$I_{a,b,c,\alpha,\beta,\gamma}$ has parameters as in Proposition \[pro:semistable\] and $s$ is an integer, then the algebras $R/I_{a,b,c,\alpha,\beta,\gamma}$ always have twin peaks and the peaks are bounded by the socle degrees. This extends the results in [@MMN Lemma 7.1] wherein the level algebras $R/I_{a,b,c,\alpha,\beta,\gamma}$ with twin peaks are identified.

\[lem:twin-peaks\] Assume the parameters of $I = I_{a,b,c,\alpha,\beta,\gamma}$ satisfy the conditions in Proposition \[pro:semistable\] and suppose $a + b + c + \alpha + \beta + \gamma \equiv 0 \pmod{3}$. Then $R/I$ has twin peaks in degrees $s$ and $s+1$. Moreover, $s+1$ is bounded above by the socle degrees of $R/I$.

The upper bounds on $A, B,$ and $C$ are exactly those required to force the ultimate and penultimate terms in the minimal free resolution of $R/I$, given in Proposition \[pro:amaci-props\](iii), to not contribute to the computation of the $h$-vector for degrees up to $s+1$. Moreover, as $A,B,C,$ and $M$ are non-negative, and using $\binom{n+1}{2} - \binom{n}{2} = n$ for $n \geq 0$, then $$\begin{split}
            h_{s+1} - h_s = & \left(\binom{s+3}{2} - \binom{A+1}{2} - \binom{B+1}{2} - \binom{C+1}{2} - \binom{M+1}{2}\right) \\
                            &  - \left(\binom{s+2}{2} - \binom{A}{2} - \binom{B}{2} - \binom{C}{2} - \binom{M}{2}\right) \\
                          = & s+2 - (A + B + C + M) 