Suppose we have the following production function:
$$F(L,K)=\max_{L_K}H(L,L_K,K)=\max_{L_K}\left[(L-L_K+1)^\alpha(L_K+K)^{1-\alpha}\right]=(L-L_K^*+1)^\alpha(L_K^*+K)^{1-\alpha}$$
With the constraint $L_K\in[0,L]$.
We know that $$\frac {dH}{dL_K}=\alpha(L-L_K+1)^{-1}H+(1-\alpha)(L_K+K)^{-1}H=0$$ Hence the value for $L_K$ at which the derivative is zero is $L_K^0=\frac {(1-\alpha)(L+1)+\alpha K}{1-2\alpha}$. And the optimal value $L_K^*$ is: $$ L_K^*=\begin{cases} L_K^0 &\text{ if } &0<L_K<L &(1)\\ L&\text { if } &L<L_K^0&(2)\\ 0 &\text { if } &L_K^0<0 &(3) \end{cases} $$
It is clear that if $L_K^*\in(0,L)$, (case $(1)$ ), then the envelope theorem holds:
$$\frac d {dL} F(L,K)=\frac \partial {\partial L}H(L,L_K^*,K)=\alpha(L-L_K^*+1)^{-1}\cdot F(L,K)$$
Moreover, in the third case (3), it is also clear to me that the envelope theorem holds. However, I am not so sure about the second case (2). I would say that the envelope theorem does not hold in this case, because if we substitute $L_K^*$ back into the original production function, we get $$F(L,K)=1^\alpha(L+K)^{1-\alpha}$$ And the derivative with respect to $L$ in this case is $$ (1-\alpha)(L+K)^{-1}\cdot F(L,K)$$
For the envelope theorem to hold in case 3, this would require $\alpha= (1-\alpha)(L+K)^{-1}$, which Almost-Always doesn't hold.
But the reason this confuses me is that in this question I was referred to this paper, which has a theorem that states:
So my question is:
Am I right that the envelope theorem doesn't hold when $L_K$ is at a corner solution?
Does this contradict the theorem, or do I misunderstand the theorem? If not, is the theorem correct?