$S/p \wedge E_0^*\mathbb{J}$ is an $H\mathbb{F}_p$-algebra.

mod $(p,v_1)$-homotopy of $THH(\ell_{K(1)}S)$ {#section 3}
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In Section \[section 2\], we reviewed the construction that takes a decreasingly filtered commutative monoid $I$ in $\mathfrak{S}$ as input and produces a May-type spectral sequence $$E^2_{s,t}=E_{s,t}THH(E_0^*I)\Rightarrow E_{s}THH(I_0)$$ for any connective generalized homology theory $E$, which we we call the $E$-THH-May spectral sequence. Also, in Section \[section 2\] we produced a Whitehead-type decreasingly filtered commutative monoid in $\mathfrak{S}$, denoted $\mathbb{J}$, associated to a cofibrant commutative ring spectrum model for $p$-complete connective image of J, which we denote $j$. We therefore have a spectral sequence $$~\label{Ethhmayj} E^2_{s,t}=E_{s,t}THH(E_0^*\mathbb{J})\Rightarrow H_{s}THH(j).$$ The purpose of this section is to compute this spectral sequence in the case $E=V(1)$.

 Computing the $H\mathbb{F}_p$-THH-May spectral sequence
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In the case where $E=H\mathbb{F}_p$, the input of the spectral sequence is calculable, and the output is already known due the work of Angeltveit-Rognes [@MR2171809]. This computation, therefore, will allow us to see the differentials in the $V(1)$-THH-May spectral sequence that are also detected in the $H\mathbb{F}_p$-THH-May spectral sequence. To begin, let us recall the computation of Angeltveit-Rognes.

 \[HFpj\] There is an isomorphism $${H\mathbb{F}_p}_*(j)\cong P(\tilde{\xi}_1^p,\tilde{\xi}_2,\bar{\xi}_3,...)\otimes E(\tilde{\tau}_2,\bar{\tau}_3, ...)\otimes E(b) \cong (\mathcal{A}//A(1))_*\otimes E(b)$$ where all the elements in $(\mathcal{A}//A(1))_*$ besides $\tilde{\tau}_2$, 