Let $A$ be a connected $E_{n+1}$-ring spectrum and let $\alpha\in\pi_k(A)$. I am having trouble showing that attaching an $E_n$-cell along $\alpha$ will necessarily not kill an element $\beta\in\pi_k(A)$ unless $\beta$ is a multiple of $\alpha$. Perhaps this is not true, but it seems plausible to me since attaching an $E_n$-cell seems like everything above the cell that kills $\alpha$ itself should be in higher dimensions.

To be precise: we can "attach an $E_n$-$A$-cell" to $A$ along $\alpha$ by taking the pushout of the following span in $E_n$-$A$-algebras: $$A\overset{\overline{0}}\leftarrow F_{E_n}(\Sigma^kA)\overset{\overline{\alpha}}\to A $$ where $F_{E_n}$ is the free $E_n$-$A$-algebra functor, $\overline{0}$ is the adjoint of the zero map $\Sigma^kA\to A$ and $\overline{\alpha}$ is the adjoint of the $A$-module map $\Sigma^kA\to A$ induced by $\alpha$.

It seems to me that there should be a "bottom layer" of $F_{E_n}(\Sigma^kA)$ that kills of $\alpha$, but that everything else (used to kill off the powers of $\alpha$ in a homotopy coherent way) should happen in higher dimensions, so cannot kill $\beta$. Is this true?

It may be useful to notice that, according to Lemma 4.4 of this paper of Antolin-Camarena and Barthel, that the above pushout is equivalent to $Ind_0^n(cof(\Sigma^k A\to A))$ where $Ind_0^n$ is the left adjoint to the forgetful functor from $E_n$-$A$-algebras to $E_0$-$A$-algebras (where $E_0$-$A$-algebras here just means unital $A$-modules).

Let $A$ be a connected $E_{n+1}$-ring spectrum and let $\alpha\in\pi_k(A)$. I am having trouble showing that attaching an $E_n$-cell along $\alpha$ will necessarily not kill an element $\beta\in\pi_k(A)$ unless $\beta$ is a multiple of $\alpha$. Perhaps this is not true, but it seems plausible to me since attaching an $E_n$-cell seems like everything above the cell that kills $\alpha$ itself should be in higher dimensions.

To be precise: we can "attach an $E_n$-$A$-cell" to $A$ along $\alpha$ by taking the pushout of the following span in $E_n$-$A$-algebras: $$A\overset{\overline{0}}\leftarrow F_{E_n}(\Sigma^kA)\overset{\overline{\alpha}}\to A $$ where $F_{E_n}$ is the free $E_n$-$A$-algebra functor, $\overline{0}$ is the adjoint of the zero map $\Sigma^kA\to A$ and $\overline{\alpha}$ is the adjoint of the $A$-module map $\Sigma^kA\to A$ induced by $\alpha$.

It seems to me that there should be a "bottom layer" of $F_{E_n}(\Sigma^kA)$ that kills of $\alpha$, but that everything else (used to kill off the powers of $\alpha$ in a homotopy coherent way) should happen in higher dimensions, so cannot kill $\beta$. Is this true?

It may be useful to notice that, according to Lemma 4.4 of this paper of Antolin-Camarena and Barthel, that the above pushout is equivalent to $Ind_0^n(cof(\Sigma^k A\to A))$ where $Ind_0^n$ is the left adjoint to the forgetful functor from $E_n$-$A$-algebras to $E_0$-$A$-algebras (where $E_0$-$A$-algebras here just means unital $A$-modules).

I should also mention that this is the pretty clearly the best we can do in general, since as Tyler pointed out in the comments it's relatively easy to attach a structured cell along something in degree $k$ and kill something in degree $k+1$. And his example is not rare either. I can construct an infinite family of ring spectra in which this occurs for arbitrarily large degrees, when only attaching $E_1$-$A$-cells.

Let $A$ be an $E_{n+1}$-ring spectrum and let $\alpha\in\pi_k(A)$. I am having trouble showing that attaching an $E_n$-cell along $\alpha$ will necessarily not kill an element $\beta\in\pi_k(A)$ unless $\beta$ is a multiple of $\alpha$. Perhaps this is not true, but it seems plausible to me since attaching an $E_n$-cell seems like everything above the cell that kills $\alpha$ itself should be in higher dimensions.

To be precise: we can "attach an $E_n$-$A$-cell" to $A$ along $\alpha$ by taking the pushout of the following span in $E_n$-$A$-algebras: $$A\overset{\overline{0}}\leftarrow F_{E_n}(\Sigma^kA)\overset{\overline{\alpha}}\to A $$ where $F_{E_n}$ is the free $E_n$-$A$-algebra functor, $\overline{0}$ is the adjoint of the zero map $\Sigma^kA\to A$ and $\overline{\alpha}$ is the adjoint of the $A$-module map $\Sigma^kA\to A$ induced by $\alpha$.

It seems to me that there should be a "bottom layer" of $F_{E_n}(\Sigma^kA)$ that kills of $\alpha$, but that everything else (used to kill off the powers of $\alpha$ in a homotopy coherent way) should happen in higher dimensions, so cannot kill $\beta$. Is this true?

It may be useful to notice that, according to Lemma 4.4 of this paper of Antolin-Camarena and Barthel, that the above pushout is equivalent to $Ind_0^n(cof(\Sigma^k A\to A))$ where $Ind_0^n$ is the left adjoint to the forgetful functor from $E_n$-$A$-algebras to $E_0$-$A$-algebras (where $E_0$-$A$-algebras here just means unital $A$-modules).

Let $A$ be a connected $E_{n+1}$-ring spectrum and let $\alpha\in\pi_k(A)$. I am having trouble showing that attaching an $E_n$-cell along $\alpha$ will necessarily not kill an element $\beta\in\pi_k(A)$ unless $\beta$ is a multiple of $\alpha$. Perhaps this is not true, but it seems plausible to me since attaching an $E_n$-cell seems like everything above the cell that kills $\alpha$ itself should be in higher dimensions.

To be precise: we can "attach an $E_n$-$A$-cell" to $A$ along $\alpha$ by taking the pushout of the following span in $E_n$-$A$-algebras: $$A\overset{\overline{0}}\leftarrow F_{E_n}(\Sigma^kA)\overset{\overline{\alpha}}\to A $$ where $F_{E_n}$ is the free $E_n$-$A$-algebra functor, $\overline{0}$ is the adjoint of the zero map $\Sigma^kA\to A$ and $\overline{\alpha}$ is the adjoint of the $A$-module map $\Sigma^kA\to A$ induced by $\alpha$.

It seems to me that there should be a "bottom layer" of $F_{E_n}(\Sigma^kA)$ that kills of $\alpha$, but that everything else (used to kill off the powers of $\alpha$ in a homotopy coherent way) should happen in higher dimensions, so cannot kill $\beta$. Is this true?

It may be useful to notice that, according to Lemma 4.4 of this paper of Antolin-Camarena and Barthel, that the above pushout is equivalent to $Ind_0^n(cof(\Sigma^k A\to A))$ where $Ind_0^n$ is the left adjoint to the forgetful functor from $E_n$-$A$-algebras to $E_0$-$A$-algebras (where $E_0$-$A$-algebras here just means unital $A$-modules).