Let $I \subset A$ be the kernel of $A \to B$. Let $\mathcal{J} \subset \mathcal{O}$ be the kernel of $\mathcal{O} \to \underline{B}$. Then $I\mathcal{O} \subset \mathcal{J}$. Set $\mathcal{K} = \mathcal{J}/I\mathcal{O}$ and $\overline{\mathcal{O}} = \mathcal{O}/I\mathcal{O}$. \medskip\noindent For every object $U = (P \to B)$ of $\mathcal{C}_{B/A}$ we can choose an isomorphism $P = A[E]$ such that the map $P \to B$ maps each $e \in E$ to zero. Then $J = \mathcal{J}(U) \subset P = \mathcal{O}(U)$ is equal to $J = IP + (e; e \in E)$. Moreover $\overline{\mathcal{O}}(U) = B[E]$ and $K = \mathcal{K}(U) = (e; e \in E)$ is the ideal generated by the variables in the polynomial ring $B[E]$. In particular it is clear that $$ K/K^2 \xrightarrow{\text{d}} \Omega_{P/A} \otimes_P B $$ is a bijection. In other words, $\Omega = \mathcal{K}/\mathcal{K}^2$ and $\text{Sym}_B^k(\Omega) = \mathcal{K}^k/\mathcal{K}^{k + 1}$. Note that $\pi_!(\Omega) = \Omega_{B/A} = 0$ (Lemma \ref{lemma-identify-H0}) as $A \to B$ is surjective (Algebra, Lemma \ref{algebra-lemma-trivial-differential-surjective}). By Lemma \ref{lemma-vanishing-symmetric-powers} we conclude that $$ H_i(\mathcal{C}_{B/A}, \mathcal{K}^k/\mathcal{K}^{k + 1}) = H_i(\mathcal{C}_{B/A}, \text{Sym}^k_{\underline{B}}(\Omega)) = 0 $$ for $i < k$. This proves the final statement of the theorem. \medskip\noindent The approach to the theorem is to note that $$ B \otimes_A^\mathbf{L} B = L\pi_!(\mathcal{O}) \otimes_A^\mathbf{L} B = L\pi_!(\mathcal{O} \otimes_{\underline{A}}^\mathbf{L} \underline{B}) = L\pi_!(\overline{\mathcal{O}}) $$ The first equality by Lemma \ref{lemma-apply-O-B-comparison}, the second equality by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-change-of-rings}, and the third equality as $\mathcal{O}$ is flat over $\underline{A}$. The sheaf $\overline{\mathcal{O}}$ has a filtration $$ \ldots \subset \mathcal{K}^3 \subset \mathcal{K}^2 \subset \mathcal{K} \subset \overline{\mathcal{O}} $$ This induces a filtration $F$ on a complex $C$ representing $L\pi_!(\overline{\mathcal{O}})$ with $F^pC$ representing $L\pi_!(\mathcal{K}^p)$ (construction of $C$ and $F$ omitted). Consider the spectral sequence of Homology, Section \ref{homology-section-filtered-complex} associated to $(C, F)$. It has $E_1$-page $$ E_1^{p, q} = H_{- p - q}(\mathcal{C}_{B/A}, \mathcal{K}^p/\mathcal{K}^{p + 1}) \quad\Rightarrow\quad H_{- p - q}(\mathcal{C}_{B/A}, \overline{\mathcal{O}}) =  \text{Tor}_{- p - q}^A(B, B) $$ and differentials $E_r^{p, q} \to E_r^{p + r, q - r + 1}$. To show convergence we will show that for every $k$ there exists a $c$ such that $H_i(\mathcal{C}_{B/A}, \mathcal{K}^n) = 0$ for $i < k$ and $n > c$\footnote{A posteriori the ``correct'' vanishing $H_i(\mathcal{C}_{B/A}, \mathcal{K}^n) = 0$ for $i < n$ can be concluded.}. \medskip\noindent Given $k \geq 0$ set $c = k^2$. We claim that $$ H_i(\mathcal{C}_{B/A}, \mathcal{K}^{n + c}) \to H_i(\mathcal{C}_{B/A}, \mathcal{K}^n) $$ is zero for $i < k$ and all $n \geq 0$. Note that $\mathcal{K}^n/\mathcal{K}^{n + c}$ has a finite filtration whose successive quotients $\mathcal{K}^m/\mathcal{K}^{m + 1}$, $n \leq m < n + c$ have $H_i(\mathcal{C}_{B/A}, \mathcal{K}^m/\mathcal{K}^{m + 1}) = 0$ for $i < n$ (see above). Hence the claim implies $H_i(\mathcal{C}_{B/A}, \mathcal{K}^{n + c}) = 0$ for $i < k$ and all $n \geq k$ which is what we need to show. \medskip\noindent Proof of the claim. Recall that for any $\mathcal{O}$-module $\mathcal{F}$ the map $\mathcal{F} \to \mathcal{F} \otimes_\mathcal{O}^\mathbf{L} B$ induces an isomorphism on applying $L\pi_!$, see Lemma \ref{lemma-O-homology-B-homology}. Consider the map $$ \mathcal{K}^{n + k} \otimes_\mathcal{O}^\mathbf{L} B \longrightarrow \mathcal{K}^n \otimes_\mathcal{O}^\mathbf{L} B $$ We claim that this map induces the zero map on cohomology sheaves in degrees $0, -1, \ldots, - k + 1$. If this second claim holds, then the $k$-fold composition $$ \mathcal{K}^{n + c} \otimes_\mathcal{O}^\mathbf{L} B \longrightarrow \mathcal{K}^n \otimes_\mathcal{O}^\mathbf{L} B $$ factors through $\tau_{\leq -k}\mathcal{K}^n \otimes_\mathcal{O}^\mathbf{L} B$ hence induces zero on $H_i(\mathcal{C}_{B/A}, -) = L_i\pi_!( - )$ for $i < k$, see Derived Categories, Lemma \ref{derived-lemma-trick-vanishing-composition}. By the remark above this means the same thing is true for $H_i(\mathcal{C}_{B/A}, \mathcal{K}^{n + c}) \to H_i(\mathcal{C}_{B/A}, \mathcal{K}^n)$ which proves the (first) claim. \medskip\noindent Proof of the second claim. The statement is local, hence we may work over an object $U = (P \to B)$ as above. We have to show the maps $$ \text{Tor}_i^P(B, K^{n + k}) \to \text{Tor}_i^P(B, K^n) $$ are zero for $i < k$. There is a spectral sequence $$ \text{Tor}_a^P(P/IP, \text{Tor}_b^{P/IP}(B, K^n)) \Rightarrow \text{Tor}_{a + b}^P(B, K^n), $$ see More on Algebra, Example \ref{more-algebra-example-tor-change-rings}. Thus it suffices to prove the maps $$ \text{Tor}_i^{P/IP}(B, K^{n + 1}) \to \text{Tor}_i^{P/IP}(B, K^n) $$ are zero for all $i$. This is Lemma \ref{lemma-map-tors-zero}.