Analytic bounds on late-time axion-scalar cosmologies

The paper states and (in App. A) rigorously proves the late‑time bounds for axion–scalar cosmologies under three geometric regimes: antialignment (γ∞·λr<0), alignment with πr≥0 and componentwise nonnegativity, and partial misalignment. In particular it establishes ϵ=(d−2)/4 (γ∞)² when (γ∞)²≤Γ(d)², and ϵ→d−1 when (γ∞)²≥Γ(d)² in the antialignment case (eqs. (II.6)–(II.7)) ; bounds ϵ≤(d−2)/4 (Γ̃)² and ϵ≤(d−2)/4 (Γ·Γ̃) under alignment hypotheses (eqs. (II.8)–(II.13)) ; and kination ϵ=d−1 under partial misalignment (eq. (II.14)) . The model’s outline reproduces these final conclusions but relies on several unjustified or incorrect steps: (i) in A1 it tries to exclude V/H²→0 by bounding 2(d−1)−g·u using |g|≥|γ∞|≤Γ(d), which goes in the wrong direction and does not control g away from the minimal face; (ii) in C it asserts a directional lower bound g·u≥(γa*+Λa*)|ua*| that does not follow from convexity and mixes the axion couplings into g·u without justification; and (iii) in B it assumes eventual sign‑definiteness u_a≥0 from a heuristic comparison argument. The paper’s statements and proofs (summarized around eqs. (II.2)–(II.5) and detailed in App. A) are consistent and complete within their stated hypotheses, whereas the model’s derivations contain gaps .