The HSIT Program: Coherence, Arithmetic, and the Riemann Hypothesis

The motivating problem is simple to state and difficult to solve. Can an explicit operator on arithmetic data expose structure relevant to the Riemann Hypothesis rather than merely translate a classical criterion into new notation? This survey records what the HSIT program has established, what later audits corrected, and what remains open. The surviving mathematical core is substantial but bounded. Denominator damping turns the infinite-degree Farey graph into a bounded operator family exactly when Re s 1/2. At finite level, the constant kernel vector and a position unitary recover the classical Mertens exponential sum. The associated commutator identity is exact, but a later algebraic audit established that it is a universal kernel-projection identity: the positive spectrum cancels. It therefore gives an exact operator-language restatement of the classical Mertens criterion, not a new proof mechanism by itself. The canonical operator has a trace-class commutator, a first positive eigenvalue bracketed between orders 1/(N^4 log N) and 1/N^4, and rigid global traces that exclude broad proposed carriers while leaving kernel-sensitive, shell-local, and off-diagonal observables open. The real Ford-damped family has a proved existence boundary at s=1/2, an explicit Hurwitz diagonal, and a fixed-level rescaled boundary moment equal to the classical totient ratio ζ(2w-1)/ζ(2w). No theorem interpolates that boundary identity to the Mertens endpoint. The SIT-inspired finite lattice companion has also been corrected. It proves a structural correspondence of restricted gauge orbits, fixed-magnitude factorization for phase/link events, and quartic pinning bounds. It does not preserve the action or Gibbs weights of the toy model, and transfer of the toy abundance constants remains open. The computational package verifies declared finite identities and negative controls; its trained Tier-3 protocol is unimplemented and reports no trained result. The program's durable contribution is thus twofold: a concrete family of arithmetic graph operators with several proved boundaries, and a source-faithful method for correcting an AI-assisted research program without erasing its valid results. No result here proves the Riemann Hypothesis, HSIT, or physical Super Information Theory.