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Whether the analogous number drops in five dimensions is unknown.","posed_since":"2010","precise":"For a unitary $d$-dimensional CFT define F~ = sin(pi d/2) log $Z$(S^d), with Z the renormalized partition function on the round sphere (Giombi and Klebanov 2014); in $d = 5$ this is F~ = log Z(S^5). Prove or refute F~_UV > F~_IR for every unitary RG flow between 5D CFTs. The 3D case was proven from strong subadditivity of entanglement entropy (Casini and Huerta 2012); that argument has not been extended to $d = 5$.","problem_ref":null,"references":"","settled_by":"A general proof for unitary $5\\mathrm{D}$ flows, or a unitary $5\\mathrm{D}$ flow (for example a $5\\mathrm{D}$ SCFT deformed into a gauge theory) with F~_UV <= F~_IR.","status_note":"Consistent with free-field, large-N and $\\epsilon$-expansion examples (2014 onward); no general proof as of 2026.","title":"Does the sphere free energy decrease along five-dimensional RG 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