TITLE: Acceptability and Soundness of J-structures.
QUESTION [7 upvotes]: I would like an example of a J-structure $(J^A,B)$ which is not acceptable and one that is not 1-sound.
Edit:Let us recall that a structure $J^A_\alpha$ is acceptable if for every limit ordinal $ \xi<\alpha $. $J^A_{\xi+\omega}\models \vert \xi\vert\leq \vert \tau\vert $, whenever $\tau<\xi$ and  satisfies ${\mathcal P}(\tau)\cap J^A_{\xi+\omega}\not \subset J^A_\xi $
A structure $J^A_\alpha$ is 1-sound if the 1-standard parameter is a very good parameter.
(I'm using the notation in Zeman's article in the handbook of set theory)
It is known that $J_\alpha$ are acceptable and sound for every ordinal $\alpha$. Moreover, being acceptable and sound is needed for almost all basic results concerning the $J^A_\alpha$ hierarchy.

REPLY [4 votes]: For an amenable $(J,B)$ which is not 1-sound, take a non-constructible real $x$ such that $\aleph_1^L = \aleph_1^{L[x]}$ (and let's say $V = L[x]$ so this is the true $\aleph_1$). Set $B = \lbrace\omega_1+n:n \in x\rbrace$. Then $(J_{\omega_1+1},B)$ is amenable and $x$ is $\Sigma_1(J_{\omega_1+1},B)$ (with parameter $\omega_1$). The $\Sigma_1$-projectum is therefore $1$, so $(J_{\omega_1+1},B)$ cannot be $1$-sound because $J_{\omega_1+1}$ is uncountable and the available parameter set is countable.
(My notes say that this example is from Lee Stanley, but they don't say where I found it. If anybody knows where this is from, please leave a comment.)