Rasmussen's s-invariant bounds smooth four-ball genus

For the standard crossingless unknot diagram, the Lee complex is the one-term vector space $A_L=\mathbb{Q}[X]\mathrm{/}(X^2-1)$ with basis $1,X$ in quantum degrees $1,-1$ and zero differential. Its induced filtration levels are therefore $s_{\max }(U)=1$ and $s_{\min }(U)=-1$, so [F2] gives $s(U)=0$.

Let $F\subset B^4$ be any connected smooth orientable proper spanning surface of genus $g$ with boundary $K$. Apply [F4] under the countable-choice hypothesis [A1]. It moves and flattens a small interior surface disk by ambient diffeomorphisms fixed on a sufficiently small restriction of the supplied outer collar, then removes an actual concentric round ball. The remaining round shell is explicitly $S^3\times I$, with unchanged boundary knot $K$, an inner standard unknot and product collars. Its surface is connected of the same genus $g$ and has Euler characteristic $-2g$; reverse its direction if needed. This uses neither a slice-only Statement for a positive-genus surface nor a claim about arbitrary embedded four-ball complements.

Apply the Lee cobordism construction to this surface and to its reverse. By [F3], each map is a filtered isomorphism of degree $-2g$. Choose a nonzero class of maximal level in $V_K$; its image is nonzero in $V_U$ and has level at least $s_{\max }(K)-2g$, which is at most $s_{\max }(U)=1$. Hence $s_{\max }(K)\le 1+2g$ and $s(K)\le 2g$. Apply the reverse map to a nonzero class of maximal level in $V_U$: its nonzero image in $V_K$ has level at least $1-2g$, so $s_{\max }(K)\ge 1-2g$ and $s(K)\ge -2g$.

Under [A1], [F1] supplies a nonempty set of smooth spanning-surface genera with a least element. Minimize $g$ over $F$ in step 1.2. Step 2.1 then gives the asserted inequality. This gives the inequality in the Statement.