Uniform Boundedness of Homogeneous Incompressible Flows in R3

The paper’s core step asserts there exists a scalar θ in the nonzero-vorticity region R^3 \ Ω_t such that ∇p = u × ∇θ, hence u · ∇p = 0, which is then used to deduce D_t |u|^2 = 0 for Euler and D_t |u|^2 ≤ μ Δ|u|^2 for Navier–Stokes, yielding an L∞ maximum principle and global regularity. However, the paper only derives the divergence identity Δp = ∇θ · ω and then (incorrectly) promotes it to the full vector identity ∇p = u × ∇θ; the necessary curl/integrability conditions for this promotion are neither verified nor generally true. This invalidates Theorems 5.1 and 6.1 and the main results (Theorems 2.1 and 2.2) claiming uniform boundedness and global continuation (citations to the paper’s statements and derivations: ; orthogonality construction and claimed conclusion u · ∇p = 0: ; Euler and Navier–Stokes ‘uniform boundedness’ theorems and their proofs rely on this: ). The candidate model correctly identifies that a scalar maximum principle for |u| fails due to the uncontrolled w · ∇Q term (with Q = p + |u|^2/2) and that establishing the paper’s claims would resolve long-standing open problems (BKM, Prodi–Serrin, Clay NS problem), which is not plausible as of the cutoff.