{
  "statement": "Prove H1: For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).",
  "starting_statement": null,
  "target_statement": "For every positive odd integer \\(\\ell\\), there exists \\(f_\\ell\\in\\mathbb Z[z]\\) such that\n\\[\nf_\\ell(s-s^{-1})f_\\ell(s^{-1}-s)\n\\equiv(-1)^{(\\ell-1)/2}\\frac{s^\\ell-s^{-\\ell}}{s-s^{-1}}\\pmod4.\n\\]\nThe quotient on the right is the Laurent polynomial\n\\(s^{\\ell-1}+s^{\\ell-3}+\\cdots+s^{1-\\ell}\\); the congruence is coefficientwise in \\((\\mathbb Z/4\\mathbb Z)[s^{\\pm1}]\\).",
  "case_id": "KT-1-H0-to-H1",
  "given_context": {
    "premise": null
  },
  "protocol": "single"
}