Note

Failure of weak chromatic splitting

Added: October 7, 2026

Publication Note

We plan to release a short expository note soon, explaining the argument using the algebraic model given by

[\operatorname{Spd}(\mathbb{F}_p)/G_n] \xleftarrow{\,f_n\,} [\operatorname{Spd}(\widehat{L})/G_{n,n-1}] \xrightarrow{\,f_{\mathrm{target}}\,} [\operatorname{Spd}(\mathbb{F}_p)/(\Gamma\times G_{n-1})].

We are making the longer document available now because OpenAI has already published a solution.

Abstract

Let $n\geq 3$ and let $p$ be any prime. We prove that the canonical map $L_{n-1}S_p\to L_{n-1}L_{K(n)}S_p$ has no left inverse. The proof follows the source Cartan class in rational degree $-3$ . A degree-three transfer of constant coefficients factors its localized image through the actual target coefficient map. Witt coefficients remove the extra principal-unit direction. A family of positive modifications detects the resulting target class: quotienting by a trivial line maps this family directly to the coheight-one correspondence. The coefficient comparison uses only low degrees, while rapid convergence supplies the passage to periodic spectra.