Log syntomic cohomology of truncated polynomials and coordinate axes

We study the logarithmic syntomic cohomology of fine and saturated log schemes and its realization in the logarithmic motivic stable homotopy category $\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{S}\mathrm{H}({\mathrm{p}\mathrm{t}}_{\mathbb{N}})$ of a log point. We prove that logarithmic prismatic and syntomic cohomology satisfy saturated descent under the sole assumption that the log structure is free, and that the presheaves $\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{T}\mathrm{H}\mathrm{H}$, $\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{T}\mathrm{C}$, $\widehat{\mathbf{\Delta }}$, and ${\mathbb{Z}}_p^{\mathrm{s}\mathrm{y}\mathrm{n}}(i)$ are representable and $\mathrm{\square }$-invariant in ${\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{S}\mathrm{H}}_{\mathrm{k}\acute{\mathrm{e}}\mathrm{t}}^{\mathrm{e}\mathrm{f}\mathrm{f}}({\mathrm{p}\mathrm{t}}_{\mathrm{N}})$. As an application, we compute ${\mathbb{Z}}_p^{\mathrm{s}\mathrm{y}\mathrm{n}}(i)$ for the projective log coordinate axes $D$ in ${\mathbb{P}}^2$, obtaining $${\mathbb{Z}}_p^{\mathrm{s}\mathrm{y}\mathrm{n}}(i)(D)\simeq {\mathbb{Z}}_p^{\mathrm{s}\mathrm{y}\mathrm{n}}(i)(k,\mathbb{N})\oplus {\mathbb{Z}}_p^{\mathrm{s}\mathrm{y}\mathrm{n}}(i-1)(k,\mathbb{N})[-2]$$ Moreover, we determine logarithmic topological cyclic homology for truncated polynomial and semistable examples, directly from the syntomic calculations.