On $\eta$-periodic Formal Ternary Laws

We study the algebraic structure underlying Sp-orientations in the $\eta$-periodic motivic stable homotopy category $SH(k)[{\eta }^{-1}]$. Borel classes determine a geometric formal ternary law, but the HW-Hurewicz map shows that its universal coefficients generate a proper subring $\mathrm{\Lambda }⊊(MSp[{\eta }^{-1}]{)}_{\ast }$, although $\mathrm{\Lambda }[1\mathrm{/}2]=(MSp[{\eta }^{-1}]{)}_{\ast }[1\mathrm{/}2]$. Thus the failure of classification is purely 2-primary. To capture part of the missing information, we introduce framed involutions. The spectrum $MSp[{\eta }^{-1}]$ carries a canonical framed involution, yielding a Quillen-type idempotent with telescope $MSL[{\eta }^{-1}]$ and a canonical splitting $(MSp[{\eta }^{-1}]{)}_{\ast }\cong {\mathcal{R}}_{fr}{\otimes }_{\mathbb{Z}}(MSL[{\eta }^{-1}]{)}_{\ast }$, where ${\mathcal{R}}_{fr}$ is the universal ring of framed involutions. We then axiomatize formal ternary laws, construct the universal Walter ring ${\mathcal{W}}^{\eta }$, and prove that ${\mathcal{W}}^{\eta }$ is isomorphic to the Lazard ring L after inverting 2. If $W(k)\cong \mathbb{Z}$, the universal geometric formal ternary law together with the canonical framed involution induces a classifying map $\phi :{\mathcal{W}}^{\eta }\to (MSp[{\eta }^{-1}]{)}_{\ast }$ that is injective and becomes an isomorphism after inverting 2. Integrally, however, additional secondary power series are needed to recover the full orientation data.