Geometric realization via unoriented bordism and a counterexample to Singer's conjecture for the sixth algebraic transfer

Let $\mathcal{A}$ be the mod-2 Steenrod algebra acting in the usual way on $P_q={\mathbb{F}}_2[x_1,\dots ,x_q]$, and let $Q{P}_q={\mathbb{F}}_2{\otimes }_{\mathcal{A}}{P}_q$. Singer’s algebraic transfer $T{r}_q$ sends the dual of $[(Q{P}_q{)}_n{]}^{GL(q,{\mathbb{F}}_2)}$ to ${\mathrm{Ext}}_{\mathcal{A}}^{q,q+n}({\mathbb{F}}_2,{\mathbb{F}}_2)$; Singer conjectured that $T{r}_q$ is always injective. We disprove this nearly forty-year-old conjecture at rank $q=6$, degree $n=36$. Verifying this requires computing $[(Q{P}_6{)}_{36}{]}^{GL(6,{\mathbb{F}}_2)}$ exactly; to handle the resulting combinatorial complexity, we build a new Julia package $\mathtt{\text{AlgebraicTransfer.jl}}$, coupling modular invariant theory with bit-level linear algebra over ${\mathbb{F}}_2$ via Steenrod-hit reductions and Kameko homomorphisms. We prove this source space is two-dimensional, strictly exceeding the known one-dimensional target ${\mathrm{Ext}}_{\mathcal{A}}^{6,42}({\mathbb{F}}_2,{\mathbb{F}}_2)$, so $T{r}_6$ is not injective. We also interpret the transfer kernel geometrically via unoriented bordism: $T{r}_q$ factors through bordism classes over $B(\mathbb{Z}\mathrm{/}2{)}^q$ whose Thom images are primitive, characterized by the vanishing of all mixed Wu numbers. Thom’s representability theorem guarantees closed $36$-manifolds realizing the homological duals of the source generators, yet we show standard models – the indecomposable Milnor hypersurface $H_{4,33}$, projective products, and Dold manifolds – cannot represent them. We further interpret the inverse Kameko map via Thom spaces of universal real line bundles. Validated by recovering classical Dickson invariant dimensions, this work delivers both a counterexample to Singer’s conjecture and a scalable methodology for the Peterson hit problem.