The finite Thom detector is a mod-two cohomology isomorphism below 2r

For k<r, the Thom-cell description gives H̃^k(T_r)=0. Every factor K(F₂,r+d_b) is (r−1)-connected, so the finite product has zero reduced cohomology in degrees below r. Thus f_r^* is an isomorphism in this range.

Now let r≤k<2r and put e=k−r, so 0≤e≤r−1. The stable-coordinate isomorphism gives H̃^k(T_r;F₂) ≅ M^e = ⊕_{b∈B(r), d_b≤e} A^{e−d_b}m_b. The last equality is the graded free-module decomposition; generators with d_b>e cannot contribute because A has no negative degrees.

The strict metastable theorem gives, strictly for 0≤i<q, H̃^{q+i}(K(F₂,q);F₂) ≅ A^i, a ↦ a(ι_q). For a factor indexed by b∈B_d, q=r+d. In total degree k<2r, its operation degree is i=k−q=e−d. When i≥0, i ≤ r−d−1 < r+d=q, so the strict Eilenberg–Mac Lane theorem applies. A product of two positive-degree polynomial generators, whether in one factor or in two factors, has degree at least 2r because every q≥r. Hence below 2r the cohomology of the finite product P_r is the direct sum of the single-factor operation classes ${\tilde{H}}^k(P_r;{\mathbb{F}}_2)\cong {⨁}_{b\in B(r),d_b\le e}{\mathcal{A}}^{e-d_b}$.

H̃^k(P_r;F₂) ≅ ⊕_{b∈B(r), d_b≤e} A^{e−d_b}. Here finite-product Künneth applies: each factor has finite-dimensional F₂ homology in every degree by the finite-type lemma, hence finite-free homology over F₂; the polynomial presentation of the Eilenberg–Mac Lane cohomology gives the stated cohomology basis. The classifying-map evaluation identity gives $f_r^{\ast }(a({\iota }_{r+d_b}))=a(m_{r,b})$.

Under the stable-coordinate identification, the right side is exactly a·m_b. The free-module decomposition above says these classes form a basis of H̃^k(T_r). Thus f_r^* is an isomorphism for every k<2r. The endpoint is intentionally excluded. At degree 2r, products of two degree-r classes can occur in P_r, and the strict Eilenberg–Mac Lane computation does not identify them by the argument above. No claim at 2r is used.