What ARG Actually Discovered: A Structured Inventory of Results in the Nyman–Beurling–Báez-Duarte Framework

This work presents a structured inventory of results obtained by the Alliance Research Group in the study of the Nyman–Beurling–Báez-Duarte (NBBD) criterion for the Riemann Hypothesis.

We reformulate convergence in NBBD as a sequence of geometric projection steps governed by alignment angles, linking the global convergence problem to the behavior of a local, measurable quantity.

The document distinguishes between:

• proven results (mathematical theorems),
• new empirical observations,
• validated predictions,
• corrections of previously reported claims,
• and precisely formulated open questions.

Key findings include:

• a new identity relating convergence to alignment angles,
• strong concentration of convergence contributions from primes (~82%),
• evidence of non-random structure partially explained by zeta zeros (~44% variance),
• and the emergence of a stable lower envelope in alignment magnitude.

We do not claim to prove or disprove the Riemann Hypothesis. Instead, we provide a new geometric and computational framework, supported by exact numerical methods, which maps previously unexplored structure and identifies critical directions for further research.

All computations are performed using an exact analytical solver (no quadrature approximation), with reproducibility validated across independent systems.