Spectrum-level identification of R^sfr with the delooped J-homomorphism fiber

Establish a homotopy equivalence of ring spectra R^sfr ≃ Σ^∞ hofib(BJ), where BJ denotes the 1-fold delooping of the real J-homomorphism.

Background

The construction of the parameterized Lagrangian Floer homotopy type yields an endomorphism ring spectrum R^sfr governing spherically framed flow categories. The authors anticipate that R^sfr encodes spherically framed bordism and is closely tied to the real J-homomorphism.

Identifying R^sfr with the suspension spectrum of the homotopy fiber of BJ would provide a conceptual and computational bridge between Floer homotopy structures and classical stable homotopy via the J-homomorphism, clarifying the intrinsic tangential structure underlying spherically framed flow categories.