Stable Thom detector coordinates commute with suspension

Fix n≥0 and the degree-n part B_n of the single global basis fixed in the detector definition. For every r≥n+2, define D_{r,n}:π_{r+n}(T_r)→V_n, V_n=F₂^{B_n}, by pairing with the stable coordinates m_{r,b}, b∈B_n. The finite-range theorem says D_{r,n} is an isomorphism. This is the same map as the π_{r+n} map of f_r after identifying π_{r+n}(P_r) with V_n by its normalized Eilenberg–Mac Lane fundamental classes.

Let β_r:S¹∧T_r→T_{r+1} be the prespectrum structure map, and let b_{r,n} be its induced stabilization map on π_{r+n}. Since m_b is an inverse-limit element, its coordinates satisfy σ⁻¹β_r^* m_{r+1,b}=m_{r,b}. For a based representative α:S^{r+n}→T_r, naturality of the Kronecker pairing gives ⟨m_{r+1,b}, (β_r∘(1∧α))*[S¹∧S^{r+n}]{F₂}⟩ =⟨β_r^* m_{r+1,b}, (1∧α)*[S¹∧S^{r+n}]{F₂}⟩ =⟨m_{r,b}, α_*[S^{r+n}]{F₂}⟩. All sphere classes in these pairings are mod-two fundamental classes, not unreduced integral classes. For the second equality, represent the cohomology suspension of m{r,b} by its external product with the degree-one generator of S¹. Under S¹∧S^{r+n}≅S^{r+n+1}, the mod-two sphere fundamental class is the external product of the two mod-two fundamental classes. Evaluation of external products on product chains is the product of the two evaluations; the S¹ factor evaluates to 1. Naturality of the Kronecker pairing then gives exactly the displayed equality. These are the published suspension, external-product/Künneth, and Kronecker naturality interfaces; coefficients are F₂, so there is no sign ambiguity. Thus $D_{r+1,n}{b}_{r,n}=D_{r,n}$ coordinate by coordinate, proving the claimed suspension compatibility.