Dissolving the Non-Unification Triangle: Asymptotically Safe Gravity and the Abelian Sector of the Cascade

This paper is a companion to From U(1) to U(5): A Bottom-Up Reconstruction of Gauge Unification (Zenodo, DOI: 10.5281/zenodo.20375387), and argues that the Standard Model non-unification triangle inherited by the cascade need not be closed, because asymptotically safe gravity dissolves it.Two independent results combine. First, by exact group theory, the fermionic contribution to every difference of one-loop beta coefficients (bᵢ − bⱼ) vanishes identically: a complete generation fills a 16 of SO(10), so fermions are concurrence-neutral, and the entire non-unification of the triangle resides in the non-Abelian gauge Casimirs and the incomplete Higgs multiplet. This singles out the Casimir-free Abelian factor — the one carrying the Landau pole — on purely group-theoretic grounds. Second, an asymptotically safe gravitational fixed point renders precisely that Abelian coupling irrelevant (hence predicted), while leaving the non-Abelian couplings relevant (hence free). Only one coupling ever needed to be fixed from the deep ultraviolet, and it is exactly the one gravity fixes; the triangle is dissolved, not closed.The paper's one new, checkable computation tests whether the cascade's Abelian charge content — one 16 of SO(10) per generation, with the exact fixed-point relation g*ₓ²/g*ᵧ² = 41/40 — is compatible with the published weak-gravity bound on the Abelian coupling. Conditionally on that bound, which is treated as an external input from a functional renormalization group truncation rather than a result derived here, the cascade falls inside it with a comfortable margin.The mechanism predicts neither sin²θ_W nor proton decay; the programme's falsifiable prediction remains the M_X-independent channel ratio R = Γ(p→ν̄K⁺)/Γ(p→e⁺π⁰) ≈ 0.16, distinct from the R ~ 1–100 of supersymmetric SU(5) and awaiting Hyper-Kamiokande.This is independent research, not peer-reviewed. The author cannot independently verify all of the theoretical physics, and the limitations of the construction are stated explicitly in the text. All numerical and group-theoretic results are reproducible via the accompanying Python scripts (exact rational arithmetic where applicable). Drafted with AI assistance (Claude, Anthropic — Claude Fable 5 during its availability, and Claude Opus 4.8); all bibliography entries were independently verified by the author against primary sources. The author bears sole responsibility for the content.