The arXiv preprint (v1, 2 Jun 2026) introduces an algebraic-diagrammatic construction (ADC) reformulation of the G3W2 vertex correction to the GW self-energy, addressing the well-known violation of positive semi-definiteness that produces negative spectral functions in standard perturbative treatments. Unlike the original G3W2, which expands the self-energy to second order in the screened interaction W, the ADC-G3W2 family enforces an exact sum-over-states structure, yielding Hermitian effective Hamiltonians whose eigenvalues guarantee physical quasiparticle and satellite spectra. This creates a formal bridge between W-based many-body perturbation theory and traditional ADC schemes that start from the bare Coulomb operator. The hierarchy—ADC-2SOSEX, ADC(3)-G3W2, and full ADC-G3W2—represents nonperturbative resummations of diagrams, extending beyond the source's claim by linking to earlier work on conserving approximations (Baym & Kadanoff, 1961) and diagrammatic resummation techniques in quantum chemistry (e.g., the ADC(3) ionization-potential studies of Schirmer et al.). The paper benchmarks valence ionization potentials but provides no statistical sample size, as it remains a formal methodological development rather than a large-scale application study; limitations include the absence of core-level or solid-state benchmarks and the computational scaling that will likely restrict full ADC-G3W2 to small molecules until further approximations are introduced. Because this is an unreviewed preprint, independent validation is still required. The advance matters for predictive materials and catalyst design, where unphysical poles have historically forced ad-hoc fixes in codes such as BerkeleyGW and Yambo.