic blueprint. The purpose of IR1† is to isolate a hard analytic estimate in a form that is (i) standard in complex/Hardyspace language and (ii) directly testable in cutoff models.