{
  "statement": "Prove H3: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]",
  "starting_statement": null,
  "target_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]",
  "case_id": "OA-2-H0-to-H3",
  "given_context": {
    "premise": null
  },
  "protocol": "single"
}