Through the Eye of the Needle: A Canonical Operator–Trace and Deterministic Prime Framework for the Riemann Hypothesis

We present a fully deterministic, operator-theoretic framework for the Riemann Hypothesis
(RH), integrating a canonical Hamiltonian construction with a Standing/Sitting Band (SSB)
method for prime propagation. By systematically seating composites and propagating standing
primes, the SSB framework guarantees no material clumping of primes, a key limitation of prior
approaches. Gaussian test functions and prime-localized quadratic forms are used to rigorously
control trace contributions, ensuring that any hypothetical off-critical zero violates trace positivity.
This humble, fully constructive proof unifies combinatorial, analytic, and operator-theoretic
techniques to provide a compelling path through the “eye of the needle” to RH.