The confusion seems to arise from your statement "the mantle of $V$ which is the smallest ground for $V$." But this not quite right.

In a bottomless model of ZFC, the mantle is not a ground. It is the intersection of all the grounds, but it isn't one of them. In a bottomless model, one can just keep going to deeper and deeper ground models, without ever landing at a minimal ground.

One can easily make a model like this by forcing over $L$, say, with the Easton product class forcing to add a Cohen subset to every regular cardinal. The model arising from any final segment of this forcing (from $\kappa$ and above) is a ground of $L[G]$, since only the set-sized forcing below $\kappa$ is missing. Every ground model of $L[G]$ will contain some final segment of the overall forcing, and one can always peel off a few more factors. So the mantle of $L[G]$ will be $L$ itself.

In the general case, we proved in the geology paper that every model of ZFC arises as the mantle of another model of ZFC.

• Fuchs, Gunter; Hamkins, Joel David; Reitz, Jonas, Set-theoretic geology, Ann. Pure Appl. Logic 166, No. 4, 464-501 (2015). ZBL1348.03051.

The confusion seems to arise from your statement "the mantle of $V$ which is the smallest ground for $V$." But this not quite right.

In a bottomless model of ZFC, the mantle is not a ground. It is the intersection of all the grounds, but it isn't one of them. In a bottomless model, one can just keep going to deeper and deeper ground models, without ever landing at a minimal ground.

One can easily make a bottomless model by forcing over $L$, say, with the Easton product class forcing to add a Cohen subset to every regular cardinal. The model arising from any final segment of this forcing (from $\kappa$ and above) is a ground of $L[G]$, since only the set-sized forcing below $\kappa$ is missing. Every ground model of $L[G]$ will contain some final segment of the overall forcing, and one can always peel off a few more factors. So the mantle of $L[G]$ will be $L$ itself.

In the general case, we proved in the geology paper that every model of ZFC arises as the mantle of another model of ZFC.

• Fuchs, Gunter; Hamkins, Joel David; Reitz, Jonas, Set-theoretic geology, Ann. Pure Appl. Logic 166, No. 4, 464-501 (2015). ZBL1348.03051.

Note that a bottomless model can have no extendible cardinal, because by Usuba's theorem the mantle would be a ground, so it wouldn't be bottomless.

In a bottomless model of ZFC, the mantle is not a ground. One can just keep going to deeper and deeper ground models, but there is no minimal ground.

One can easily make a model like this by forcing over $L$, say, with the Easton product class forcing to add a Cohen subset to every regular cardinal. The model arising from any final segment of this forcing (from $\kappa$ and above) is a ground of $L[G]$, since only the set-sized forcing below $\kappa$ is missing. Every ground model of $L[G]$ will contain some final segment of the overall forcing, and one can always peel off a few more factors. So the mantle of $L[G]$ will be $L$ itself.

In the general case, we proved in the geology paper that every model of ZFC arises as the mantle of another model of ZFC.

• Fuchs, Gunter; Hamkins, Joel David; Reitz, Jonas, Set-theoretic geology, Ann. Pure Appl. Logic 166, No. 4, 464-501 (2015). ZBL1348.03051.

The confusion seems to arise from your statement "the mantle of $V$ which is the smallest ground for $V$." But this not quite right.

In a bottomless model of ZFC, the mantle is not a ground. It is the intersection of all the grounds, but it isn't one of them. In a bottomless model, one can just keep going to deeper and deeper ground models, without ever landing at a minimal ground.

One can easily make a model like this by forcing over $L$, say, with the Easton product class forcing to add a Cohen subset to every regular cardinal. The model arising from any final segment of this forcing (from $\kappa$ and above) is a ground of $L[G]$, since only the set-sized forcing below $\kappa$ is missing. Every ground model of $L[G]$ will contain some final segment of the overall forcing, and one can always peel off a few more factors. So the mantle of $L[G]$ will be $L$ itself.

In the general case, we proved in the geology paper that every model of ZFC arises as the mantle of another model of ZFC.

• Fuchs, Gunter; Hamkins, Joel David; Reitz, Jonas, Set-theoretic geology, Ann. Pure Appl. Logic 166, No. 4, 464-501 (2015). ZBL1348.03051.