Actual EM orientation and projective bundle maps on that model

Shift the Picard map in [F1] by2. Since $\mathrm{P}\mathrm{i}\mathrm{c}(Y)\simeq {\mathrm{\Omega }}^{\mathrm{\infty }}R{\mathrm{\Gamma }}_{\mathrm{N}\mathrm{i}\mathrm{s}}(Y,{\mathbb{G}}_m)[1]$, this gives a pointed map $\mathrm{P}\mathrm{i}\mathrm{c}\to {\mathrm{\Omega }}^{\mathrm{\infty }}{A}_1$. Multiplication gives the associated module map $\mathrm{P}\mathrm{i}\mathrm{c}\otimes {\mathcal{E}}_S\to {\mathbb{P}}^1\otimes {\mathcal{E}}_S$. On the universal $\mathcal{O}(1)$ this is exactly the bonding class used in the model; by its Ω-condition it restricts to the identity under P¹ suspension. Thus it is the linear orientation of [F2], with the original EM first Chern class, not an arbitrarily chosen group isomorphism.

The actual complex PBF in [F1] is compatible with multiplication and the bonding class. Levelwise its reduced P-space formula is the source PBF for the stable coefficient model. Negative level indices that occur are zero: reduced weight-zero P¹ cohomology is zero by [F1], so looping $A_0$ by P¹ is zero, and further negative shifts remain zero. Thus the levelwise formulas also cover the low-level boundary, not only sufficiently large levels.

Orientability and [F2] identify the space and stable coefficient models on fundamental spectra, so the formula of step 2.1 is the required equivalence of oriented motivic modules. Its products and tautological class are those in step 1.1. The general bundle version follows from the source PBF or its local descent statement, preserving the same class. No affine-torsor detection or homotopy invariance has been assumed.