complexity for $S/p$. Therefore, $\tau_{\ge 0}L_{K(1)}S$ has telescopic complexity one. The red-shift conjecture may therefore be phrased in this case as follows: For any $V$ in $\mathfrak{S}_{1}$, the map $$\label{eq case of red shift} V\wedge K(\tau_{\ge 0}L_{K(1)}S)\rightarrow v_2^{-1}V\wedge K(\tau_{\ge 0}L_{K(1)}S)$$ induces an isomorphism in homotopy in sufficiently high degrees. A conjecture of Adams states that all objects in $\mathfrak{S}_1$ can be constructed as finite cell $V(1)$-modules [@MR2707393]. Assuming the validity of Adams’ conjecture, proving the map (\[eq case of red shift\]) induces an isomorphism in homotopy for $V=V(1)$ implies that $K(\tau_{\ge 0}L_{K(1)}S)$ has telescopic complexity $2$. The author plans to address this question in future work as a natural extension of the results of this paper.

In the present paper, we compute topological Hochschild homology of the connective $K(1)$-local sphere, after smashing with the Smith-Toda complex $V(1)$, as follows.

 \[main thm\] There is an isomorphism of graded rings $$V(1)_*(THH(\tau_{\ge 0}L_{K(1)}S))\cong P(\mu_2)\otimes \Gamma(\sigma b) \otimes \mathbb{F}_p\{\alpha_1, \lambda_1', \lambda_2\alpha_1, \lambda_2\lambda_1', \lambda_2\lambda_1'\alpha_1 \}$$ where the products between the classes $$\{\alpha_1, \lambda_1', \lambda_2\alpha_1, \lambda_2\lambda_1', \lambda_2\lambda_1'\alpha_1 \}$$ are zero except for $$\alpha_1\cdot \lambda_2\lambda_1'=\lambda_1'\cdot \lambda_2\alpha_1 =\lambda_2\lambda_1'\alpha_1.$$

The paper is divided into two sections. Section \[section 2\] gives a brief summary of the construction of the THH-May 