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  "conventions": {
    "edge_direction": "dependency -> dependent",
    "logical_edges": "Curated sufficient proof routes for the case contracts and central reusable results; not claimed to enumerate every implicit use of classical mathematics.",
    "reference_edges": "Syntactic references are indexed separately. They may point forward, to examples or to alternate routes, and are not logical dependencies.",
    "status": "A result proved under a withdrawn assertion remains conditional; withdrawal is not proof that the asserted conclusion is false.",
    "powers": "A^n has dimension 6n. Full-row powers and residual-model powers are distinct.",
    "statement_encoding": "LaTeX statements are preserved, including assumptions; case contracts also contain a concise English strategy.",
    "source_locations": "One-based line numbers in the frozen original source; PDF pages are physical one-based page numbers from the compiled editions."
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    "checked_on": "2026-10-11",
    "source": "https://github.com/openai/math/blob/main/history.md",
    "withdrawn_assertions": [
      "EXT:Weil8",
      "EXT:KS",
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    "cm_assertion": "Not in the official three-item withdrawal list checked on this date; not independently validated."
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  "nodes": [
    {
      "id": "p:isogeny",
      "kind": "prelemma",
      "number": "P.1",
      "title_latex": "Prelemma P.1",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Rational Hodge classes, their algebraicity, and the Hodge conjecture on all\npowers are invariant under isogeny.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 241,
        "statement_end_line": 244,
        "proof_start_line": 245,
        "end_line": 251,
        "latex_labels": [
          "p:isogeny"
        ],
        "original_pdf_page": 4,
        "ai_pdf_page": 6
      },
      "citation_keys": [],
      "logical_dependencies": [
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:fft",
      "kind": "prelemma",
      "number": "P.2",
      "title_latex": "Prelemma P.2",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Suppose that the complexified Hodge group acts on $V_\\C$ as a product of full\nsymplectic groups on their standard representations, allowing repeated\ncopies. Then every Hodge class on every power is generated by divisor\nclasses. The same conclusion holds for a product of full general linear\ngroups acting on standard representations and their duals.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 270,
        "statement_end_line": 276,
        "proof_start_line": 277,
        "end_line": 294,
        "latex_labels": [
          "p:fft"
        ],
        "original_pdf_page": 4,
        "ai_pdf_page": 7
      },
      "citation_keys": [
        "Moonen"
      ],
      "logical_dependencies": [
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:small",
      "kind": "prelemma",
      "number": "P.3",
      "title_latex": "Prelemma P.3",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "An irreducible symplectic weight-one constituent of dimension $2$, $4$, or\n$6$, acted on by a connected semisimple image, has full image $\\Sp_2$,\n$\\Sp_4$, or $\\Sp_6$, respectively. An irreducible orthogonal constituent of\ndimension $6$ has full image $\\SO_6$.\nFor a symplectic constituent of dimension $12$, the only possible\ncomplex semisimple Lie-algebra images are $\\mathfrak{sp}_{12}$ and\n$\\mathfrak{sl}_2\\oplus\\mathfrak{sl}_4$, the latter on\n$\\C^2\\otimes\\bigwedge^2\\C^4$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 307,
        "statement_end_line": 316,
        "proof_start_line": 317,
        "end_line": 346,
        "latex_labels": [
          "p:small"
        ],
        "original_pdf_page": 5,
        "ai_pdf_page": 7
      },
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      "logical_dependencies": [
        "EXT:Classical"
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      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:goursat",
      "kind": "prelemma",
      "number": "P.4",
      "title_latex": "Prelemma P.4",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Let $G\\subseteq\\prod_{i=1}^e\\Sp(W_i)$ be connected, with $\\dim W_i=2r$\nindependent of $i$ and with surjective projections. If the $W_i$ are\npairwise nonisomorphic $G$-modules, then $G$ is the full product.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 348,
        "statement_end_line": 352,
        "proof_start_line": 353,
        "end_line": 360,
        "latex_labels": [
          "p:goursat"
        ],
        "original_pdf_page": 5,
        "ai_pdf_page": 7
      },
      "citation_keys": [],
      "logical_dependencies": [
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:realconstruction",
      "kind": "preproposition",
      "number": "P.5",
      "title_latex": "Preproposition P.5",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Let $F$ be totally real of degree $e$, let $r\\geq1$, and choose\n$Z_\\sigma\\in\\mathfrak H_r$ for every real embedding $\\sigma:F\\hookrightarrow\\R$.\nIn $\\C^{er}$ take the lattice which is the image of\n$\\mathcal O_F^r\\oplus\\mathfrak d_F^{-1\\,r}$ under\n\\[\n (x,y)\\longmapsto(\\sigma(x)+Z_\\sigma\\sigma(y))_\\sigma .\n\\]\nThe quotient is a polarized abelian variety of dimension $er$ with an\n$F$-action. At a very general point its Hodge group is the restriction of\nscalars of $\\Sp_{2r}$ and its endomorphism algebra is $F$; in particular it\nis simple.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 368,
        "statement_end_line": 380,
        "proof_start_line": 381,
        "end_line": 398,
        "latex_labels": [
          "p:realconstruction"
        ],
        "original_pdf_page": 6,
        "ai_pdf_page": 7
      },
      "citation_keys": [
        "Moonen"
      ],
      "logical_dependencies": [
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:quaternionconstruction",
      "kind": "preproposition",
      "number": "P.6",
      "title_latex": "Preproposition P.6",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Let $D_0=(-1,3)_\\Q$, with generators $i^2=-1$, $j^2=3$, $ij=-ji$.\nLet $F=\\Q$ or $\\Q(2\\cos(2\\pi/7))$, and set $D_F=D_0\\otimes_\\Q F$.\nFor $n=3/[F:\\Q]$, on $D_F^n$ put\n\\[\n \\psi(x,y)=\\Tr_{F/\\Q}\\Trd_{D_F/F}\\left(\\sum_k\\bar x_k i y_k\\right).\n\\]\nChoose an order lattice and multiply $\\psi$ by a positive integer to make\nit integral. The compatible complex structures commuting with left $D_F$\nform a nonempty PEL period domain $\\mathfrak H_n^{[F:\\Q]}$. Very general\nmembers are simple sixfolds of Type II with endomorphism algebra $D_F$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 400,
        "statement_end_line": 411,
        "proof_start_line": 412,
        "end_line": 430,
        "latex_labels": [
          "p:quaternionconstruction"
        ],
        "original_pdf_page": 6,
        "ai_pdf_page": 8
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:realconstruction",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:products",
      "kind": "preproposition",
      "number": "P.7",
      "title_latex": "Preproposition P.7",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Fix a partition $6=d_1+\\cdots+d_k$. In the product of Siegel moduli spaces\n$\\prod_i\\mathcal A_{d_i}(N)$ choose a very general tuple $(B_1,\\ldots,B_k)$,\nwith $N\\geq3$, and put $A=\\prod_i B_i$. Each $B_i$ is simple,\n$\\End^0(B_i)=\\Q$, the factors are pairwise nonisogenous, and\n\\[\n \\Hg(A)=\\prod_i\\Sp_{2d_i},\\qquad\n \\End^0(A)=\\Q^k,\\qquad\n P_A(t)=\\prod_i(1+t+\\cdots+t^{d_i}).\n\\]\nAll Hodge classes on all powers of $A$ are divisor-generated.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 432,
        "statement_end_line": 443,
        "proof_start_line": 444,
        "end_line": 455,
        "latex_labels": [
          "p:products"
        ],
        "original_pdf_page": 6,
        "ai_pdf_page": 8
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:fft",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:markman",
      "kind": "pretheorem",
      "number": "P.8",
      "title_latex": "Markman",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "For a polarized abelian sixfold of Weil type over an imaginary quadratic\nfield, the Weil classes are algebraic if its Hermitian discriminant is\n$-1$ in $\\Q^\\times/N_{K/\\Q}K^\\times$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 459,
        "statement_end_line": 463,
        "proof_start_line": null,
        "end_line": 463,
        "latex_labels": [
          "p:markman"
        ],
        "original_pdf_page": 7,
        "ai_pdf_page": 8
      },
      "citation_keys": [],
      "logical_dependencies": [
        "EXT:Markman6"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:openai-inputs",
      "kind": "pretheorem",
      "number": "P.9",
      "title_latex": "New external inputs",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "We additionally use the following two statements, with the preprint\nstatus specified above.\n\\begin{enumerate}\n\\item If $X$ is an abelian eightfold with an imaginary quadratic\n$k$-action, compatible polarization, and hyperbolic rational\n$k$-Hermitian form of signature $(4,4)$, then its entire rational\nWeil plane $W_k(X)\\subset H^8(X,\\Q)$ is algebraic. This includes\nevery compatible polarization type and all special periods.\n\\item Every rational Hodge class on every complex CM abelian variety\nis algebraic, in every dimension and codimension.\n\\end{enumerate}",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 470,
        "statement_end_line": 482,
        "proof_start_line": null,
        "end_line": 482,
        "latex_labels": [
          "p:openai-inputs"
        ],
        "original_pdf_page": 7,
        "ai_pdf_page": 8
      },
      "citation_keys": [],
      "logical_dependencies": [
        "EXT:Weil8",
        "EXT:CM"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:CM",
        "EXT:Weil8"
      ],
      "curated_transitive_conditional_inputs": [
        "EXT:CM",
        "EXT:Weil8"
      ]
    },
    {
      "id": "p:hyperbolic-test",
      "kind": "prelemma",
      "number": "P.10",
      "title_latex": "Rational hyperbolicity",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Let $k/\\Q$ be imaginary quadratic. A nondegenerate $k$-Hermitian\nform of rank $2r$ and signature $(r,r)$ is hyperbolic if and only if\nits determinant class is $[(-1)^r]$ in\n$\\Q^\\times/N_{k/\\Q}(k^\\times)$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 488,
        "statement_end_line": 493,
        "proof_start_line": 494,
        "end_line": 510,
        "latex_labels": [
          "p:hyperbolic-test"
        ],
        "original_pdf_page": 7,
        "ai_pdf_page": 8
      },
      "citation_keys": [
        "OpenAIWeilPowers"
      ],
      "logical_dependencies": [
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:all-sixfold-weil",
      "kind": "pretheorem",
      "number": "P.11",
      "title_latex": "All sixfold Weil planes from the eightfold input",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Assume the split-eightfold input in Theorem~\\ref{p:openai-inputs}.\nLet $A$ be any abelian sixfold of Weil type over an imaginary quadratic\nfield $k$, with a compatible polarization and signature $(3,3)$.\nThen every class in $W_k(A)\\subset H^6(A,\\Q)$ is algebraic,\nwithout a discriminant restriction.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 512,
        "statement_end_line": 519,
        "proof_start_line": 520,
        "end_line": 575,
        "latex_labels": [
          "p:all-sixfold-weil"
        ],
        "original_pdf_page": 7,
        "ai_pdf_page": 9
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:hyperbolic-test",
        "EXT:Weil8",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:Weil8"
      ],
      "curated_transitive_conditional_inputs": [
        "EXT:Weil8"
      ]
    },
    {
      "id": "p:CMproducts-new",
      "kind": "prelemma",
      "number": "P.12",
      "title_latex": "Products of CM factors",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Assume the CM input in Theorem~\\ref{p:openai-inputs}. Every finite\nproduct of CM abelian varieties satisfies the rational Hodge conjecture\non all its powers, without a Hodge-group independence assumption.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 577,
        "statement_end_line": 581,
        "proof_start_line": 582,
        "end_line": 587,
        "latex_labels": [
          "p:CMproducts-new"
        ],
        "original_pdf_page": 8,
        "ai_pdf_page": 9
      },
      "citation_keys": [],
      "logical_dependencies": [
        "EXT:CM",
        "p:isogeny"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:CM",
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": [
        "EXT:CM"
      ]
    },
    {
      "id": "p:polarization",
      "kind": "prelemma",
      "number": "P.13",
      "title_latex": "Prelemma P.13",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Let $A$ have dimension $6$, imaginary quadratic multiplication by $K$, and\n$\\Hg(A)_\\C=\\SL_6$ on $W\\oplus W^*$. If\n$W_K(A)=\\bigwedge_K^6 H^1(A,\\Q)$ is algebraic, then the Hodge conjecture\nholds for every power $A^n$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 589,
        "statement_end_line": 594,
        "proof_start_line": 595,
        "end_line": 615,
        "latex_labels": [
          "p:polarization"
        ],
        "original_pdf_page": 8,
        "ai_pdf_page": 9
      },
      "citation_keys": [],
      "logical_dependencies": [
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:HLsix",
      "kind": "prelemma",
      "number": "P.14",
      "title_latex": "Prelemma P.14",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "For a smooth projective sixfold, the rational Hodge conjecture follows\nfrom its cases in codimensions two and three.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 619,
        "statement_end_line": 622,
        "proof_start_line": 623,
        "end_line": 632,
        "latex_labels": [
          "p:HLsix"
        ],
        "original_pdf_page": 9,
        "ai_pdf_page": 9
      },
      "citation_keys": [],
      "logical_dependencies": [
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:lowdim",
      "kind": "pretheorem",
      "number": "P.15",
      "title_latex": "Low-dimensional inputs",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "We use the following established results with rational coefficients.\n\\begin{enumerate}\n\\item A simple abelian variety of prime dimension has full Lefschetz\nHodge group and divisor-generated Hodge rings on every power\n\\cite[Theorem (2.7)]{MZ} (Tankeev--Ribet).\n\\item If $\\Hom(E,X)=0$ for an elliptic curve $E$, failure of\n$\\Hg(X\\times E)=\\Hg(X)\\times\\Hg(E)$ requires that $E$ have CM by an\nimaginary quadratic field embedding in a simple field component of\nthe center of $\\End^0(X)$\n\\cite[Proposition (3.8)]{MZ}.\n\\item The Hodge conjecture holds for products of abelian varieties of\ndimension at most two \\cite[Proposition 2.18 in the preprint]{RamonMari}.\n\\item All Weil classes on abelian fourfolds of Weil type are algebraic,\nfor every imaginary quadratic field and every discriminant\n\\cite[proof of Corollary 1.6.1]{Markman}.\n\\end{enumerate}",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 633,
        "statement_end_line": 650,
        "proof_start_line": null,
        "end_line": 650,
        "latex_labels": [
          "p:lowdim"
        ],
        "original_pdf_page": 9,
        "ai_pdf_page": 9
      },
      "citation_keys": [
        "MZ",
        "Markman",
        "RamonMari"
      ],
      "logical_dependencies": [
        "EXT:LowDim"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:oddscale",
      "kind": "prelemma",
      "number": "P.16",
      "title_latex": "Odd-rank polarization adjustment",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Suppose an imaginary-quadratic Weil sixfold is $K$-linearly isogenous to\n$X\\times Y$, where the $K$-rank $r$ of $H_1(X,\\Q)$ is odd.\nThen its Weil classes are algebraic.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 657,
        "statement_end_line": 662,
        "proof_start_line": 663,
        "end_line": 679,
        "latex_labels": [
          "p:oddscale"
        ],
        "original_pdf_page": 9,
        "ai_pdf_page": 10
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:markman",
        "p:isogeny"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:quartic-scale",
      "kind": "prelemma",
      "number": "P.17",
      "title_latex": "Real-quadratic polarization adjustment",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Let a sixfold $A$ be of Weil type for an imaginary quadratic field $k$.\nSuppose its endomorphisms contain $K=kF$, where $F=\\Q(\\sqrt d)$ is\nreal quadratic, $d>0$, and a polarization has Rosati involution equal\nto complex conjugation on $K$. Then $A$ admits a $k$-compatible\npolarization of discriminant $-1$. Consequently its $k$-Weil classes\nare algebraic.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 681,
        "statement_end_line": 688,
        "proof_start_line": 689,
        "end_line": 726,
        "latex_labels": [
          "p:quartic-scale"
        ],
        "original_pdf_page": 10,
        "ai_pdf_page": 10
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:markman"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:CMsplit",
      "kind": "prelemma",
      "number": "P.18",
      "title_latex": "A split trace polarization for CM Weil sixfolds",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Let $A$ be a simple CM sixfold with $K=\\End^0(A)$, and let\n$k\\subset K$ be imaginary quadratic with signature $(3,3)$ on $A$.\nThen $A$ admits a compatible split $k$-Weil polarization. Consequently\n$W_k(A)$ consists of algebraic classes.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 728,
        "statement_end_line": 734,
        "proof_start_line": 735,
        "end_line": 782,
        "latex_labels": [
          "p:CMsplit"
        ],
        "original_pdf_page": 10,
        "ai_pdf_page": 10
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:markman"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:prime-ell",
      "kind": "prelemma",
      "number": "P.19",
      "title_latex": "A prime-dimensional factor and an elliptic curve",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Let $B$ be simple of dimension $g\\in\\{3,5\\}$ and let $E$ be elliptic.\nAll Hodge classes on $B\\times E$ are divisor-generated except possibly\nfor a single rational Weil plane in codimension $(g+1)/2$.\nThe exceptional case occurs precisely when $E$ has CM by a field\n$k\\subset\\End^0(B)$ whose signature on $B$ is\n$((g+1)/2,(g-1)/2)$, up to interchange. In that case\n\\[\n B^\\bullet(B\\times E)=D^\\bullet(B\\times E)+W_k(B\\times E),\n\\]\nwhere the $k$-action on $E$ is chosen to balance the signature.\nEvery class in the displayed space is algebraic.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 784,
        "statement_end_line": 797,
        "proof_start_line": 798,
        "end_line": 846,
        "latex_labels": [
          "p:prime-ell"
        ],
        "original_pdf_page": 11,
        "ai_pdf_page": 11
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:lowdim",
        "p:oddscale"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:three-ell",
      "kind": "prelemma",
      "number": "P.20",
      "title_latex": "A threefold and repeated elliptic factors",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Let $B$ be a simple threefold, and $P$ a product of elliptic curves.\nThen $B\\times P$ satisfies the Hodge conjecture. Its Hodge ring is\ngenerated by divisors and pullbacks of the fourfold Weil plane on\n$B\\times E$, if there is an elliptic isogeny class $E$ in $P$ with\n$k=\\End^0(E)\\subset\\End^0(B)$ imaginary quadratic.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 848,
        "statement_end_line": 855,
        "proof_start_line": 856,
        "end_line": 881,
        "latex_labels": [
          "p:three-ell"
        ],
        "original_pdf_page": 12,
        "ai_pdf_page": 11
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:prime-ell",
        "p:lowdim"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:surface-three",
      "kind": "prelemma",
      "number": "P.21",
      "title_latex": "A surface separates from a threefold and elliptic curves",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "If $S$ is a simple abelian surface, $B$ a simple threefold, and $P$ a\nproduct of elliptic curves, then\n$\\Hg(S\\times B\\times P)=\\Hg(S)\\times\\Hg(B\\times P)$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 883,
        "statement_end_line": 888,
        "proof_start_line": 889,
        "end_line": 915,
        "latex_labels": [
          "p:surface-three"
        ],
        "original_pdf_page": 12,
        "ai_pdf_page": 11
      },
      "citation_keys": [
        "MZ"
      ],
      "logical_dependencies": [
        "p:lowdim"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:smallunitary",
      "kind": "prelemma",
      "number": "P.22",
      "title_latex": "Small unitary constituents",
      "row": "Infrastructure",
      "assumptions_and_conclusion_latex": "Let $A$ be a simple sixfold with $\\End^0(A)=K$ a CM field of degree\n$2e$, where $e=2$ or $3$, and put $q=6/e$. Then\n\\[\n \\Hg(A)_\\C^{\\dd}=\\SL_q^e.\n\\]\nIts determinant image $T\\subset U_K$ has rank either $e$ or $e-1$.\nIn the latter case there is an imaginary quadratic subfield $k\\subset K$\nwith $T=\\ker(N_{K/k}:U_K\\to U_k)$, and $A$ is of $k$-Weil type.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 918,
        "statement_end_line": 927,
        "proof_start_line": 928,
        "end_line": 986,
        "latex_labels": [
          "p:smallunitary"
        ],
        "original_pdf_page": 13,
        "ai_pdf_page": 12
      },
      "citation_keys": [
        "MZ"
      ],
      "logical_dependencies": [
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "t:S1",
      "kind": "theorem",
      "number": "1.1",
      "title_latex": "Theorem 1.1",
      "row": "S1",
      "assumptions_and_conclusion_latex": "If $A$ is a simple abelian sixfold and $\\End^0(A)=\\Q$, then\n$\\Hg(A)=\\Sp_{12}$, and $\\B^\\bullet(A^n)=\\D^\\bullet(A^n)$ for every $n\\geq1$.\nOn $A$ itself $P_A(t)=1+t+\\cdots+t^6$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 993,
        "statement_end_line": 997,
        "proof_start_line": 998,
        "end_line": 1020,
        "latex_labels": [
          "t:S1"
        ],
        "original_pdf_page": 14,
        "ai_pdf_page": 12
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:small",
        "p:fft"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:1.2",
      "kind": "example",
      "number": "1.2",
      "title_latex": "Example 1.2",
      "row": "S1",
      "assumptions_and_conclusion_latex": "Use Proposition~\\ref{p:realconstruction} with $F=\\Q$, $r=6$.\nExplicitly $A_Z=\\C^6/(\\Z^6+Z\\Z^6)$ for $Z\\in\\mathfrak H_6$ very general.\nThe construction proves simplicity and $\\End^0(A_Z)=\\Q$. The seven\none-dimensional Hodge spaces are generated by $1,\\theta,\\ldots,\\theta^6$;\non powers add the Poincare divisors between the copies.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1021,
        "statement_end_line": 1027,
        "proof_start_line": null,
        "end_line": 1027,
        "latex_labels": [],
        "original_pdf_page": 14,
        "ai_pdf_page": 13
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:2.1",
      "kind": "theorem",
      "number": "2.1",
      "title_latex": "Theorem 2.1",
      "row": "S2",
      "assumptions_and_conclusion_latex": "Let $A$ be a simple sixfold with $\\End^0(A)=F$ totally real of degree $2$.\nThen\n\\[\n \\Hg(A)=\\Res_{F/\\Q}\\Sp_{6},\\qquad\n P_A(t)=(1+t+\\cdots+t^{3})^{2},\n\\]\nand $\\B^\\bullet(A^n)=\\D^\\bullet(A^n)$ for every $n$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1031,
        "statement_end_line": 1039,
        "proof_start_line": 1040,
        "end_line": 1053,
        "latex_labels": [],
        "original_pdf_page": 14,
        "ai_pdf_page": 13
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:small",
        "p:goursat",
        "p:fft"
      ],
      "dependency_coverage": "curated_case_main_theorem",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:2.2",
      "kind": "example",
      "number": "2.2",
      "title_latex": "Example 2.2",
      "row": "S2",
      "assumptions_and_conclusion_latex": "Set $F=\\Q(\\sqrt5)$ and $r=3$ in\nProposition~\\ref{p:realconstruction}. This specifies the lattice, period\nmatrices, and polarization and proves that a very general member belongs\nto this simple row. Choose a rational basis $a_1,\\ldots,a_{2}$ of $F$.\nThe rational $(1,1)$ classes defined by the forms\n$\\psi(a_jx,y)$ are divisor classes and generate $\\B^\\bullet(A)$.\nAfter splitting $F$, the ring is generated by $\\theta_\\sigma$ with\nrelations $\\theta_\\sigma^{4}=0$; its dimension in codimension $p$ is the\ncoefficient of $t^p$ in the polynomial above. Endomorphism-translated\nPoincare divisors complete the generators on powers.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1054,
        "statement_end_line": 1065,
        "proof_start_line": null,
        "end_line": 1065,
        "latex_labels": [],
        "original_pdf_page": 15,
        "ai_pdf_page": 13
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:3.1",
      "kind": "theorem",
      "number": "3.1",
      "title_latex": "Theorem 3.1",
      "row": "S3",
      "assumptions_and_conclusion_latex": "Let $A$ be a simple sixfold with $\\End^0(A)=F$ totally real of degree $3$.\nThen\n\\[\n \\Hg(A)=\\Res_{F/\\Q}\\Sp_{4},\\qquad\n P_A(t)=(1+t+\\cdots+t^{2})^{3},\n\\]\nand $\\B^\\bullet(A^n)=\\D^\\bullet(A^n)$ for every $n$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1069,
        "statement_end_line": 1077,
        "proof_start_line": 1078,
        "end_line": 1091,
        "latex_labels": [],
        "original_pdf_page": 15,
        "ai_pdf_page": 13
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:small",
        "p:goursat",
        "p:fft"
      ],
      "dependency_coverage": "curated_case_main_theorem",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:3.2",
      "kind": "example",
      "number": "3.2",
      "title_latex": "Example 3.2",
      "row": "S3",
      "assumptions_and_conclusion_latex": "Set $F=\\Q(2\\cos(2\\pi/7))$ and $r=2$ in\nProposition~\\ref{p:realconstruction}. This specifies the lattice, period\nmatrices, and polarization and proves that a very general member belongs\nto this simple row. Choose a rational basis $a_1,\\ldots,a_{3}$ of $F$.\nThe rational $(1,1)$ classes defined by the forms\n$\\psi(a_jx,y)$ are divisor classes and generate $\\B^\\bullet(A)$.\nAfter splitting $F$, the ring is generated by $\\theta_\\sigma$ with\nrelations $\\theta_\\sigma^{3}=0$; its dimension in codimension $p$ is the\ncoefficient of $t^p$ in the polynomial above. Endomorphism-translated\nPoincare divisors complete the generators on powers.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1092,
        "statement_end_line": 1103,
        "proof_start_line": null,
        "end_line": 1103,
        "latex_labels": [],
        "original_pdf_page": 15,
        "ai_pdf_page": 13
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:4.1",
      "kind": "theorem",
      "number": "4.1",
      "title_latex": "Theorem 4.1",
      "row": "S4",
      "assumptions_and_conclusion_latex": "Let $A$ be a simple sixfold with $\\End^0(A)=F$ totally real of degree $6$.\nThen\n\\[\n \\Hg(A)=\\Res_{F/\\Q}\\Sp_{2},\\qquad\n P_A(t)=(1+t+\\cdots+t^{1})^{6},\n\\]\nand $\\B^\\bullet(A^n)=\\D^\\bullet(A^n)$ for every $n$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1107,
        "statement_end_line": 1115,
        "proof_start_line": 1116,
        "end_line": 1129,
        "latex_labels": [],
        "original_pdf_page": 16,
        "ai_pdf_page": 14
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:small",
        "p:goursat",
        "p:fft"
      ],
      "dependency_coverage": "curated_case_main_theorem",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:4.2",
      "kind": "example",
      "number": "4.2",
      "title_latex": "Example 4.2",
      "row": "S4",
      "assumptions_and_conclusion_latex": "Set $F=\\Q(\\zeta_{13}+\\zeta_{13}^{-1})$ and $r=1$ in\nProposition~\\ref{p:realconstruction}. This specifies the lattice, period\nmatrices, and polarization and proves that a very general member belongs\nto this simple row. Choose a rational basis $a_1,\\ldots,a_{6}$ of $F$.\nThe rational $(1,1)$ classes defined by the forms\n$\\psi(a_jx,y)$ are divisor classes and generate $\\B^\\bullet(A)$.\nAfter splitting $F$, the ring is generated by $\\theta_\\sigma$ with\nrelations $\\theta_\\sigma^{2}=0$; its dimension in codimension $p$ is the\ncoefficient of $t^p$ in the polynomial above. Endomorphism-translated\nPoincare divisors complete the generators on powers.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1130,
        "statement_end_line": 1141,
        "proof_start_line": null,
        "end_line": 1141,
        "latex_labels": [],
        "original_pdf_page": 16,
        "ai_pdf_page": 14
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:5.1",
      "kind": "theorem",
      "number": "5.1",
      "title_latex": "Theorem 5.1",
      "row": "S5",
      "assumptions_and_conclusion_latex": "Let $A$ be a simple sixfold whose endomorphism algebra is an indefinite\nquaternion division algebra over $\\Q$. Then $\\Hg(A)_\\C=\\Sp_6$ on\n$V_\\C=U\\otimes W$, $\\dim U=2$, $\\dim W=6$, where the group acts trivially\non $U$. Every Hodge class on every power is divisor-generated, and\n\\[\n P_A(t)=1+3t+6t^2+10t^3+6t^4+3t^5+t^6.\n\\]",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1146,
        "statement_end_line": 1154,
        "proof_start_line": 1155,
        "end_line": 1173,
        "latex_labels": [],
        "original_pdf_page": 16,
        "ai_pdf_page": 14
      },
      "citation_keys": [
        "Vial"
      ],
      "logical_dependencies": [
        "p:small",
        "p:fft"
      ],
      "dependency_coverage": "curated_case_main_theorem",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:5.2",
      "kind": "example",
      "number": "5.2",
      "title_latex": "Example 5.2",
      "row": "S5",
      "assumptions_and_conclusion_latex": "Take $D=(-1,3)_\\Q$, $n=3$, and the explicit trace polarization of\nProposition~\\ref{p:quaternionconstruction}. Very general compatible\ncomplex structures give simple Type II(1) sixfolds. The three-dimensional\n$\\NS(A)_\\Q$ corresponds to the self-adjoint subspace of the orthogonal\nRosati involution. Its divisor classes generate all Hodge spaces with\nrespective dimensions $(1,3,6,10,6,3,1)$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1174,
        "statement_end_line": 1181,
        "proof_start_line": null,
        "end_line": 1181,
        "latex_labels": [],
        "original_pdf_page": 16,
        "ai_pdf_page": 14
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:6.1",
      "kind": "theorem",
      "number": "6.1",
      "title_latex": "Theorem 6.1",
      "row": "S6",
      "assumptions_and_conclusion_latex": "Let $A$ be a simple sixfold of Type II with real center $F$ of degree $3$.\nThe complexified Hodge group is $\\SL_2^3$, acting on\n$\\bigoplus_\\sigma U_\\sigma\\otimes W_\\sigma$ with\n$\\dim U_\\sigma=\\dim W_\\sigma=2$. All Hodge classes on all powers are\ndivisor-generated, and $P_A(t)=(1+3t+t^2)^3$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1186,
        "statement_end_line": 1192,
        "proof_start_line": 1193,
        "end_line": 1205,
        "latex_labels": [],
        "original_pdf_page": 17,
        "ai_pdf_page": 14
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:small",
        "p:goursat",
        "p:fft"
      ],
      "dependency_coverage": "curated_case_main_theorem",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:6.2",
      "kind": "example",
      "number": "6.2",
      "title_latex": "Example 6.2",
      "row": "S6",
      "assumptions_and_conclusion_latex": "Set $F=\\Q(2\\cos(2\\pi/7))$ and $D=(-1,3)_\\Q\\otimes F$. This is an\nindefinite quaternion division algebra by the odd-degree argument.\nProposition~\\ref{p:quaternionconstruction} with $n=1$ gives a nonempty\nthree-dimensional family and simple very general members.\nHere $\\rho(A)=9$ and the Hodge-space dimensions are\n$(1,9,30,45,30,9,1)$. The nine rational divisor generators come from a\nbasis of the Rosati-fixed subspace of $D$; Poincare divisors and $D$\nendomorphisms provide generators on powers.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1206,
        "statement_end_line": 1215,
        "proof_start_line": null,
        "end_line": 1215,
        "latex_labels": [],
        "original_pdf_page": 17,
        "ai_pdf_page": 15
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "t:S7structure",
      "kind": "theorem",
      "number": "7.1",
      "title_latex": "Theorem 7.1",
      "row": "S7",
      "assumptions_and_conclusion_latex": "For a simple Type III(1) sixfold,\n$V_\\C=U\\otimes W$, $\\dim U=2$, $\\dim W=6$, and\n$\\Hg(A)_\\C=\\SO(W)$. Moreover\n\\[\n P_A(t)=1+t+t^2+8t^3+t^4+t^5+t^6,\n \\qquad\\dim_\\Q\\bigl(\\B^3(A)/\\D^3(A)\\bigr)=7.\n\\]\nOn powers, the invariant algebra is generated by degree-two pairings and\nsix-linear determinant invariants on the multiplicity space.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1223,
        "statement_end_line": 1233,
        "proof_start_line": 1234,
        "end_line": 1254,
        "latex_labels": [
          "t:S7structure"
        ],
        "original_pdf_page": 17,
        "ai_pdf_page": 15
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:small",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "t:S7complete",
      "kind": "theorem",
      "number": "7.2",
      "title_latex": "Complete Type III row",
      "row": "S7",
      "assumptions_and_conclusion_latex": "Assuming the split-eightfold input, every simple Type III(1) sixfold\nsatisfies the rational Hodge conjecture on every power.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1255,
        "statement_end_line": 1258,
        "proof_start_line": 1259,
        "end_line": 1282,
        "latex_labels": [
          "t:S7complete"
        ],
        "original_pdf_page": 18,
        "ai_pdf_page": 15
      },
      "citation_keys": [],
      "logical_dependencies": [
        "t:S7structure",
        "p:all-sixfold-weil",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:Weil8"
      ],
      "curated_transitive_conditional_inputs": [
        "EXT:Weil8"
      ]
    },
    {
      "id": "t:S7split",
      "kind": "theorem",
      "number": "7.3",
      "title_latex": "The earlier split-subfield case",
      "row": "S7",
      "assumptions_and_conclusion_latex": "In the setting of Theorem~\\ref{t:S7structure}, suppose that a subfield\n$K\\subset D$ makes the polarized variety a sixfold of split Weil type.\nThen all Hodge classes on all powers of $A$ are algebraic.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1284,
        "statement_end_line": 1288,
        "proof_start_line": 1289,
        "end_line": 1303,
        "latex_labels": [
          "t:S7split"
        ],
        "original_pdf_page": 18,
        "ai_pdf_page": 15
      },
      "citation_keys": [],
      "logical_dependencies": [
        "t:S7structure",
        "p:markman"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:S7onecycle",
      "kind": "corollary",
      "number": "7.4",
      "title_latex": "A single exceptional cycle in Type III",
      "row": "S7",
      "assumptions_and_conclusion_latex": "For a simple Type III(1) sixfold $A$, the Hodge conjecture holds on\nall powers if and only if there exists $Z\\in\\CH^3(A)_\\Q$ with\n$\\mathrm{cl}(Z)\\notin\\Q\\theta^3$, where $\\theta$ is a polarization.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1304,
        "statement_end_line": 1309,
        "proof_start_line": 1310,
        "end_line": 1334,
        "latex_labels": [
          "p:S7onecycle"
        ],
        "original_pdf_page": 18,
        "ai_pdf_page": 15
      },
      "citation_keys": [],
      "logical_dependencies": [
        "t:S7structure",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "ex:hamilton",
      "kind": "example",
      "number": "7.5",
      "title_latex": "Example 7.5",
      "row": "S7",
      "assumptions_and_conclusion_latex": "Let $D=(-1,-1)_\\Q$ with $i^2=j^2=-1$, and $V=D^3$. Put\n\\[\n \\psi(x,y)=\\Trd\\left(\\sum_{k=1}^3\\bar x_k y_k i\\right).\n\\]\nOn an order lattice, clear denominators. Right multiplication by $-i$ is\na compatible complex structure, since\n$\\psi(x,x(-i))=\\Trd(\\sum_k\\bar x_kx_k)>0$ for $x\\ne0$.\nThe left $D$-action has Rosati adjoint $a\\mapsto\\bar a$.\nThe compatible PEL domain is $\\SO^*(6)/\\mathrm U(3)$, of complex dimension\n$3$. Its group is noncompact and $\\Q$-simple; arithmetic monodromy is dense\nin its connected $\\SO_6$ complex form. The generic commutant is $D$;\nvery general members are therefore simple Type III(1) sixfolds.\nTake $K=\\Q(i)$ acting on the left. The associated Hermitian form can be\nwritten as $h(x,y)=\\psi(ix,y)+i\\psi(x,y)$. On the $K$-basis $(1,j)$ of\neach copy of $D$, it has matrix $\\mathrm{diag}(2,-2)$.\nThus on $D^3$ it is three hyperbolic planes; the span of the three vectors\n$1+j$ in separate summands is a three-dimensional totally isotropic\n$K$-subspace. Its discriminant is $-64$, the class of $-1$ modulo norms.\nThis computation uses the fixed rational polarization, so it remains valid\nthroughout the compatible family. The signature is $(3,3)$: left $j$\ninterchanges the two $K$ eigenspaces and commutes with the complex\nstructure. Theorem~\\ref{t:S7split} applies. The eight middle Hodge classes\nconsist of $\\theta^3$ and seven volume classes; the latter are spanned by\n$D$-translates of the algebraic Weil plane.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1336,
        "statement_end_line": 1361,
        "proof_start_line": null,
        "end_line": 1361,
        "latex_labels": [
          "ex:hamilton"
        ],
        "original_pdf_page": 19,
        "ai_pdf_page": 16
      },
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    },
    {
      "id": "result:7.6",
      "kind": "remark",
      "number": "7.6",
      "title_latex": "Remark 7.6",
      "row": "S7",
      "assumptions_and_conclusion_latex": "The following criterion determines exactly when this split-subfield\nmethod applies. The subsequent example shows that it does not apply\nto every Type III sixfold.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1362,
        "statement_end_line": 1366,
        "proof_start_line": null,
        "end_line": 1366,
        "latex_labels": [],
        "original_pdf_page": 19,
        "ai_pdf_page": 16
      },
      "citation_keys": [],
      "logical_dependencies": [],
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    },
    {
      "id": "p:quaternion-split-test",
      "kind": "proposition",
      "number": "7.7",
      "title_latex": "An arithmetic split-subfield criterion",
      "row": "S7",
      "assumptions_and_conclusion_latex": "Let $A$ be a simple Type III(1) sixfold with definite quaternion\nendomorphism algebra $D$. Diagonalize its rational polarization on\n$H_1(A,\\Q)=D^3$ as\n\\[\n \\psi(x,y)=\\Trd\\!\\left(\\sum_{s=1}^3\\bar x_s y_s q_s\\right),\n \\qquad \\bar q_s=-q_s,\\quad q_s\\ne0,\n \\qquad N=\\prod_{s=1}^3\\operatorname{Nrd}(q_s)>0.\n\\]\nThere exists an imaginary quadratic subfield $k\\subset D$ for which\nthis polarization has $k$-Weil discriminant $-1$ if and only if\n$\\Q(\\sqrt{-N})$ embeds in $D$. When it does, the Hodge conjecture\nholds on every power of $A$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1367,
        "statement_end_line": 1381,
        "proof_start_line": 1382,
        "end_line": 1417,
        "latex_labels": [
          "p:quaternion-split-test"
        ],
        "original_pdf_page": 19,
        "ai_pdf_page": 16
      },
      "citation_keys": [],
      "logical_dependencies": [
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
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    },
    {
      "id": "ex:quaternion-nonsplit",
      "kind": "example",
      "number": "7.8",
      "title_latex": "A quaternionic family outside every split subfield",
      "row": "S7",
      "assumptions_and_conclusion_latex": "Take $D=(-2,-5)_\\Q$, with generators $i,j$, and\n$(q_1,q_2,q_3)=(i,j,ij)$. Their reduced norms are $2,5,10$, hence\n$N=100$ and $\\Q(\\sqrt{-N})=\\Q(\\sqrt{-1})$.\nThe algebra $D$ is division at $5$: the local Hilbert symbol is\n$(-2,-5)_5=(\\frac{-2}{5})=-1$. But $\\Q(\\sqrt{-1})$ splits at $5$,\nso it cannot embed in $D$. Indeed tensoring such an embedding with\n$\\Q_5$ would embed $\\Q_5\\times\\Q_5$ in a division algebra.\nThus Proposition~\\ref{p:quaternion-split-test} excludes every split\nimaginary quadratic subfield for this polarization.\n\nThis is a nonempty polarized datum. On the $s$th real quaternionic\ncoordinate take right multiplication by\n$-q_s/\\sqrt{\\operatorname{Nrd}(q_s)}$ as complex structure. Its square\nis $-1$ and\n\\[\n \\psi(x,Jx)=2\\sum_s\\sqrt{\\operatorname{Nrd}(q_s)}\\,\n                  \\operatorname{Nrd}(x_s)>0\\quad(x\\ne0).\n\\]\nThe Riemann relations and an order lattice give abelian varieties.\nThe same arithmetic-monodromy and commutant argument as in\nExample~\\ref{ex:hamilton} gives simple very general members with\n$\\End^0=D$ in a three-dimensional PEL family.\nFor such a member every rational polarization is a positive rational\nmultiple of this one, because the Rosati-fixed subspace of $D$ is\n$\\Q$. A multiplier $c$ replaces $N$ by $c^6N$, which defines the\nsame quadratic field. Hence changing polarization cannot remove this\nobstruction. This is an obstruction to the split-subfield method,\nnot a nonalgebraicity assertion. Nevertheless,\nTheorem~\\ref{t:S7complete} now proves the Hodge conjecture on all\npowers of every simple member, using the eightfold input. For example,\nchoose the subfield $k=\\Q(\\sqrt{-2})\\subset D$ generated by $i$ and stabilize its\nsixfold Weil structure as in Theorem~\\ref{p:all-sixfold-weil};\nno split imaginary quadratic subfield inside $D$ is required.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1418,
        "statement_end_line": 1453,
        "proof_start_line": null,
        "end_line": 1453,
        "latex_labels": [
          "ex:quaternion-nonsplit"
        ],
        "original_pdf_page": 20,
        "ai_pdf_page": 16
      },
      "citation_keys": [],
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    },
    {
      "id": "t:S8",
      "kind": "theorem",
      "number": "8.1",
      "title_latex": "Theorem 8.1",
      "row": "S8",
      "assumptions_and_conclusion_latex": "Suppose $A$ is a simple sixfold with $\\End^0(A)=K$ imaginary quadratic.\nFor the $K$-action on $H^{1,0}$ write its signature as $(r,6-r)$, with\n$1\\leq r\\leq5$. If $r\\ne3$, then $\\Hg(A)$ is the full unitary group\nand every Hodge class on every power is divisor-generated.\nIf $r=3$, then $\\Hg(A)$ is the special unitary group,\n\\[\n P_A(t)=1+t+t^2+3t^3+t^4+t^5+t^6,\n\\]\nand the Hodge conjecture on all powers is equivalent to algebraicity of\n$W_K(A)$. Under the split-eightfold input it holds for every\ndiscriminant, on all powers.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1460,
        "statement_end_line": 1472,
        "proof_start_line": 1473,
        "end_line": 1509,
        "latex_labels": [
          "t:S8"
        ],
        "original_pdf_page": 20,
        "ai_pdf_page": 17
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:all-sixfold-weil",
        "p:polarization",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:Weil8"
      ],
      "curated_transitive_conditional_inputs": [
        "EXT:Weil8"
      ]
    },
    {
      "id": "result:8.2",
      "kind": "example",
      "number": "8.2",
      "title_latex": "Example 8.2",
      "row": "S8",
      "assumptions_and_conclusion_latex": "Take $K=\\Q(i)$, $V=K^6$, and\n$h=\\mathrm{diag}(1,1,1,-1,-1,-1)$ with alternating form\n$\\Tr_{K/\\Q}(i h)$ on an order lattice. Compatible positive complex\nstructures form the nonempty unitary domain of signature $(3,3)$, of\ncomplex dimension $9$. The generic monodromy is the full special unitary\ngroup, whose commutant is $K$; very general members are simple with\n$\\End^0=K$. The Hermitian form is hyperbolic and has discriminant $-1$.\nThe middle Hodge space has dimension $3$, generated by $\\theta^3$ and an\nalgebraic rational basis of $W_K(A)$. Their pullbacks under $K$-linear sum\nmaps, together with divisors, generate the Hodge classes on every power.\nFor a comparison of boundaries, replacing the last diagonal entry by\n$-3$ gives determinant $-3$. The positive number $3$ is not a norm from\n$\\Q(i)$ (its $3$-adic valuation is odd), so the split theorem does not\napply directly to that family. Its product with two oppositely oriented\n$\\Q(i)$-CM elliptic curves with Hermitian coefficients $1,-3$\nhas signature $(4,4)$ and determinant $9$, hence is split.\nTheorem~\\ref{p:all-sixfold-weil} contracts its algebraic Weil plane\nto the original sixfold. Lemma~\\ref{p:polarization} then proves\nalgebraicity on every power, including this formerly unresolved family.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1510,
        "statement_end_line": 1530,
        "proof_start_line": null,
        "end_line": 1530,
        "latex_labels": [],
        "original_pdf_page": 21,
        "ai_pdf_page": 17
      },
      "citation_keys": [],
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    },
    {
      "id": "result:9.1",
      "kind": "theorem",
      "number": "9.1",
      "title_latex": "Theorem 9.1",
      "row": "S9",
      "assumptions_and_conclusion_latex": "Let $A$ be simple with CM endomorphism field $K$ of degree $4$.\nIts derived Hodge group has complex form $\\SL_{3}^{2}$.\nThere are exactly the following alternatives for its determinant image.\n\\begin{enumerate}\n\\item The image is all of $U_K$. Then $\\Hg(A)_\\C=\\GL_{3}^{2}$,\nall powers are divisor-generated, and $P_A(t)=(1+t+t^2+t^3)^2$.\n\\item The image is $\\ker N_{K/k}$ for an imaginary quadratic subfield\n$k\\subset K$. Then $A$ is of $k$-Weil type,\n\\[\n B^\\bullet(A)=D^\\bullet(A)+W_k(A),\\qquad\n P_A(t)=(1+t+t^2+t^3)^2+2t^3.\n\\]\nThe sum adds a two-dimensional space only in codimension three.\nEvery class is algebraic, and the Hodge conjecture holds on all powers.\n\\end{enumerate}",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1536,
        "statement_end_line": 1552,
        "proof_start_line": 1553,
        "end_line": 1568,
        "latex_labels": [],
        "original_pdf_page": 21,
        "ai_pdf_page": 17
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:smallunitary",
        "p:quartic-scale",
        "p:coupled-det"
      ],
      "dependency_coverage": "curated_case_main_theorem",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
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    },
    {
      "id": "result:9.2",
      "kind": "example",
      "number": "9.2",
      "title_latex": "Example 9.2",
      "row": "S9",
      "assumptions_and_conclusion_latex": "Let $K=\\Q(\\zeta_5)$, $F=\\Q(\\sqrt5)$, $V=K^3$, and\n$h=\\mathrm{diag}(1,1,-1)$. Choose a nonzero totally imaginary\n$\\delta\\in K$ and the trace alternating form $\\Tr_{K/\\Q}(\\delta h)$,\nwith compatible positive complex structures and an order lattice.\nThe signatures at the two real embeddings of $F$ are $(2,1)$.\nThis gives a nonempty PEL domain of complex dimension $4$.\nArithmetic monodromy gives generic derived group\n$\\Res_{F/\\Q}\\SU(h)$, whose complex form is $\\SL_3^2$.\nThe determinant Hodge cocharacter is nonzero. On the two-dimensional\nanti-conjugate character space, a generator of the cyclic Galois group\n$\\mathrm{Gal}(K/\\Q)$ acts by a signed rotation of order $4$; this\nrepresentation is irreducible over $\\Q$. The orbit of any nonzero\ndeterminant vector therefore spans that entire space. The generic\ncentral torus is full, so the generic Hodge group is the full unitary group.\nIts centralizer is $K$; very general members are simple with $\\End^0=K$.\nTheir Hodge dimensions are $(1,2,3,4,3,2,1)$, generated by the two\nrational Rosati-fixed divisor classes, with the usual Poincare generators\non powers. This verifies the group hypothesis rather than assuming it.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1569,
        "statement_end_line": 1588,
        "proof_start_line": null,
        "end_line": 1588,
        "latex_labels": [],
        "original_pdf_page": 22,
        "ai_pdf_page": 18
      },
      "citation_keys": [],
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    },
    {
      "id": "p:coupled-det",
      "kind": "proposition",
      "number": "9.3",
      "title_latex": "Proposition 9.3",
      "row": "S9",
      "assumptions_and_conclusion_latex": "Suppose a sixfold has a CM-field action by $K$ of degree\n$2e$, $q=6/e$, $e\\in\\{2,3,6\\}$, and\n\\[\n \\Hg(A)_\\C^{\\dd}=\\prod_{j=1}^e\\SL(W_j),\\quad \\dim W_j=q.\n\\]\nChoose one embedding in each conjugate pair. Suppose the determinant\nimage of $\\Hg(A)_\\C$ is the connected torus\n$\\{(z_1,\\ldots,z_e):\\prod z_j=1\\}$, and these selected embeddings are\nexactly those restricting to a fixed embedding of an imaginary quadratic\nsubfield $k\\subset K$. If $W_k(A)$ consists of algebraic classes,\nthe Hodge conjecture holds for $A^n$ for every $n$. In particular\nthis holds when the induced $k$-Weil structure has discriminant $-1$.\nOn $A$,\n\\[\n P_A(t)=(1+t+\\cdots+t^q)^e+2t^3.\n\\]",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1590,
        "statement_end_line": 1607,
        "proof_start_line": 1608,
        "end_line": 1642,
        "latex_labels": [
          "p:coupled-det"
        ],
        "original_pdf_page": 22,
        "ai_pdf_page": 18
      },
      "citation_keys": [
        "Markman"
      ],
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        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
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    },
    {
      "id": "result:9.4",
      "kind": "example",
      "number": "9.4",
      "title_latex": "A split coupled quartic-CM family",
      "row": "S9",
      "assumptions_and_conclusion_latex": "Take $K=\\Q(i,\\sqrt2)$, $F=\\Q(\\sqrt2)$, $k=\\Q(\\sqrt{-2})$,\n$V=K^3$, and $h_K=\\operatorname{diag}(1,1,-1)$. Use the alternating form\n$\\psi=\\Tr_{K/\\Q}(i h_K)$ on an order lattice. The signatures relative\nto $\\Q(i)$ are $(2,1),(2,1)$; the compatible PEL domain has dimension\nfour. Its generic derived group is $\\Res_{F/\\Q}\\SU(h_K)$, with complex\nform $\\SL_3^2$, by arithmetic monodromy. The determinant cocharacter\nrelative to these two embeddings is proportional to $(1,1)$. The\nbiquadratic Galois group interchanges its coordinates and negates them\nsimultaneously, so its orbit spans exactly this line. This determines the\ngeneric central determinant torus.\n\nRelative instead to the two embeddings restricting to one embedding of\n$k$, one $\\Q(i)$-embedding is conjugated. Thus the signatures are\n$(2,1),(1,2)$, the central relation is $z_1z_2=1$, and the total $k$\nsignature is $(3,3)$. Writing $\\delta'=i\\sqrt2$, the $k$-Hermitian form\nassociated with the same $\\psi$ is\n\\[\n h_k=\\Tr_{K/k}\\bigl((1/\\sqrt2)h_K\\bigr).\n\\]\nOn the $k$-basis $(1,\\sqrt2)$ its first two blocks are\n$\\left(\\begin{smallmatrix}0&2\\\\2&0\\end{smallmatrix}\\right)$, and the\nthird is the negative of this block. The span of the first vector in each\nblock is a three-dimensional isotropic $k$-subspace. This proves splitness\nexplicitly. The generic centralizer on $V$ is $K$: the two derived\nstandard modules are inequivalent and each is distinguished from its dual\nby the nontrivial center. Hence a very general member is simple with\n$\\End^0(A)=K$. Proposition~\\ref{p:coupled-det} applies: every power\nsatisfies the Hodge conjecture, while\n$P_A(t)=1+2t+3t^2+6t^3+3t^4+2t^5+t^6$.\nIn particular its two exceptional middle classes are genuinely present\nand algebraic; this example is not divisor-generated.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1643,
        "statement_end_line": 1675,
        "proof_start_line": null,
        "end_line": 1675,
        "latex_labels": [],
        "original_pdf_page": 23,
        "ai_pdf_page": 18
      },
      "citation_keys": [],
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    },
    {
      "id": "result:9.5",
      "kind": "example",
      "number": "9.5",
      "title_latex": "A nonsplit polarization that can be corrected",
      "row": "S9",
      "assumptions_and_conclusion_latex": "Let $k=\\Q(i)$, $F=\\Q(\\sqrt3)$, $K=kF$, and\n$H=\\operatorname{diag}(1,-1,1+\\sqrt3)$ on $K^3$.\nUse $\\psi=\\Tr_{K/\\Q}(iH)$ and a compatible very general complex structure.\nThe signatures are $(2,1)$ and $(1,2)$. The product derived monodromy\nand the determinant-circle relation give the group in the second\nalternative of S9; its commutant is $K$ and the sixfold is simple.\nHere $b=-1-\\sqrt3$, $N_{F/\\Q}(b)=-2$, and the initial $k$-determinant is\n\\[\n D=12^3(-2)=-3456=-6\\cdot24^2.\n\\]\nIt is not the class of $-1$: the factor $3$ occurs to odd order and\nis inert in $\\Q(i)$. But $a=3+\\sqrt3$ is totally positive and has\nnorm $6$. Replacing $H$ by $aH$ gives\n\\[\n D'=D\\cdot6^3=-746496=-864^2.\n\\]\nThis is a polarization of discriminant $-1$ on the same complex abelian\nvariety. Its Weil plane is unchanged and is therefore algebraic.\nThe Hodge dimensions are $(1,2,3,6,3,2,1)$.\nThis example tests the nontrivial step of Lemma~\\ref{p:quartic-scale}:\nscaling by a totally positive real-quadratic endomorphism can change\nthe norm class even though a rational scalar on a sixfold cannot.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1677,
        "statement_end_line": 1700,
        "proof_start_line": null,
        "end_line": 1700,
        "latex_labels": [],
        "original_pdf_page": 23,
        "ai_pdf_page": 18
      },
      "citation_keys": [],
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    },
    {
      "id": "result:10.1",
      "kind": "theorem",
      "number": "10.1",
      "title_latex": "Theorem 10.1",
      "row": "S10",
      "assumptions_and_conclusion_latex": "Let $A$ be simple with CM endomorphism field $K$ of degree $6$.\nIts derived Hodge group has complex form $\\SL_{2}^{3}$.\nThere are exactly the following alternatives for its determinant image.\n\\begin{enumerate}\n\\item The image is all of $U_K$. Then $\\Hg(A)_\\C=\\GL_{2}^{3}$,\nall powers are divisor-generated, and $P_A(t)=(1+t+t^2)^3$.\n\\item The image is $\\ker N_{K/k}$ for an imaginary quadratic subfield\n$k\\subset K$. Then $A$ is of $k$-Weil type,\n\\[\n B^\\bullet(A)=D^\\bullet(A)+W_k(A),\\qquad\n P_A(t)=(1+t+t^2)^3+2t^3.\n\\]\nThe sum adds a two-dimensional space only in codimension three.\nUnder the split-eightfold input, every class on every power is\nalgebraic, for arbitrary induced $k$-Hermitian discriminant.\n\\end{enumerate}",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1708,
        "statement_end_line": 1725,
        "proof_start_line": 1726,
        "end_line": 1740,
        "latex_labels": [],
        "original_pdf_page": 24,
        "ai_pdf_page": 19
      },
      "citation_keys": [],
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        "p:smallunitary",
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        "p:coupled-det"
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      "dependency_coverage": "curated_case_main_theorem",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
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    },
    {
      "id": "result:10.2",
      "kind": "example",
      "number": "10.2",
      "title_latex": "Example 10.2",
      "row": "S10",
      "assumptions_and_conclusion_latex": "Set $K=\\Q(\\zeta_7)$, $F=\\Q(\\alpha)$ with\n$\\alpha=2\\cos(2\\pi/7)$, and $V=K^2$. Choose\n$h=\\mathrm{diag}(1,\\alpha-1)$ and an integral multiple of\n$\\Tr_{K/\\Q}(\\delta h)$ for a nonzero totally imaginary $\\delta$.\nThe conjugates of $\\alpha$ are approximately\n$1.24698,-0.44504,-1.80194$. Thus the signatures are $(2,0),(1,1),(1,1)$,\nand the nonempty compatible period domain has complex dimension $2$.\nThe rational derived unitary group is simple and has noncompact real\nfactors. Arithmetic monodromy is dense in it, including its compact real\nfactor, so its generic complex form is $\\SL_2^3$.\nThe determinant cocharacter has just one nonzero component in the\nthree-dimensional anti-conjugate character space. The cyclic Galois group\nof $K$ permutes its three coordinate directions with signs; that vector's\norbit spans all three directions. The central determinant torus is full,\nso the generic Hodge group is $\\GL_2^3$ after complexification.\nThe commutant is $K$, proving simplicity and $\\End^0=K$ for very general\nmembers. Their Hodge dimensions are $(1,3,6,7,6,3,1)$; a basis of $F$\ngives three rational divisor generators. Although the label is Type IV,\nthese members are not CM points: their Hodge group has nontrivial derived\npart. Calling the entire row toric would be incorrect.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1741,
        "statement_end_line": 1762,
        "proof_start_line": null,
        "end_line": 1762,
        "latex_labels": [],
        "original_pdf_page": 24,
        "ai_pdf_page": 19
      },
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    },
    {
      "id": "result:10.3",
      "kind": "example",
      "number": "10.3",
      "title_latex": "A split coupled sextic-CM family",
      "row": "S10",
      "assumptions_and_conclusion_latex": "Let $K=\\Q(\\zeta_7)=kF$, where $k=\\Q(\\sqrt{-7})$ and\n$F=\\Q(\\alpha)$, $\\alpha=2\\cos(2\\pi/7)$,\n$\\alpha^3+\\alpha^2-2\\alpha-1=0$. Set\n\\[\n a=\\alpha^2+\\alpha-1,\\qquad b=\\alpha-1,\\qquad\n V=K^2,\\qquad h_K=\\operatorname{diag}(a,b),\n\\]\nand use $\\Tr_{K/\\Q}(\\sqrt{-7}\\,h_K)$ on an order lattice.\nAt the three real embeddings the signs of $a$ are $(+,-,+)$ and those\nof $b$ are $(+,-,-)$, so the signatures are $(2,0),(0,2),(1,1)$.\nThe compatible domain has dimension one and generic derived group\n$\\Res_{F/\\Q}\\SU(h_K)$, of complex form $\\SL_2^3$.\nThe determinant cocharacter is proportional to $(1,-1,0)$.\nThe cyclic Galois action on $F$ generates the plane of coordinate sum\nzero; conjugation negates it. Thus the determinant image is exactly\n$z_1z_2z_3=1$. The three standard derived modules are distinct and their\ncentral characters are nonzero, so none is identified with its dual;\nthe generic centralizer is $K$. Very general members are simple with\n$\\End^0(A)=K$.\n\nThe induced $k$-Hermitian form is $h_k=\\Tr_{K/k}(h_K)$ and has signature\n$(3,3)$. In the basis $(1,\\alpha,\\alpha^2)$ the trace-form determinant\nof $F$ is $49$. The multiplication matrices are\n\\[ M_a=\\begin{pmatrix}-1&1&0\\\\1&1&1\\\\1&0&1\\end{pmatrix},\\qquad\nM_b=\\begin{pmatrix}-1&0&1\\\\1&-1&2\\\\0&1&-2\\end{pmatrix}. \\]\nTheir determinants are $-1$ and $1$, respectively. Therefore\n\\[\n \\det(h_k)=49^2\\operatorname{Norm}_{F/\\Q}(ab)=-49^2.\n\\]\nIn $\\Q^*/\\operatorname{Norm}_{k/\\Q}(k^*)$ this is $-1$, since\n$49^2$ is the norm of the rational number $49$.\nThis is exactly the discriminant hypothesis of\n\\cite[Theorem 1.5.1]{Markman}; no additional identification of Hermitian\nforms is needed. Proposition~\\ref{p:coupled-det} gives the Hodge\nconjecture on all powers and\n\\[ P_A(t)=1+3t+6t^2+9t^3+6t^4+3t^5+t^6. \\]\nThe two additional degree-six classes here are the algebraic $k$-Weil\nplane. The full-unitary example above has middle dimension seven, so\nendomorphism degree alone demonstrably does not determine Hodge dimensions.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1764,
        "statement_end_line": 1804,
        "proof_start_line": null,
        "end_line": 1804,
        "latex_labels": [],
        "original_pdf_page": 25,
        "ai_pdf_page": 19
      },
      "citation_keys": [
        "Markman"
      ],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:fermat18",
      "kind": "lemma",
      "number": "10.4",
      "title_latex": "All powers of the Fermat-eighteen Jacobian",
      "row": "S10",
      "assumptions_and_conclusion_latex": "Let $F_{18}$ be the Fermat curve of degree eighteen. The Hodge\nconjecture holds for every power of $J(F_{18})$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1806,
        "statement_end_line": 1809,
        "proof_start_line": 1810,
        "end_line": 1826,
        "latex_labels": [
          "p:fermat18"
        ],
        "original_pdf_page": 25,
        "ai_pdf_page": 20
      },
      "citation_keys": [
        "MilneTateStandard"
      ],
      "logical_dependencies": [
        "EXT:MilneCMproducts",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:MilneCMproducts"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:S10cyclic18",
      "kind": "theorem",
      "number": "10.5",
      "title_latex": "A nonsplit cyclic-eighteen family",
      "row": "S10",
      "assumptions_and_conclusion_latex": "For $t\\in\\C\\setminus\\{0,1\\}$ let $C_t$ be the smooth projective\nmodel of\n\\[\n y^{18}=x(x-1)^4(x-t)^4,\n\\]\nand let $B_t$ be its primitive $K=\\Q(\\zeta_{18})$ Jacobian factor.\nThen $\\dim B_t=6$, its $k=\\Q(\\sqrt{-3})$ action has signature\n$(3,3)$, and its Weil plane $W_k(B_t)$ is algebraic for every $t$.\nThe Hodge conjecture holds for every power of every $B_t$.\nFor very general $t$, $B_t$ is simple with $\\End^0(B_t)=K$, and\n\\[\n \\Hg(B_t)=\\det_K^{-1}(\\ker(N_{K/k}:U_K\\to U_k)),\\qquad\n P_{B_t}(z)=(1+z+z^2)^3+2z^3.\n\\]\nThe inherited polarization has $k$-Hermitian determinant class\n$-2$, not $-1$. For these very general members no change of\npolarization makes this Weil structure split.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 1828,
        "statement_end_line": 1846,
        "proof_start_line": 1847,
        "end_line": 1992,
        "latex_labels": [
          "p:S10cyclic18"
        ],
        "original_pdf_page": 25,
        "ai_pdf_page": 20
      },
      "citation_keys": [
        "MilneTateStandard",
        "MoonenCyclic",
        "YuZheng"
      ],
      "logical_dependencies": [
        "p:fermat18",
        "p:coupled-det",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:MilneCMproducts"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "t:CM",
      "kind": "proposition",
      "number": "11.1",
      "title_latex": "Proposition 11.1",
      "row": "S11",
      "assumptions_and_conclusion_latex": "Let $A$ be a CM sixfold with field $K$ of degree $12$. If its Hodge group\nis the full norm-one torus\n$T_K=\\ker(\\Res_{K/\\Q}\\mathbf G_m\\to\\Res_{K^+/\\Q}\\mathbf G_m)$,\nthen all Hodge classes on all powers are divisor-generated and\n$P_A(t)=(1+t)^6$.\nFor a specified CM type, $\\Hg(A)$ is the smallest rational subtorus\ncontaining its Hodge circle; its cocharacter space is the rational span of\nthe Galois translates of the signed CM-type vector in a normal closure.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2009,
        "statement_end_line": 2018,
        "proof_start_line": 2019,
        "end_line": 2032,
        "latex_labels": [
          "t:CM"
        ],
        "original_pdf_page": 28,
        "ai_pdf_page": 21
      },
      "citation_keys": [],
      "logical_dependencies": [
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:CMtest",
      "kind": "proposition",
      "number": "11.2",
      "title_latex": "An exact finite test for every specified CM type",
      "row": "S11",
      "assumptions_and_conclusion_latex": "Let $A$ be simple with $\\End^0(A)=K$ of degree twelve, and fix its CM\ntype. Form a rational matrix $M$ whose rows are all Galois translates\nof its signed CM-type vector in six conjugate-pair coordinates. For\n$\\delta\\in\\{-1,0,1\\}^6$ write $w(\\delta)$ for its support size.\nLet $n_j$ count the vectors with $M\\delta=0$ and $w(\\delta)=j$.\nThen\n\\[\n \\dim B^2(A)=15+n_4,\\qquad\n \\dim B^3(A)=20+2n_4+n_6.\n\\]\nThe divisor-generated dimensions are $15$ and $20$ respectively.\nThus $n_4=n_6=0$ proves the Hodge conjecture for $A$ even without a\nprior full-rank assertion about $M$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2034,
        "statement_end_line": 2049,
        "proof_start_line": 2050,
        "end_line": 2074,
        "latex_labels": [
          "p:CMtest"
        ],
        "original_pdf_page": 28,
        "ai_pdf_page": 21
      },
      "citation_keys": [],
      "logical_dependencies": [
        "t:CM",
        "p:HLsix"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:11.3",
      "kind": "remark",
      "number": "11.3",
      "title_latex": "Remark 11.3",
      "row": "S11",
      "assumptions_and_conclusion_latex": "This is an exhaustive computation for a specified field and CM type,\nusing at most $3^6=729$ exponent vectors. A character-neutral exceptional\nmonomial is not by itself an algebraic cycle. Nonzero $n_4$ or $n_6$\nstill requires a geometric algebraicity argument for its rational\nGalois-stable span. Theorem~\\ref{p:CMcomplete} below in fact shows\nthat for a simple CM sixfold $n_4=0$ and $n_6\\in\\{0,2\\}$;\nthe test remains a useful independent check on the supplied CM data.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2075,
        "statement_end_line": 2083,
        "proof_start_line": null,
        "end_line": 2083,
        "latex_labels": [],
        "original_pdf_page": 29,
        "ai_pdf_page": 22
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:CMquartic",
      "kind": "corollary",
      "number": "11.4",
      "title_latex": "A checkable degenerate CM stratum",
      "row": "S11",
      "assumptions_and_conclusion_latex": "In Proposition~\\ref{p:CMtest}, suppose $n_4=0$ and every weight-six\nrelation, up to sign, selects precisely the six embeddings above one\nembedding of an imaginary quadratic subfield $k\\subset K$.\nThen all Hodge classes on $A$ are algebraic; no subfield condition\non $K^+$ is needed.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2085,
        "statement_end_line": 2091,
        "proof_start_line": 2092,
        "end_line": 2100,
        "latex_labels": [
          "p:CMquartic"
        ],
        "original_pdf_page": 29,
        "ai_pdf_page": 22
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:CMtest",
        "p:CMsplit",
        "p:fft",
        "p:HLsix"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:CMrankfive",
      "kind": "corollary",
      "number": "11.5",
      "title_latex": "All CM sixfolds of Hodge rank at least five",
      "row": "S11",
      "assumptions_and_conclusion_latex": "If $A$ is a simple CM sixfold and $\\dim\\Hg(A)\\ge5$, the Hodge\nconjecture holds for every power of $A$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2102,
        "statement_end_line": 2106,
        "proof_start_line": 2107,
        "end_line": 2118,
        "latex_labels": [
          "p:CMrankfive"
        ],
        "original_pdf_page": 29,
        "ai_pdf_page": 22
      },
      "citation_keys": [
        "MZ"
      ],
      "logical_dependencies": [
        "p:CMsplit",
        "p:coupled-det",
        "EXT:LowDim"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:CMsixrank",
      "kind": "lemma",
      "number": "11.6",
      "title_latex": "A rank bound without a Galois hypothesis",
      "row": "S11",
      "assumptions_and_conclusion_latex": "Every simple CM sixfold has Hodge rank at least five.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2119,
        "statement_end_line": 2121,
        "proof_start_line": 2122,
        "end_line": 2194,
        "latex_labels": [
          "p:CMsixrank"
        ],
        "original_pdf_page": 29,
        "ai_pdf_page": 22
      },
      "citation_keys": [],
      "logical_dependencies": [
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:CMcomplete",
      "kind": "theorem",
      "number": "11.7",
      "title_latex": "All simple CM sixfolds",
      "row": "S11",
      "assumptions_and_conclusion_latex": "For every simple CM abelian sixfold $A$ and every $n\\ge1$, the\nrational Hodge conjecture holds for $A^n$.\nOn $A$ itself the only possible Hodge-dimension polynomials are\n\\[\n P_A(t)=(1+t)^6\\quad\\hbox{or}\\quad (1+t)^6+2t^3.\n\\]\nIn the first case all classes on every power are divisor-generated.\nIn the second case divisors together with pullbacks of the algebraic\nWeil plane, under $K$-linear sum maps, generate them.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2200,
        "statement_end_line": 2210,
        "proof_start_line": 2211,
        "end_line": 2217,
        "latex_labels": [
          "p:CMcomplete"
        ],
        "original_pdf_page": 30,
        "ai_pdf_page": 23
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:CMsixrank",
        "p:CMrankfive"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:11.8",
      "kind": "remark",
      "number": "11.8",
      "title_latex": "Independent finite cross-check",
      "row": "S11",
      "assumptions_and_conclusion_latex": "The rank-four exclusion also has a small exact check. Normalize\n$f=(1,\\ldots,1)$ and complete it to four independent signed orbit\nvectors. Distinct Hodge weights make the six columns distinct among\nthe eight vectors $(1,a,b,c)$, $a,b,c\\in\\{1,-1\\}$. The two omitted\nvertices have Hamming distance one, two, or three. Up to cube\nsymmetry these are all possibilities. For their row spaces, exact\nsigned-permutation stabilizer computations give, respectively,\n\\[\n\\begin{array}{c|c|c|c}\n\\text{omitted distance}&|\\operatorname{Aut}_{\\pm}(W)|&\n \\text{coordinate orbit sizes}&\n \\dim\\langle\\operatorname{Aut}_{\\pm}(W)f\\rangle\\\\\\hline\n1&96&6&3\\\\\n2&32&4+2&4\\\\\n3&96&6&1\n\\end{array}\n\\]\nThe middle model is not transitive; the others cannot be generated\nby the orbit of $f$. The script \\texttt{verify\\_cm\\_sixfold\\_rank.py}\nchecks all $6!2^6$ signed permutations in each model using rational\nlinear algebra. This verifies the rank lemma independently of\nlisting Galois groups, but is not used in place of its proof.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2219,
        "statement_end_line": 2242,
        "proof_start_line": null,
        "end_line": 2242,
        "latex_labels": [],
        "original_pdf_page": 30,
        "ai_pdf_page": 23
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:CMabelian",
      "kind": "corollary",
      "number": "11.9",
      "title_latex": "All abelian Galois CM fields of degree twelve",
      "row": "S11",
      "assumptions_and_conclusion_latex": "Let $A$ be a simple CM sixfold. If $K=\\End^0(A)$ is Galois and\nabelian over $\\Q$, the Hodge conjecture holds for every power of $A$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2244,
        "statement_end_line": 2248,
        "proof_start_line": 2249,
        "end_line": 2293,
        "latex_labels": [
          "p:CMabelian"
        ],
        "original_pdf_page": 31,
        "ai_pdf_page": 23
      },
      "citation_keys": [
        "Yanai"
      ],
      "logical_dependencies": [
        "p:CMrankfive",
        "t:CM",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:CMgalois",
      "kind": "corollary",
      "number": "11.10",
      "title_latex": "All Galois CM fields of degree twelve",
      "row": "S11",
      "assumptions_and_conclusion_latex": "Let $A$ be a simple CM sixfold whose endomorphism field $K$ is Galois\nover $\\Q$. The Hodge conjecture holds for every power of $A$.\nIf its Galois group is nonabelian, $\\Hg(A)=U_K$ and all these\nHodge classes are divisor-generated.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2295,
        "statement_end_line": 2301,
        "proof_start_line": 2302,
        "end_line": 2366,
        "latex_labels": [
          "p:CMgalois"
        ],
        "original_pdf_page": 31,
        "ai_pdf_page": 23
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:CMabelian",
        "t:CM",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:11.11",
      "kind": "example",
      "number": "11.11",
      "title_latex": "Exact matrix certificate for the trace construction",
      "row": "S11",
      "assumptions_and_conclusion_latex": "Let $F=\\Q(\\theta)$ with\n$f(T)=T^6+T^5-5T^4-4T^3+6T^2+3T-1$; one can take\n$\\theta=2\\cos(2\\pi/13)$. This is the real sextic subfield of\n$\\Q(\\zeta_{13})$. For $K=\\Q(i)F$ and $b'=1/f'(\\theta)$ the trace\nHermitian form, in the $\\Q(i)$-basis $1,\\theta,\\ldots,\\theta^5$, is\n\\[\n \\left(\\Tr_{F/\\Q}\\frac{\\theta^{u+v}}{f'(\\theta)}\\right)_{u,v=0}^5\n =\\begin{pmatrix}\n 0&0&0&0&0&1\\\\\n 0&0&0&0&1&-1\\\\\n 0&0&0&1&-1&6\\\\\n 0&0&1&-1&6&-7\\\\\n 0&1&-1&6&-7&27\\\\\n 1&-1&6&-7&27&-35\n \\end{pmatrix}.\n\\]\nIts determinant is $-1$ and its upper-left $3\\times3$ block is zero.\nChoose the CM type by the positivity signs of $i/f'(\\theta)$; after\nclearing denominators this trace form polarizes the associated CM\nabelian variety. The example verifies the split-form construction.\nFor an arbitrary prescribed balanced CM type, the ordering step in\nLemma~\\ref{p:CMsplit} is essential; this particular $\\theta$ is not\nasserted to fit every type.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2368,
        "statement_end_line": 2392,
        "proof_start_line": null,
        "end_line": 2392,
        "latex_labels": [],
        "original_pdf_page": 32,
        "ai_pdf_page": 24
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "ex:CM13",
      "kind": "example",
      "number": "11.12",
      "title_latex": "Example 11.12",
      "row": "S11",
      "assumptions_and_conclusion_latex": "Let $C$ be the smooth projective curve $y^2=x^{13}-1$ and $A=J(C)$.\nThere are $14$ branch points of the hyperelliptic map including infinity,\nso $g(C)=6$. The automorphism $(x,y)\\mapsto(\\zeta_{13}x,y)$ acts on the\nbasis $x^{a-1}dx/y$, $1\\leq a\\leq6$, with eigenvalues $\\zeta_{13}^a$.\nIt gives an embedding $K=\\Q(\\zeta_{13})\\hookrightarrow\\End^0(A)$ and\nCM type $\\Phi=\\{1,2,3,4,5,6\\}$ modulo $13$.\nIn the basis of six conjugate pairs, six Galois translates of the signed\nCM vector have matrix\n\\[\n M=\\begin{pmatrix}\n1&1&1&1&1&1\\\\\n1&1&1&-1&-1&-1\\\\\n1&1&-1&-1&1&1\\\\\n1&-1&-1&1&-1&-1\\\\\n1&-1&1&-1&-1&1\\\\\n1&-1&1&-1&1&-1\n\\end{pmatrix},\\qquad\\det(M)=-160.\n\\]\nThe row with multiplier $a$ has entry $+1$ at column $b$ if\n$ab\\bmod13\\in\\Phi$, and $-1$ otherwise; an inverse-multiplier convention\ngives the same orbit span. Exact elimination gives the displayed nonzero\ndeterminant, so the Hodge group is the full six-dimensional torus.\nIts rational centralizer is $K$ (over $\\C$ the twelve distinct character\nspaces have diagonal centralizer). Therefore $\\End^0(A)=K$ and $A$ is\nsimple. Proposition~\\ref{t:CM} applies, with dimensions\n$(1,6,15,20,15,6,1)$ and six rational divisor generators.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2394,
        "statement_end_line": 2421,
        "proof_start_line": null,
        "end_line": 2421,
        "latex_labels": [
          "ex:CM13"
        ],
        "original_pdf_page": 33,
        "ai_pdf_page": 24
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:11.13",
      "kind": "remark",
      "number": "11.13",
      "title_latex": "Remark 11.13",
      "row": "S11",
      "assumptions_and_conclusion_latex": "For a proper CM torus the vanishing of the total character must be tested\non that smaller torus, not on $T_K$. Additional monomials may survive.\nTheir exact degrees, rational dimensions, and algebraic representatives\nmust be supplied before a degenerate type is counted as solved.\nAbsolute Hodge classes are not thereby algebraic. Andre's reduction to\nsplit Weil classes also does not keep all auxiliary varieties in dimension\nsix; see \\cite[Theorem 1.4]{MarkmanSurvey}.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2422,
        "statement_end_line": 2430,
        "proof_start_line": null,
        "end_line": 2430,
        "latex_labels": [],
        "original_pdf_page": 33,
        "ai_pdf_page": 25
      },
      "citation_keys": [
        "MarkmanSurvey"
      ],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:11.14",
      "kind": "example",
      "number": "11.14",
      "title_latex": "A degenerate CM sixfold with algebraic exceptional classes",
      "row": "S11",
      "assumptions_and_conclusion_latex": "Let $F=\\Q(\\zeta_{13}+\\zeta_{13}^{-1})$ and $K=F(i)$.\nThen $[K:\\Q]=12$ and $\\mathrm{Gal}(K/\\Q)=C_6\\times C_2$,\nwhere complex conjugation is the second factor. Order the real\nembeddings of $F$ cyclically under a generator of $C_6$. Choose the\nCM type whose signs on the six embeddings of $i$ are\n\\[\n f=(1,1,-1,1,-1,-1).\n\\]\nThe complex torus $A=\\C^6/\\Phi(\\mathcal O_K)$ is polarizable:\nchoose $b\\in F$ with prescribed signs by weak approximation so that\n$ib$ defines a positive CM Riemann form. Clear denominators in\n\\[\\psi(x,y)=\\Tr_{K/\\Q}(ib\\,x\\overline y).\\]\nThe signed CM matrix consists of the six cyclic shifts of $f$.\nIts rows sum to zero. A maximal minor is\n\\[\n \\det\\begin{pmatrix}\n 1&1&-1&1&-1\\\\\n 1&-1&1&-1&-1\\\\\n -1&1&-1&-1&1\\\\\n 1&-1&-1&1&1\\\\\n -1&-1&1&1&-1\n \\end{pmatrix}=-16.\n\\]\nThus its rational kernel is exactly $\\Q(1,1,1,1,1,1)$ and its rank\nis five. All twelve restricted eigencharacters are distinct: their\npairwise differences cannot be multiples of this all-ones vector.\nThe commutant is therefore $K$, which proves $\\End^0(A)=K$ and simplicity.\nIts Hodge group is $\\ker(N_{K/\\Q(i)}:U_K\\to U_{\\Q(i)})$.\nThe finite test gives $n_4=0$, $n_6=2$, and\n\\[\n P_A(t)=(1+t)^6+2t^3,\n \\qquad (1,6,15,22,15,6,1).\n\\]\nThe real field $F$ contains $\\Q(\\sqrt{13})$, so\nLemma~\\ref{p:quartic-scale} applied to $\\Q(i,\\sqrt{13})$ makes the\ntwo exceptional $\\Q(i)$-Weil classes algebraic.\nDivisors and these classes exhaust $A$; Proposition~\\ref{p:coupled-det}\nwith $e=6$, $q=1$ additionally proves algebraicity on every power.\nThis is a degenerate CM example, not merely a full-torus example.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2432,
        "statement_end_line": 2472,
        "proof_start_line": null,
        "end_line": 2472,
        "latex_labels": [],
        "original_pdf_page": 33,
        "ai_pdf_page": 25
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
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      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:12.1",
      "kind": "theorem",
      "number": "12.1",
      "title_latex": "Theorem 12.1",
      "row": "N1",
      "assumptions_and_conclusion_latex": "Let $A\\sim B\\times E$ with $B$ a simple fivefold and $E$ elliptic.\nEvery rational Hodge class on $A$ is algebraic. Its Hodge ring is generated\nby divisors, except that a rational two-dimensional Weil space in\ncodimension three must be added if $E$ has CM by a field acting on $B$\nwith signature $(3,2)$, up to interchange.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2495,
        "statement_end_line": 2501,
        "proof_start_line": 2502,
        "end_line": 2509,
        "latex_labels": [],
        "original_pdf_page": 34,
        "ai_pdf_page": 25
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:prime-ell",
        "p:oddscale"
      ],
      "dependency_coverage": "curated_case_main_theorem",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:12.2",
      "kind": "example",
      "number": "12.2",
      "title_latex": "Example 12.2",
      "row": "N1",
      "assumptions_and_conclusion_latex": "Choose $(B_1,\\ldots,B_{2})$ jointly very generally in\n$\\prod_i\\mathcal A_{d_i}(N)$ with $d_i=(5,1)$,\nas constructed in Proposition~\\ref{p:products}. Then each factor is\nsimple with $\\End^0(B_i)=\\Q$, and the factors are pairwise nonisogenous.\nTheir product is an actual sixfold of this partition, with\n$\\End^0(A)=\\Q^{2}$ and the proved joint group above.\nThe Hodge-space dimensions in codimensions $0$ through $6$ are\n$(1,2,2,2,2,2,1)$.\nAn explicit monomial basis over $\\Q$ on $A$ is\n$\\theta_1^{a_1}\\cdots\\theta_{2}^{a_{2}}$ with\n$0\\leq a_i\\leq d_i$; every such monomial is an intersection of divisors.\nOn powers add Poincare divisors between the copies of each individual\n$B_i$. There are no cross-factor divisors or higher mixed invariants in\nthis verified jointly generic example.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2510,
        "statement_end_line": 2525,
        "proof_start_line": null,
        "end_line": 2525,
        "latex_labels": [],
        "original_pdf_page": 35,
        "ai_pdf_page": 25
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:12.3",
      "kind": "example",
      "number": "12.3",
      "title_latex": "A genuinely exceptional $5+1$ product",
      "row": "N1",
      "assumptions_and_conclusion_latex": "Let $k=\\Q(i)$ and take a very general polarized fivefold $B$ with\n$k$-Hermitian datum $(k^5,\\operatorname{diag}(1,1,1,-1,-1))$.\nThe unitary PEL domain has dimension six; its dense derived monodromy\nand nonzero determinant circle give $\\Hg(B)=U(3,2)$ and\n$\\End^0(B)=k$. Thus $B$ is simple. Let $E:y^2=x^3-x$ with the opposite\n$k$-action. Then $A=B\\times E$ has signature $(3,3)$ and determinant\n$-1$. Lemma~\\ref{p:prime-ell} gives\n\\[\n P_A(t)=(1+t+\\cdots+t^5)(1+t)+2t^3,\n \\qquad (\\dim B^p(A))_{p=0}^6=(1,2,2,4,2,2,1).\n\\]\nProducts of the two polarization classes span the divisor part;\nthe remaining middle plane is $W_k(A)$, algebraic by the split Weil\ntheorem. This example explicitly exhibits the mixed classes absent\nfrom the jointly symplectic example.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2526,
        "statement_end_line": 2542,
        "proof_start_line": null,
        "end_line": 2542,
        "latex_labels": [],
        "original_pdf_page": 35,
        "ai_pdf_page": 26
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N2CM-new",
      "kind": "corollary",
      "number": "13.1",
      "title_latex": "All CM--CM products in N2",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Assume the CM input in Theorem~\\ref{p:openai-inputs}.\nIf $B$ is a simple CM fourfold and $S$ a simple CM surface, then\nthe Hodge conjecture holds on every power of $B\\times S$.\nIn particular every coupled case in Proposition~\\ref{p:N2CMstructure}\nis algebraic; no independence of the two Hodge groups is required.\nTogether with Proposition~\\ref{p:N2strata}, this settles every\nsixfold in N2 whose fourfold factor is CM.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2554,
        "statement_end_line": 2562,
        "proof_start_line": 2563,
        "end_line": 2570,
        "latex_labels": [
          "p:N2CM-new"
        ],
        "original_pdf_page": 35,
        "ai_pdf_page": 26
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:CMproducts-new",
        "p:N2strata"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:CM",
        "EXT:Classical",
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": [
        "EXT:CM"
      ]
    },
    {
      "id": "ex:N2CM-new",
      "kind": "example",
      "number": "13.2",
      "title_latex": "A CM fourfold times a CM surface",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Let $L=\\Q(\\zeta_{15})$ and take the CM type with embedding\nexponents $\\Phi=\\{1,2,4,7\\}$ modulo $15$.\nIt selects one embedding from each conjugate pair.\nIts multiplicative stabilizer in $(\\Z/15\\Z)^\\times$ is trivial:\nany stabilizing element belongs to $\\Phi$, and direct multiplication\nexcludes $2,4,7$. Since $L/\\Q$ is Galois, the type is primitive.\nThe standard CM construction therefore gives a simple fourfold\n$B=\\C^4/\\Phi(\\mathcal O_L)$.\nFor its polarization, choose $\\delta\\in L$ with\n$\\bar\\delta=-\\delta$ and $\\operatorname{Im}\\phi(\\delta)>0$\nfor every $\\phi\\in\\Phi$, by weak approximation.\nAn integral multiple of $\\Tr_{L/\\Q}(\\delta x\\bar y)$ on\n$\\mathcal O_L$ is a Riemann form. Primitivity of the CM type\ngives simplicity.\nTake $S=J(y^2=x^5-1)$, the simple CM surface with field\n$\\Q(\\zeta_5)$. Corollary~\\ref{p:N2CM-new} proves the Hodge\nconjecture for every $(B\\times S)^n$ by the external CM theorem.\nThis argument does not require computing the joint torus or assuming\nthat its exceptional classes are divisor-generated.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2571,
        "statement_end_line": 2591,
        "proof_start_line": null,
        "end_line": 2591,
        "latex_labels": [
          "ex:N2CM-new"
        ],
        "original_pdf_page": 35,
        "ai_pdf_page": 26
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:13.3",
      "kind": "proposition",
      "number": "13.3",
      "title_latex": "Proposition 13.3",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Let $A\\sim B_1\\times\\cdots\\times B_{2}$ with dimensions\n$(4,2)$. Assume the actual joint Hodge group is the\nindependent product of the full symplectic groups on these factors.\nThen all Hodge classes on all powers of $A$ are divisor-generated,\nand $P_A(t)=(1+t+\\cdots+t^{4})(1+t+t^2)$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2593,
        "statement_end_line": 2599,
        "proof_start_line": 2600,
        "end_line": 2611,
        "latex_labels": [],
        "original_pdf_page": 36,
        "ai_pdf_page": 26
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:isogeny",
        "p:fft"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N2strata",
      "kind": "proposition",
      "number": "13.4",
      "title_latex": "Two further complete strata",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Let $A=B\\times S$, with $B$ a simple fourfold and $S$ a simple surface.\nThe Hodge conjecture holds for $A$ if either $S$ is not CM, or the center\nof $\\End^0(B)$ is totally real or imaginary quadratic.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2612,
        "statement_end_line": 2616,
        "proof_start_line": 2617,
        "end_line": 2630,
        "latex_labels": [
          "p:N2strata"
        ],
        "original_pdf_page": 36,
        "ai_pdf_page": 26
      },
      "citation_keys": [
        "MZ",
        "Markman"
      ],
      "logical_dependencies": [
        "p:lowdim"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N2galois",
      "kind": "proposition",
      "number": "13.5",
      "title_latex": "A splitting-field separation test",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Let $A=B\\times S$, where $B$ is a simple fourfold and $S$ a simple\nCM surface with field $F$. Suppose the center $K$ of $\\End^0(B)$\nis CM. Let $L_F,L_K\\subset\\overline\\Q$ be their normal closures.\nIf $L_F\\not\\subset L_K$, then\n$\\Hg(A)=\\Hg(B)\\times\\Hg(S)$ and the Hodge conjecture holds for $A$.\nIn particular this holds when $K/\\Q$ is multiquadratic, including\nevery biquadratic quartic center.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2631,
        "statement_end_line": 2639,
        "proof_start_line": 2640,
        "end_line": 2651,
        "latex_labels": [
          "p:N2galois"
        ],
        "original_pdf_page": 36,
        "ai_pdf_page": 26
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:lowdim"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N2nonCMstructure",
      "kind": "proposition",
      "number": "13.6",
      "title_latex": "All residual non-CM-fourfold groups in N2",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Let $A=B\\times S$ be an N2 case with $B$ not CM and not already\ncovered by Propositions~\\ref{p:N2strata}--\\ref{p:N2galois}.\nThen $\\End^0(B)=K$ is a primitive quartic CM field,\n$\\End^0(S)=F$ is a primitive quartic CM field, and the two\n$K$-signatures on $B$ are $(2,0),(1,1)$ after orientation.\nIn particular $\\Hg(B)$ is the full $K$-unitary group.\nOrient the two complex coordinates on $U_F$ at the CM type of $S$.\nThere are precisely two alternatives:\n\\begin{enumerate}\n\\item There is no rational torus morphism $\\nu:U_F\\to U_K$\nwhose complex coordinate expression is $(u,v)\\mapsto(uv,u/v)$,\nup to conjugating the second $K$ embedding. Then the joint group\nis the independent product and the Hodge conjecture holds.\n\\item Such a morphism exists. It is an isogeny, and the joint group is\n\\[\n \\{(g,s)\\in\\Hg(B)\\times U_F:\\det_K g=\\nu(s)\\}.\n\\]\nIts Hodge ring and polynomial are exactly those in\nLemma~\\ref{p:quartic-mixed}. Algebraicity is equivalent to the\nexistence of one codimension-two algebraic cycle with nonzero\nprojection to the four-dimensional space $Q$ in that lemma.\n\\end{enumerate}\nThe torus-morphism condition is a finite Galois-equivariance test\non the displayed integral matrix; it is not an algebraicity assumption.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2652,
        "statement_end_line": 2678,
        "proof_start_line": 2679,
        "end_line": 2729,
        "latex_labels": [
          "p:N2nonCMstructure"
        ],
        "original_pdf_page": 36,
        "ai_pdf_page": 27
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:N2strata",
        "p:N2galois",
        "p:quartic-mixed",
        "TOOL:determinant-projector"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N2CMrankthree",
      "kind": "proposition",
      "number": "13.7",
      "title_latex": "Rank-three CM fourfolds are independent of surfaces",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Let $B$ be a simple CM fourfold with $L=\\End^0(B)$.\nIts Hodge rank is three or four. In rank three, $\\Hg(B)$ is a\n$\\Q$-simple torus. Consequently, for every simple CM surface $S$,\n\\[\n \\Hg(B\\times S)=\\Hg(B)\\times\\Hg(S),\n\\]\nand the Hodge conjecture holds for $B\\times S$. In this case\n\\[\n P_{B\\times S}(t)=(1+t)^6+2t^2(1+t)^2\n =1+6t+17t^2+24t^3+17t^4+6t^5+t^6.\n\\]",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2731,
        "statement_end_line": 2744,
        "proof_start_line": 2745,
        "end_line": 2775,
        "latex_labels": [
          "p:N2CMrankthree"
        ],
        "original_pdf_page": 37,
        "ai_pdf_page": 27
      },
      "citation_keys": [
        "MZ",
        "Markman"
      ],
      "logical_dependencies": [
        "p:lowdim"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N2CMstructure",
      "kind": "proposition",
      "number": "13.8",
      "title_latex": "All CM-fourfold couplings in N2",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Let $B$ be a simple CM fourfold, $L=\\End^0(B)$, and let $S$ be\na simple CM surface with field $F$. If their joint Hodge group\nis not the independent product, then $\\Hg(B)=U_L$ and there is\na primitive quartic CM subfield $K\\subset L$ such that the\n$K$-signatures on $B$ are $(2,0),(1,1)$ after orientation.\nThere is a rational torus isogeny $\\nu:U_F\\to U_K$ with complex\ncoordinates $(u,v)\\mapsto(uv,u/v)$, and the actual joint group is\n\\[\n H=\\{(b,s)\\in U_L\\times U_F:N_{L/K}(b)=\\nu(s)\\}.\n\\]\nConversely, these field, signature, and equivariant-isogeny data\nwith $\\Hg(B)=U_L$ imply this group equality. Put\n\\[\n T=\\bigwedge\\nolimits_K^2H^1(B,\\Q),\\qquad\n Q=\\bigl(T\\otimes H^2(S,\\Q)\\bigr)\\cap H^{2,2}(B\\times S),\n\\]\nusing the rational-summand convention of Lemma~\\ref{p:cyclic-transfer}.\nThen $\\dim_\\Q Q=4$, divisors and $Q$ generate the Hodge ring, and\n\\[\n P_{B\\times S}(t)=(1+t)^6+4t^2(1+t)^2\n =1+6t+19t^2+28t^3+19t^4+6t^5+t^6.\n\\]\nThe Hodge conjecture is equivalent to the existence of a\ncodimension-two algebraic cycle whose projection to $Q$ is nonzero.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2777,
        "statement_end_line": 2803,
        "proof_start_line": 2804,
        "end_line": 2893,
        "latex_labels": [
          "p:N2CMstructure"
        ],
        "original_pdf_page": 38,
        "ai_pdf_page": 27
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:N2CMrankthree",
        "p:quartic-mixed",
        "TOOL:determinant-projector"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:13.9",
      "kind": "remark",
      "number": "13.9",
      "title_latex": "A finite-kernel obstruction to an automatic reduction",
      "row": "N2",
      "assumptions_and_conclusion_latex": "In either coupled N2 model, projection of the joint group onto\n$\\Hg(B)$ has kernel $\\mu_2$: its nontrivial element acts trivially\non $H^1(B)$ and as $-1$ on $H^1(S)$. Thus $H^1(S)$ is not an\nobject of the tensor category generated by $H^1(B)$ and Tate\nstructures. Even an all-powers theorem for $B$ does not make $S$\nan available factor by tensor operations. The even-degree\nmixed Hodge tensors still require an algebraic correspondence;\nthe projection criterion above records exactly that missing step.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2903,
        "statement_end_line": 2912,
        "proof_start_line": null,
        "end_line": 2912,
        "latex_labels": [],
        "original_pdf_page": 39,
        "ai_pdf_page": 28
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:quartic-mixed",
      "kind": "lemma",
      "number": "13.10",
      "title_latex": "The quartic mixed invariant calculation",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Let $B$ be a fourfold and $P$ an abelian surface. Suppose that over\n$\\C$ their actual joint Hodge representation is\n\\[\n H^1(B)_\\C=V_1\\oplus V_1^*\\oplus V_2\\oplus V_2^*,\\quad\n H^1(P)_\\C=\\C_u\\oplus\\C_{u^{-1}}\\oplus\\C_v\\oplus\\C_{v^{-1}},\n \\qquad \\dim V_j=2,\n\\]\nwith group\n\\[\n H=\\{(g_1,g_2,u,v):\\det g_1=uv,\\ \\det g_2=u/v\\}\n \\subset\\GL_2^2\\times\\mathbf G_m^2.\n\\]\nSuppose also that the sum $T_\\C$ of the four determinant lines\n$\\bigwedge^2V_j,\\bigwedge^2V_j^*$ descends to\n$T\\subset H^2(B,\\Q)$. There is a four-dimensional rational space\n\\[\n Q=\\bigl(T\\otimes H^2(P,\\Q)\\bigr)\\cap H^{2,2}(B\\times P)\n \\subset\\B^2(B\\times P)\n\\]\nsuch that the Hodge ring is generated by divisors and $Q$. More precisely,\n\\[\n P_{B\\times P}(t)=(1+t+t^2)^2(1+t)^2+4t^2(1+t+t^2)\n =1+4t+12t^2+14t^3+12t^4+4t^5+t^6.\n\\]\nThus the Hodge conjecture on this sixfold is equivalent to the\nalgebraicity of $Q$, not merely of a middle-degree Weil plane.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2914,
        "statement_end_line": 2941,
        "proof_start_line": 2942,
        "end_line": 2960,
        "latex_labels": [
          "p:quartic-mixed"
        ],
        "original_pdf_page": 40,
        "ai_pdf_page": 28
      },
      "citation_keys": [],
      "logical_dependencies": [
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:mixed-allpowers",
      "kind": "proposition",
      "number": "13.11",
      "title_latex": "Tensor contraction for every residual mixed model",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Let $A=B\\times P$ be a residual coupled case of N2 or N3 in\nPropositions~\\ref{p:N2nonCMstructure}, \\ref{p:N2CMstructure}, or\n\\ref{p:N3residual}; thus $B$ is simple and $P$ is a CM surface or a\nproduct of the two specified CM elliptic curves. The following are\nequivalent:\n\\begin{enumerate}\n\\item some codimension-two algebraic cycle on $A$ has nonzero\nprojection to the four-dimensional mixed space $Q$;\n\\item the Hodge conjecture holds for $A$;\n\\item the Hodge conjecture holds for $A^n$ for every $n\\ge1$.\n\\end{enumerate}",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 2962,
        "statement_end_line": 2975,
        "proof_start_line": 2976,
        "end_line": 3033,
        "latex_labels": [
          "p:mixed-allpowers"
        ],
        "original_pdf_page": 40,
        "ai_pdf_page": 29
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:quartic-mixed",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:cyclic-transfer",
      "kind": "lemma",
      "number": "13.12",
      "title_latex": "A Fermat correspondence for weighted cyclic covers",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Let $m\\in\\{5,8,10,12\\}$ and let $\\boldsymbol a=(a_1,a_2,a_3,a_4)$\nsatisfy $1\\le a_i<m$, $\\sum a_i\\equiv0\\pmod m$, and\n$\\gcd(m,a_1,a_2,a_3,a_4)=1$. Assume the four primitive multiplicities\n\\[\n d_j=-1+\\sum_{i=1}^4\\left\\langle-\\frac{ja_i}{m}\\right\\rangle,\n \\qquad j\\in(\\Z/m\\Z)^\\times,\n\\]\nhave multiset $\\{2,1,1,0\\}$. For $t\\in\\C\\setminus\\{0,1\\}$ let\n$C_{m,\\boldsymbol a,t}$ be the smooth projective model of\n\\[\n y^m=x^{a_1}(x-1)^{a_2}(x-t)^{a_3},\n\\]\nand let $B_{m,\\boldsymbol a,t}$ be the primitive $K=\\Q(\\zeta_m)$\nisogeny factor of its Jacobian. This factor has dimension four;\nthe entire Jacobian need not have dimension four. Put\n\\[\n T=\\bigwedge\\nolimits_K^2 H^1(B_{m,\\boldsymbol a,t},\\Q)\n \\subset H^2(B_{m,\\boldsymbol a,t},\\Q),\n\\]\nwhere the inclusion denotes the rational summand comprising the four\nsame-embedding determinant lines. For every abelian surface $P$, all\nrational Hodge classes in $T\\otimes H^2(P,\\Q)$ are algebraic on\n$B_{m,\\boldsymbol a,t}\\times P$, for every allowed $t$.\nFor $m=5,8,12$ and $\\boldsymbol a=(1,1,1,m-3)$ retain the shorthand\n$C_{m,t},B_{m,t}$ used below.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3035,
        "statement_end_line": 3062,
        "proof_start_line": 3063,
        "end_line": 3205,
        "latex_labels": [
          "p:cyclic-transfer"
        ],
        "original_pdf_page": 41,
        "ai_pdf_page": 29
      },
      "citation_keys": [
        "MilneLefschetz",
        "MoonenCyclic"
      ],
      "logical_dependencies": [
        "EXT:Classical",
        "p:lowdim"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N2cyclic",
      "kind": "theorem",
      "number": "13.13",
      "title_latex": "A complete cyclic-five-cover family in N2",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Let $B_t=J(C_{5,t})$ with $\\End^0(B_t)=\\Q(\\zeta_5)$, and let\n$S=J(v^2=z^5-1)$. Then the Hodge conjecture holds for $B_t\\times S$.\nIts Hodge dimensions are $(1,4,12,14,12,4,1)$, including four\nexceptional degree-four classes. The endomorphism hypothesis holds\nfor very general $t$, so this is a nonconstant family of sixfolds\nwith a simple fourfold and a simple CM surface.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3207,
        "statement_end_line": 3214,
        "proof_start_line": 3215,
        "end_line": 3241,
        "latex_labels": [
          "p:N2cyclic"
        ],
        "original_pdf_page": 43,
        "ai_pdf_page": 31
      },
      "citation_keys": [
        "MoonenCyclic"
      ],
      "logical_dependencies": [
        "p:cyclic-transfer",
        "p:mixed-allpowers"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:weighted-families",
      "kind": "theorem",
      "number": "13.14",
      "title_latex": "All weighted quartic-cover families",
      "row": "N2",
      "assumptions_and_conclusion_latex": "For any branch data in Lemma~\\ref{p:cyclic-transfer}, put\n$B_t=B_{m,\\boldsymbol a,t}$ and choose\n\\[\n P=\\begin{cases}\n J(v^2=z^5-1),&m=5,10,\\\\\n E_i\\times E_{\\sqrt{-2}},&m=8,\\\\\n E_i\\times E_{\\sqrt{-3}},&m=12,\n \\end{cases}\n\\]\nwhere $E_{\\sqrt{-d}}$ has CM by $\\Q(\\sqrt{-d})$ and $E_i$ by\n$\\Q(i)$. Then the Hodge conjecture holds for $B_t\\times P$ for every\n$t\\ne0,1$. Whenever $B_t$ is simple, it holds for every power of\n$B_t\\times P$. Very general parameters have simple $B_t$ with\n$\\End^0(B_t)=K$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3243,
        "statement_end_line": 3258,
        "proof_start_line": 3259,
        "end_line": 3295,
        "latex_labels": [
          "p:weighted-families"
        ],
        "original_pdf_page": 44,
        "ai_pdf_page": 31
      },
      "citation_keys": [
        "MoonenCyclic"
      ],
      "logical_dependencies": [
        "p:cyclic-transfer",
        "p:mixed-allpowers"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:curve-specialization",
      "kind": "lemma",
      "number": "13.15",
      "title_latex": "Specializing a fixed Hodge subvariation over a curve",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Let $f:\\mathcal X\\to C$ be a smooth projective family over a smooth\nconnected projective complex curve, and let $\\mathcal V\\subset R^{2p}f_*\\Q$\nbe a rational polarizable subvariation everywhere of type $(p,p)$.\nIf every class of $\\mathcal V_c$ is algebraic for very general $c$,\nthe same holds at every $c\\in C$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3297,
        "statement_end_line": 3304,
        "proof_start_line": 3305,
        "end_line": 3328,
        "latex_labels": [
          "p:curve-specialization"
        ],
        "original_pdf_page": 44,
        "ai_pdf_page": 31
      },
      "citation_keys": [],
      "logical_dependencies": [
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:PELcompletion",
      "kind": "theorem",
      "number": "13.16",
      "title_latex": "Completion of the represented one-dimensional PEL types",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Fix the marked rational polarized PEL datum of any family in\nTheorem~\\ref{p:weighted-families}, including its signatures and the\nmatching fixed surface $P$. Every simple fourfold $B$ with this datum\nsatisfies the Hodge conjecture on all powers of $B\\times P$.\nLikewise, every simple sixfold with the marked rational PEL datum of\nTheorem~\\ref{p:S10cyclic18} satisfies the Hodge conjecture on all powers.\nHere equality of data means a $K$-linear identification of the rational\nHermitian spaces up to compatible positive conformal scaling, in the\nsame period-domain component. No new rational Hermitian form is added\nby this completion step.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3330,
        "statement_end_line": 3342,
        "proof_start_line": 3343,
        "end_line": 3394,
        "latex_labels": [
          "p:PELcompletion"
        ],
        "original_pdf_page": 45,
        "ai_pdf_page": 32
      },
      "citation_keys": [
        "MilneShimura"
      ],
      "logical_dependencies": [
        "p:weighted-families",
        "p:S10cyclic18",
        "p:curve-specialization",
        "p:mixed-allpowers",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:MilneCMproducts"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "ex:weighted-newtypes",
      "kind": "example",
      "number": "13.17",
      "title_latex": "Two additional marked rational Hermitian types",
      "row": "N2",
      "assumptions_and_conclusion_latex": "The branch vectors\n\\[\n (m,\\boldsymbol a)=(10,(1,2,2,5)),\\qquad(12,(1,1,4,6))\n\\]\ngive the curves\n$y^{10}=x(x-1)^2(x-t)^2$ and\n$y^{12}=x(x-1)(x-t)^4$. Their genera are six and seven, respectively,\nby $g=1+m-\\frac12\\sum_i\\gcd(m,a_i)$. Their primitive factors both\nhave dimension four. The preceding theorem proves the claimed N2\nand N3 products at every parameter, and all powers when that factor\nis simple. At non-CM parameters their Hodge polynomial is\n$1+4t+12t^2+14t^3+12t^4+4t^5+t^6$.\n\nThese extend the rational Hermitian types in the earlier examples.\nFor $x=\\zeta_m^{a_1},y=\\zeta_m^{a_2},z=\\zeta_m^{a_3}$, the geometric\nskew-Hermitian intersection matrix of \\cite[Proposition 7.4]{YuZheng}\nhas determinant\n\\[\n D(\\boldsymbol a)=\\frac{1-xyz}{(1-x)(1-y)(1-z)}.\n\\]\nUse the same imaginary scalar to turn each compared pair of skew\nforms into Hermitian forms, so this scalar cancels in their ratio.\nFor the first comparison identify $\\zeta_5=\\zeta_{10}^2$; for the\nsecond write $\\sqrt3=2\\zeta_{12}-\\zeta_{12}^3$. Exact calculation gives\n\\[\n \\frac{D_{10}(1,2,2,5)}{D_5(1,1,1,2)}=2,\\qquad\n \\frac{D_{12}(1,1,4,6)}{D_{12}(1,1,1,9)}\n       =1-\\frac{\\sqrt3}{3}.\n\\]\nAt the prime over $2$ of $\\Q(\\sqrt5)$, the extension\n$\\Q(\\zeta_5)/\\Q(\\sqrt5)$ is unramified quadratic and inert:\n$2$ has order four modulo five, and its residue degree in the real\nsubfield is two. The first ratio has valuation one. At\n$\\pi=\\sqrt3$ in $\\Q(\\sqrt3)$, the extension obtained by adjoining\n$i$ is unramified and inert, since $X^2+1$ is irreducible over\n$\\mathbf F_3$. The second ratio equals $(\\pi-1)/\\pi$ and has\nvaluation $-1$. Local norms in an unramified quadratic extension\nhave even valuation, so neither ratio is a relative norm.\nA basis change multiplies a Hermitian determinant by a relative norm;\nconformal scaling in rank two multiplies it by a square, also a norm.\nThus each new marked rational type differs from the indicated old\none. Theorem~\\ref{p:PELcompletion} covers all simple members of each\nrepresented marked rational PEL type, not only the explicitly\nparametrized fibers. This is a comparison within this manuscript,\nnot a claim of priority or of distinctness of every branch orbit.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3396,
        "statement_end_line": 3443,
        "proof_start_line": null,
        "end_line": 3443,
        "latex_labels": [
          "ex:weighted-newtypes"
        ],
        "original_pdf_page": 46,
        "ai_pdf_page": 32
      },
      "citation_keys": [
        "YuZheng"
      ],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:13.18",
      "kind": "remark",
      "number": "13.18",
      "title_latex": "Finite branch catalogue and its precise scope",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Modulo permutations of the four branch points and multiplication of\nall exponents by a unit modulo $m$, the hypotheses of\nLemma~\\ref{p:cyclic-transfer} give the following representatives:\n\\begin{center}\\small\n\\begin{tabular}{cl}\n\\toprule\n$m$ & Branch representatives\\\\\\midrule\n5 & $(1,1,1,2)$\\\\\n8 & $(1,1,1,5),\\ (1,1,2,4),\\ (1,2,2,3)$\\\\\n10 & $(1,1,2,6),\\ (1,1,3,5),\\ (1,2,2,5),\\ (1,2,3,4)$\\\\\n12 & $(1,1,1,9),\\ (1,1,2,8),\\ (1,1,3,7),\\ (1,1,4,6),$\\\\\n & $(1,2,3,6),\\ (1,2,4,5),\\ (1,3,4,4),$\\\\\n & $(1,3,10,10),\\ (2,3,3,4)$\\\\\\bottomrule\n\\end{tabular}\n\\end{center}\nThe enumeration ranges over $1\\le a_1,a_2,a_3<m$, sets\n$a_4=-a_1-a_2-a_3\\pmod m$, and applies exactly the nonzero, gcd,\nand multiplicity tests in the lemma. Canonicalizing by all units and\nsorting gives $1,3,4,9$ orbits, from $16,112,240,496$ ordered types.\nThe script \\texttt{verify\\_weighted\\_quartic.py} independently checks\nunordered quadruples and reconstructs all ordered orbits, and verifies\nthe two determinant ratios. All seventeen branch families are covered\nby Theorem~\\ref{p:weighted-families}; they are not claimed to represent\nseventeen distinct rational PEL types. Exhaustion of these finite\ncovering data does not exhaust all forms in N2 or N3.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3445,
        "statement_end_line": 3471,
        "proof_start_line": null,
        "end_line": 3471,
        "latex_labels": [],
        "original_pdf_page": 46,
        "ai_pdf_page": 32
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N2auxiliary",
      "kind": "theorem",
      "number": "13.19",
      "title_latex": "Residual N2 from the auxiliary-spin tensor",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Assume (Aux) and the CM Hodge theorem\n\\cite[Theorem 1.1]{OpenAICM}. For every coupled non-CM case\n$A=B\\times P$ of Proposition~\\ref{p:N2nonCMstructure}, there is\n$Y\\in\\CH^2(A)_\\Q$ with nonzero projection to its mixed space $Q$.\nConsequently the rational Hodge conjecture holds for every $A^n$,\n$n\\geq1$. There is no restriction on the primitive quartic field\nor on the rational Hermitian form.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3502,
        "statement_end_line": 3511,
        "proof_start_line": 3512,
        "end_line": 3679,
        "latex_labels": [
          "p:N2auxiliary"
        ],
        "original_pdf_page": 47,
        "ai_pdf_page": 33
      },
      "citation_keys": [
        "OpenAICM"
      ],
      "logical_dependencies": [
        "p:N2nonCMstructure",
        "p:quartic-mixed",
        "p:mixed-allpowers",
        "EXT:Aux",
        "EXT:CM",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Aux",
        "EXT:CM",
        "EXT:Classical",
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": [
        "EXT:Aux",
        "EXT:CM"
      ]
    },
    {
      "id": "result:13.20",
      "kind": "example",
      "number": "13.20",
      "title_latex": "Example 13.20",
      "row": "N2",
      "assumptions_and_conclusion_latex": "Choose $(B_1,\\ldots,B_{2})$ jointly very generally in\n$\\prod_i\\mathcal A_{d_i}(N)$ with $d_i=(4,2)$,\nas constructed in Proposition~\\ref{p:products}. Then each factor is\nsimple with $\\End^0(B_i)=\\Q$, and the factors are pairwise nonisogenous.\nTheir product is an actual sixfold of this partition, with\n$\\End^0(A)=\\Q^{2}$ and the proved joint group above.\nThe Hodge-space dimensions in codimensions $0$ through $6$ are\n$(1,2,3,3,3,2,1)$.\nAn explicit monomial basis over $\\Q$ on $A$ is\n$\\theta_1^{a_1}\\cdots\\theta_{2}^{a_{2}}$ with\n$0\\leq a_i\\leq d_i$; every such monomial is an intersection of divisors.\nOn powers add Poincare divisors between the copies of each individual\n$B_i$. There are no cross-factor divisors or higher mixed invariants in\nthis verified jointly generic example.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3680,
        "statement_end_line": 3695,
        "proof_start_line": null,
        "end_line": 3695,
        "latex_labels": [],
        "original_pdf_page": 49,
        "ai_pdf_page": 34
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N3CM-new",
      "kind": "corollary",
      "number": "14.1",
      "title_latex": "All remaining CM-fourfold cases in N3",
      "row": "N3",
      "assumptions_and_conclusion_latex": "Assume the CM input in Theorem~\\ref{p:openai-inputs}.\nIf $B,E_1,E_2$ are CM, then the Hodge conjecture holds on all\npowers of $B\\times E_1\\times E_2$, whether or not the elliptic\nfields coincide or the joint torus splits as a product.\nTogether with Proposition~\\ref{p:N3repeated}, this settles every\nsixfold in N3 whose fourfold factor is CM.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3707,
        "statement_end_line": 3714,
        "proof_start_line": 3715,
        "end_line": 3721,
        "latex_labels": [
          "p:N3CM-new"
        ],
        "original_pdf_page": 49,
        "ai_pdf_page": 34
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:CMproducts-new",
        "p:N3strata",
        "p:N3repeated"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:CM",
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": [
        "EXT:CM"
      ]
    },
    {
      "id": "result:14.2",
      "kind": "example",
      "number": "14.2",
      "title_latex": "Distinct elliptic fields with a CM fourfold",
      "row": "N3",
      "assumptions_and_conclusion_latex": "Take the simple CM fourfold $B$ of Example~\\ref{ex:N2CM-new},\nand elliptic curves $E_1,E_2$ with CM by\n$k_1=\\Q(\\sqrt{-3})$ and $k_2=\\Q(\\sqrt{-15})$, respectively.\nBoth fields are contained in $\\Q(\\zeta_{15})$ and are distinct.\nFor the type $\\{1,2,4,7\\}$, the multiplicities for the four\nembeddings of $k_1k_2$, indexed by signs, are\n$(n_{++},n_{+-},n_{-+},n_{--})=(2,1,1,0)$ after orientation.\nThus the fourfold has the pure--mixed quartic pattern rather than\ntwo balanced quartic signatures. Nevertheless all powers of\n$B\\times E_1\\times E_2$ satisfy the Hodge conjecture by\nCorollary~\\ref{p:N3CM-new}, including their mixed classes.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3722,
        "statement_end_line": 3734,
        "proof_start_line": null,
        "end_line": 3734,
        "latex_labels": [],
        "original_pdf_page": 49,
        "ai_pdf_page": 35
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:14.3",
      "kind": "proposition",
      "number": "14.3",
      "title_latex": "Proposition 14.3",
      "row": "N3",
      "assumptions_and_conclusion_latex": "Let $A\\sim B_1\\times\\cdots\\times B_{3}$ with dimensions\n$(4,1,1)$. Assume the actual joint Hodge group is the\nindependent product of the full symplectic groups on these factors.\nThen all Hodge classes on all powers of $A$ are divisor-generated,\nand $P_A(t)=(1+t+\\cdots+t^{4})(1+t)(1+t)$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3736,
        "statement_end_line": 3742,
        "proof_start_line": 3743,
        "end_line": 3754,
        "latex_labels": [],
        "original_pdf_page": 50,
        "ai_pdf_page": 35
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:isogeny",
        "p:fft"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N3strata",
      "kind": "proposition",
      "number": "14.4",
      "title_latex": "Quadratic-center and independent strata",
      "row": "N3",
      "assumptions_and_conclusion_latex": "Let $A=B\\times E_1\\times E_2$, with $B$ a simple fourfold.\nThe Hodge conjecture holds if no CM field of an $E_i$ embeds in the\ncenter of $\\End^0(B)$. It also holds whenever $\\End^0(B)$ itself is\nan imaginary quadratic field $k$, including both $E_i$ with CM by $k$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3755,
        "statement_end_line": 3760,
        "proof_start_line": 3761,
        "end_line": 3799,
        "latex_labels": [
          "p:N3strata"
        ],
        "original_pdf_page": 50,
        "ai_pdf_page": 35
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:lowdim",
        "p:oddscale"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N3repeated",
      "kind": "proposition",
      "number": "14.5",
      "title_latex": "All repeated elliptic factors",
      "row": "N3",
      "assumptions_and_conclusion_latex": "For every simple fourfold $B$ and elliptic curve $E$, the Hodge\nconjecture holds for $B\\times E^2$, without a restriction on\n$\\End^0(B)$. More generally, for $B\\times E_1\\times E_2$ the\nonly cases not covered here have $E_1,E_2$ nonisogenous CM curves\nwith both CM fields embedding in the center of $\\End^0(B)$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3800,
        "statement_end_line": 3806,
        "proof_start_line": 3807,
        "end_line": 3847,
        "latex_labels": [
          "p:N3repeated"
        ],
        "original_pdf_page": 50,
        "ai_pdf_page": 35
      },
      "citation_keys": [
        "MZ"
      ],
      "logical_dependencies": [
        "p:N3strata",
        "p:lowdim",
        "p:oddscale"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N3residual",
      "kind": "proposition",
      "number": "14.6",
      "title_latex": "The full residual structure in N3",
      "row": "N3",
      "assumptions_and_conclusion_latex": "Let $A=B\\times E_1\\times E_2$ be an N3 case not already covered\nby Proposition~\\ref{p:N3repeated}. Write $k_i=\\End^0(E_i)$.\nThen $k_1\\ne k_2$ are imaginary quadratic fields in the center\nof $\\End^0(B)$, and $K=k_1k_2$ acts on $H^1(B)$ with signatures\n$(2,0),(1,1)$, after orienting the two actions. Put\n\\[\n T=\\bigwedge\\nolimits_K^2 H^1(B,\\Q),\\qquad\n Q=\\bigl(T\\otimes H^1(E_1,\\Q)\\otimes H^1(E_2,\\Q)\\bigr)\n       \\cap H^{2,2}(A).\n\\]\nHere $T$ has the same summand interpretation as in\nLemma~\\ref{p:cyclic-transfer}. There are exactly two alternatives:\n\\begin{enumerate}\n\\item $\\End^0(B)=K$, $\\Hg(B)_\\C=\\GL_2^2$, and\n$P_A(t)=1+4t+12t^2+14t^3+12t^4+4t^5+t^6$;\n\\item $B$ is simple CM with field $L\\supset K$ of degree eight,\n$\\Hg(B)=U_L$, and\n\\[\n P_A(t)=(1+t)^6+4t^2(1+t)^2\n =1+6t+19t^2+28t^3+19t^4+6t^5+t^6.\n\\]\n\\end{enumerate}\nIn both cases the actual joint group is the connected group\n\\[\n H=\\{(g,u,v)\\in\\Hg(B)\\times U_{k_1}\\times U_{k_2}:\n                         \\det_K g=uv\\text{ in }U_K\\}.\n\\]\nThe rational space $Q$ has dimension four and, together with\ndivisors, generates the entire Hodge ring of $A$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3849,
        "statement_end_line": 3879,
        "proof_start_line": 3880,
        "end_line": 3937,
        "latex_labels": [
          "p:N3residual"
        ],
        "original_pdf_page": 51,
        "ai_pdf_page": 36
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:N3repeated",
        "p:quartic-mixed",
        "TOOL:quartic-signature-group",
        "TOOL:determinant-projector"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N3onecycle",
      "kind": "corollary",
      "number": "14.7",
      "title_latex": "A single-cycle criterion for all residual N3 cases",
      "row": "N3",
      "assumptions_and_conclusion_latex": "In Proposition~\\ref{p:N3residual}, the Hodge conjecture holds for\n$A$ if and only if there is a cycle $Z\\in\\CH^2(A)_\\Q$ whose\ncohomology class has nonzero projection to $Q$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3940,
        "statement_end_line": 3945,
        "proof_start_line": 3946,
        "end_line": 3960,
        "latex_labels": [
          "p:N3onecycle"
        ],
        "original_pdf_page": 52,
        "ai_pdf_page": 36
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:N3residual",
        "TOOL:determinant-projector"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N3cyclic",
      "kind": "theorem",
      "number": "14.8",
      "title_latex": "Two coupled cyclic-cover families in N3",
      "row": "N3",
      "assumptions_and_conclusion_latex": "For $m=8$ or $12$, let $B_{m,t}$ be as in\nLemma~\\ref{p:cyclic-transfer}, with\n$\\End^0(B_{m,t})=\\Q(\\zeta_m)$. Let $E_1$ have CM by $k_1=\\Q(i)$,\nand let $E_2$ have CM by\n\\[\n k_2=\\Q(\\sqrt{-2})\\quad(m=8),\\qquad\n k_2=\\Q(\\sqrt{-3})\\quad(m=12).\n\\]\nThen the Hodge conjecture holds for $B_{m,t}\\times E_1\\times E_2$,\nwith Hodge dimensions $(1,4,12,14,12,4,1)$. The endomorphism\nhypothesis holds for very general $t$ in each family.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3962,
        "statement_end_line": 3974,
        "proof_start_line": 3975,
        "end_line": 4003,
        "latex_labels": [
          "p:N3cyclic"
        ],
        "original_pdf_page": 52,
        "ai_pdf_page": 36
      },
      "citation_keys": [
        "MoonenCyclic"
      ],
      "logical_dependencies": [
        "p:N2cyclic",
        "p:quartic-mixed",
        "p:cyclic-transfer",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:cyclic-allparameters",
      "kind": "corollary",
      "number": "14.9",
      "title_latex": "All parameters in the three cyclic-cover families",
      "row": "N3",
      "assumptions_and_conclusion_latex": "The Hodge conjecture holds for all $t\\in\\C\\setminus\\{0,1\\}$\nin the products of Theorems~\\ref{p:N2cyclic} and\n\\ref{p:N3cyclic}, without the assumption\n$\\End^0(B_{m,t})=\\Q(\\zeta_m)$. If the fourfold factor remains\nsimple but becomes CM, the Hodge dimensions are\n$(1,6,19,28,19,6,1)$ instead of $(1,4,12,14,12,4,1)$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4006,
        "statement_end_line": 4014,
        "proof_start_line": 4015,
        "end_line": 4054,
        "latex_labels": [
          "p:cyclic-allparameters"
        ],
        "original_pdf_page": 53,
        "ai_pdf_page": 37
      },
      "citation_keys": [
        "MZ"
      ],
      "logical_dependencies": [
        "p:N2cyclic",
        "p:N3cyclic",
        "p:N3residual",
        "p:cyclic-transfer",
        "N5",
        "N6",
        "N7",
        "N8",
        "N9",
        "N10",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N3spin-descent",
      "kind": "lemma",
      "number": "14.10",
      "title_latex": "Connected Spin descent",
      "row": "N3",
      "assumptions_and_conclusion_latex": "Let $G$ be a geometrically connected algebraic group over a field $F$\nof characteristic zero and $r:G\\to\\SO(W,q)$ an $F$-morphism. If\n$r_{\\overline F}$ lifts to $\\operatorname{Spin}(W,q)$, then that lift is\nunique and descends to $F$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4067,
        "statement_end_line": 4072,
        "proof_start_line": 4073,
        "end_line": 4077,
        "latex_labels": [
          "p:N3spin-descent"
        ],
        "original_pdf_page": 54,
        "ai_pdf_page": 37
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N3KS",
      "kind": "theorem",
      "number": "14.11",
      "title_latex": "Residual N3 under the specified KS correspondence",
      "row": "N3",
      "assumptions_and_conclusion_latex": "Assume the external Kuga--Satake assertion above. Let\n$A=B\\times E_1\\times E_2$ be in the non-CM alternative of\nProposition~\\ref{p:N3residual}: $B$ is simple,\n$\\End^0(B)=K=k_1k_2$ for distinct imaginary quadratic fields,\n$E_i$ has CM by $k_i$, and the $K$-multiplicities are $(2,1,1,0)$.\nThen there is a cycle $Z\\in\\CH^2(A)_\\Q$ with nonzero projection to\nthe four-dimensional space $Q$ of that proposition. In particular\nthe rational Hodge conjecture holds for every $A^n$, $n\\geq1$.\nNo Hermitian discriminant or covering-family restriction is imposed.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4079,
        "statement_end_line": 4090,
        "proof_start_line": 4091,
        "end_line": 4280,
        "latex_labels": [
          "p:N3KS"
        ],
        "original_pdf_page": 54,
        "ai_pdf_page": 37
      },
      "citation_keys": [
        "HuybrechtsK3"
      ],
      "logical_dependencies": [
        "p:N3residual",
        "p:N3onecycle",
        "p:mixed-allpowers",
        "p:N3spin-descent",
        "EXT:KS",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:KS",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": [
        "EXT:KS"
      ]
    },
    {
      "id": "result:14.12",
      "kind": "remark",
      "number": "14.12",
      "title_latex": "Logical scope of Theorem~\\ref{p:N3KS}",
      "row": "N3",
      "assumptions_and_conclusion_latex": "The proof uses the \\emph{specified} full even-Clifford KS map as an\nexternal algebraicity input. It does not turn a complex half-spinor\ninto a rational abelian factor, assume the Hodge isometry (N3.1)\nalgebraic, or claim that an individual complex line in (N3.6) is a\nrational cycle. The rational descent occurs in the full regular module\nand the finite rational correspondence spaces (N3.5).",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4282,
        "statement_end_line": 4289,
        "proof_start_line": null,
        "end_line": 4289,
        "latex_labels": [],
        "original_pdf_page": 56,
        "ai_pdf_page": 39
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:14.13",
      "kind": "example",
      "number": "14.13",
      "title_latex": "Example 14.13",
      "row": "N3",
      "assumptions_and_conclusion_latex": "Choose $(B_1,\\ldots,B_{3})$ jointly very generally in\n$\\prod_i\\mathcal A_{d_i}(N)$ with $d_i=(4,1,1)$,\nas constructed in Proposition~\\ref{p:products}. Then each factor is\nsimple with $\\End^0(B_i)=\\Q$, and the factors are pairwise nonisogenous.\nTheir product is an actual sixfold of this partition, with\n$\\End^0(A)=\\Q^{3}$ and the proved joint group above.\nThe Hodge-space dimensions in codimensions $0$ through $6$ are\n$(1,3,4,4,4,3,1)$.\nAn explicit monomial basis over $\\Q$ on $A$ is\n$\\theta_1^{a_1}\\cdots\\theta_{3}^{a_{3}}$ with\n$0\\leq a_i\\leq d_i$; every such monomial is an intersection of divisors.\nOn powers add Poincare divisors between the copies of each individual\n$B_i$. There are no cross-factor divisors or higher mixed invariants in\nthis verified jointly generic example.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4290,
        "statement_end_line": 4305,
        "proof_start_line": null,
        "end_line": 4305,
        "latex_labels": [],
        "original_pdf_page": 56,
        "ai_pdf_page": 39
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:14.14",
      "kind": "example",
      "number": "14.14",
      "title_latex": "A mixed Weil plane requiring both elliptic factors",
      "row": "N3",
      "assumptions_and_conclusion_latex": "Take a very general simple $k=\\Q(i)$ fourfold with Hermitian matrix\n$\\operatorname{diag}(1,1,1,-1)$, and let $E:y^2=x^3-x$ have the opposite\n$k$-action. The PEL argument gives $\\Hg(B)=U(3,1)$ and $\\End^0(B)=k$.\nOn $A=B\\times E^2$ the total signature is $(3,3)$ and the determinant\nis $-1$. Proposition~\\ref{p:N3strata} yields\n\\[\n P_A(t)=(1+t+\\cdots+t^4)(1+4t+t^2)+2t^3,\n \\qquad (1,5,6,8,6,5,1).\n\\]\nThe five divisor classes consist of one on $B$ and four on $E^2$.\nThey generate the ordinary part; the two additional middle classes\nare the algebraic Weil plane. With only one elliptic factor the weight\n$2$ of the fourfold volume could not be cancelled.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4306,
        "statement_end_line": 4320,
        "proof_start_line": null,
        "end_line": 4320,
        "latex_labels": [],
        "original_pdf_page": 56,
        "ai_pdf_page": 39
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:15.1",
      "kind": "proposition",
      "number": "15.1",
      "title_latex": "Proposition 15.1",
      "row": "N4",
      "assumptions_and_conclusion_latex": "Let $A\\sim B_1\\times\\cdots\\times B_{2}$ with dimensions\n$(3,3)$. Assume the actual joint Hodge group is the\nindependent product of the full symplectic groups on these factors.\nThen all Hodge classes on all powers of $A$ are divisor-generated,\nand $P_A(t)=(1+t+\\cdots+t^{3})(1+t+\\cdots+t^{3})$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4326,
        "statement_end_line": 4332,
        "proof_start_line": 4333,
        "end_line": 4344,
        "latex_labels": [],
        "original_pdf_page": 57,
        "ai_pdf_page": 39
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:isogeny",
        "p:fft"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:CMthreepair",
      "kind": "lemma",
      "number": "15.2",
      "title_latex": "Two sextic CM factors: the actual Hodge-circle test",
      "row": "N4",
      "assumptions_and_conclusion_latex": "Let $B,C$ be nonisogenous simple CM threefolds with endomorphism fields\n$F_B,F_C$. Their joint Hodge group is $U_{F_B}\\times U_{F_C}$ unless\nthe fields contain a common imaginary quadratic field $k$.\nIn that case orient their $k$-signatures oppositely; the joint group is\n\\[\n \\ker\\bigl(N_{F_B/k}N_{F_C/k}:U_{F_B}\\times U_{F_C}\\longrightarrow U_k\\bigr).\n\\]",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4345,
        "statement_end_line": 4354,
        "proof_start_line": 4355,
        "end_line": 4452,
        "latex_labels": [
          "p:CMthreepair"
        ],
        "original_pdf_page": 57,
        "ai_pdf_page": 39
      },
      "citation_keys": [
        "MZ"
      ],
      "logical_dependencies": [
        "EXT:LowDim",
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N4strata",
      "kind": "proposition",
      "number": "15.3",
      "title_latex": "Threefold strata",
      "row": "N4",
      "assumptions_and_conclusion_latex": "Let $B,C$ be simple threefolds. The Hodge conjecture holds for $B\\times C$\nin each of the following cases: $B,C$ are isogenous; at least one has\nType I; or one has imaginary quadratic endomorphism algebra and the\nother is CM. It also holds if their joint group is a product, or is the\nfull determinant kernel\n\\[\n \\{(g_1,g_2)\\in U_k(B)\\times U_k(C):\\det_k(g_1)\\det_k(g_2)=1\\},\n\\]\nwhere their $k$-signatures have been oriented as $(2,1)$ and $(1,2)$.\nThe last displayed group equality is a hypothesis of this intermediate\nproposition; Theorem~\\ref{p:N4complete} below removes the need to\nassume it in the classification of the whole row.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4454,
        "statement_end_line": 4467,
        "proof_start_line": 4468,
        "end_line": 4504,
        "latex_labels": [
          "p:N4strata"
        ],
        "original_pdf_page": 58,
        "ai_pdf_page": 40
      },
      "citation_keys": [
        "MZ"
      ],
      "logical_dependencies": [
        "p:lowdim",
        "p:oddscale"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:N4complete",
      "kind": "theorem",
      "number": "15.4",
      "title_latex": "The complete $3+3$ row",
      "row": "N4",
      "assumptions_and_conclusion_latex": "For any two simple abelian threefolds $B,C$, all Hodge classes on\n$B\\times C$ are algebraic. If they are nonisogenous, their joint group\nis a product except possibly for the single determinant relation\nassociated with a common imaginary quadratic field. In that exceptional\ncase the Hodge ring on the sixfold is generated by divisors and\n$W_k(B\\times C)$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4505,
        "statement_end_line": 4512,
        "proof_start_line": 4513,
        "end_line": 4565,
        "latex_labels": [
          "p:N4complete"
        ],
        "original_pdf_page": 59,
        "ai_pdf_page": 40
      },
      "citation_keys": [
        "MZ"
      ],
      "logical_dependencies": [
        "p:CMthreepair",
        "p:N4strata",
        "p:oddscale"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:15.5",
      "kind": "example",
      "number": "15.5",
      "title_latex": "Example 15.5",
      "row": "N4",
      "assumptions_and_conclusion_latex": "Choose $(B_1,\\ldots,B_{2})$ jointly very generally in\n$\\prod_i\\mathcal A_{d_i}(N)$ with $d_i=(3,3)$,\nas constructed in Proposition~\\ref{p:products}. Then each factor is\nsimple with $\\End^0(B_i)=\\Q$, and the factors are pairwise nonisogenous.\nTheir product is an actual sixfold of this partition, with\n$\\End^0(A)=\\Q^{2}$ and the proved joint group above.\nThe Hodge-space dimensions in codimensions $0$ through $6$ are\n$(1,2,3,4,3,2,1)$.\nAn explicit monomial basis over $\\Q$ on $A$ is\n$\\theta_1^{a_1}\\cdots\\theta_{2}^{a_{2}}$ with\n$0\\leq a_i\\leq d_i$; every such monomial is an intersection of divisors.\nOn powers add Poincare divisors between the copies of each individual\n$B_i$. There are no cross-factor divisors or higher mixed invariants in\nthis verified jointly generic example.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4566,
        "statement_end_line": 4581,
        "proof_start_line": null,
        "end_line": 4581,
        "latex_labels": [],
        "original_pdf_page": 60,
        "ai_pdf_page": 41
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:15.6",
      "kind": "example",
      "number": "15.6",
      "title_latex": "Two nonisogenous unitary threefolds",
      "row": "N4",
      "assumptions_and_conclusion_latex": "Over $k=\\Q(i)$ use Hermitian matrices\n$h_B=\\operatorname{diag}(1,1,-1)$ and\n$h_C=\\operatorname{diag}(1,-1,-1)$. Choose a jointly very general point\nof the product of their two-dimensional PEL domains.\nDerived monodromy is the independent product $\\SU(h_B)\\times\\SU(h_C)$.\nThe determinant circle has coordinates $(1,-1)$, so the joint group\nis exactly the determinant kernel in Proposition~\\ref{p:N4strata}.\nIts commutant is $k\\times k$, proving that the factors are simple\nand nonisogenous. The sixfold has split determinant $-1$, and\n\\[\n P_A(t)=(1+t+t^2+t^3)^2+2t^3,\n \\qquad (1,2,3,6,3,2,1).\n\\]\nThe two polarizations generate the ordinary part and $W_k(A)$ supplies\nthe two algebraic middle classes. The group hypothesis has been verified\nfor this family; it is not inferred merely from the partition $3+3$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4582,
        "statement_end_line": 4599,
        "proof_start_line": null,
        "end_line": 4599,
        "latex_labels": [],
        "original_pdf_page": 60,
        "ai_pdf_page": 41
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:15.7",
      "kind": "example",
      "number": "15.7",
      "title_latex": "Two CM threefolds with a mixed Weil plane",
      "row": "N4",
      "assumptions_and_conclusion_latex": "Let $F_1=\\Q(\\zeta_7+\\zeta_7^{-1})$,\n$F_2=\\Q(\\zeta_9+\\zeta_9^{-1})$, and $K_j=\\Q(i)F_j$.\nEach $K_j$ is cyclic sextic. Choose signed CM types $(1,1,-1)$ and\n$(-1,-1,1)$ relative to $\\Q(i)$, respectively, and let $B,C$ be the\ncorresponding CM threefolds. These nonconstant types are primitive:\nthe only proper CM subfield is $\\Q(i)$ and an induced type would be\nconstant. Thus the factors are simple with endomorphism fields $K_j$.\nThey are nonisogenous, since these fields are nonisomorphic (their\nramification at $7$ differs). Lemma~\\ref{p:CMthreepair} gives the joint\nrank-five norm kernel. The six opposite-coordinate pairings and the\ntwo mixed volumes give\n\\[\n P_{B\\times C}(t)=(1+t)^6+2t^3,\n \\qquad (1,6,15,22,15,6,1).\n\\]\nThe twenty ordinary middle classes are divisor products; the two\nadditional classes form the algebraic plane $W_{\\Q(i)}(B\\times C)$.\nThis is a coupled CM example of the complete theorem, rather than\nan independence assumption.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4600,
        "statement_end_line": 4620,
        "proof_start_line": null,
        "end_line": 4620,
        "latex_labels": [],
        "original_pdf_page": 60,
        "ai_pdf_page": 41
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:16.1",
      "kind": "theorem",
      "number": "16.1",
      "title_latex": "Theorem 16.1",
      "row": "N5",
      "assumptions_and_conclusion_latex": "Let $A\\sim B\\times S\\times E$, where $B,S$ are simple of dimensions\nthree and two and $E$ is elliptic. Then the Hodge conjecture holds for $A$.\nIts Hodge ring is generated by divisors and the pullback of the Weil\nplane on $B\\times E$, when the matching imaginary-quadratic case occurs.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4625,
        "statement_end_line": 4630,
        "proof_start_line": 4631,
        "end_line": 4639,
        "latex_labels": [],
        "original_pdf_page": 61,
        "ai_pdf_page": 41
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:surface-three",
        "p:three-ell"
      ],
      "dependency_coverage": "curated_case_main_theorem",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:16.2",
      "kind": "example",
      "number": "16.2",
      "title_latex": "Example 16.2",
      "row": "N5",
      "assumptions_and_conclusion_latex": "Choose $(B_1,\\ldots,B_{3})$ jointly very generally in\n$\\prod_i\\mathcal A_{d_i}(N)$ with $d_i=(3,2,1)$,\nas constructed in Proposition~\\ref{p:products}. Then each factor is\nsimple with $\\End^0(B_i)=\\Q$, and the factors are pairwise nonisogenous.\nTheir product is an actual sixfold of this partition, with\n$\\End^0(A)=\\Q^{3}$ and the proved joint group above.\nThe Hodge-space dimensions in codimensions $0$ through $6$ are\n$(1,3,5,6,5,3,1)$.\nAn explicit monomial basis over $\\Q$ on $A$ is\n$\\theta_1^{a_1}\\cdots\\theta_{3}^{a_{3}}$ with\n$0\\leq a_i\\leq d_i$; every such monomial is an intersection of divisors.\nOn powers add Poincare divisors between the copies of each individual\n$B_i$. There are no cross-factor divisors or higher mixed invariants in\nthis verified jointly generic example.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4640,
        "statement_end_line": 4655,
        "proof_start_line": null,
        "end_line": 4655,
        "latex_labels": [],
        "original_pdf_page": 61,
        "ai_pdf_page": 42
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "ex:N5weil",
      "kind": "example",
      "number": "16.3",
      "title_latex": "A fourfold Weil plane inside the $3+2+1$ row",
      "row": "N5",
      "assumptions_and_conclusion_latex": "Take a very general $k=\\Q(i)$ threefold $B$ with Hermitian datum\n$\\operatorname{diag}(1,1,-1)$, the curve $E:y^2=x^3-x$ with opposite\n$k$-action, and a very general simple surface $S$ with $\\End^0(S)=\\Q$.\nThen $\\Hg(B)=U(2,1)$, and Lemma~\\ref{p:surface-three} separates $S$.\nThe Hodge ring is generated by divisors and the pullback of the\nalgebraic plane $W_k(B\\times E)$. The dimension polynomial is\n\\[\n P_A(t)=(1+2t+4t^2+2t^3+t^4)(1+t+t^2),\n \\qquad (1,3,7,8,7,3,1).\n\\]\nThus this example includes exceptional classes and their products\nwith divisors on the surface.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4656,
        "statement_end_line": 4669,
        "proof_start_line": null,
        "end_line": 4669,
        "latex_labels": [
          "ex:N5weil"
        ],
        "original_pdf_page": 61,
        "ai_pdf_page": 42
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:17.1",
      "kind": "theorem",
      "number": "17.1",
      "title_latex": "Theorem 17.1",
      "row": "N6",
      "assumptions_and_conclusion_latex": "If $A\\sim B\\times E_1\\times E_2\\times E_3$, where $B$ is a simple\nthreefold and the $E_i$ are elliptic, the Hodge conjecture holds for $A$.\nThe generators are divisors and pullbacks of fourfold Weil classes from\n$B$ times a matching CM elliptic factor, as in Lemma~\\ref{p:three-ell}.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4674,
        "statement_end_line": 4679,
        "proof_start_line": 4680,
        "end_line": 4684,
        "latex_labels": [],
        "original_pdf_page": 61,
        "ai_pdf_page": 42
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:three-ell"
      ],
      "dependency_coverage": "curated_case_main_theorem",
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:17.2",
      "kind": "example",
      "number": "17.2",
      "title_latex": "Example 17.2",
      "row": "N6",
      "assumptions_and_conclusion_latex": "Choose $(B_1,\\ldots,B_{4})$ jointly very generally in\n$\\prod_i\\mathcal A_{d_i}(N)$ with $d_i=(3,1,1,1)$,\nas constructed in Proposition~\\ref{p:products}. Then each factor is\nsimple with $\\End^0(B_i)=\\Q$, and the factors are pairwise nonisogenous.\nTheir product is an actual sixfold of this partition, with\n$\\End^0(A)=\\Q^{4}$ and the proved joint group above.\nThe Hodge-space dimensions in codimensions $0$ through $6$ are\n$(1,4,7,8,7,4,1)$.\nAn explicit monomial basis over $\\Q$ on $A$ is\n$\\theta_1^{a_1}\\cdots\\theta_{4}^{a_{4}}$ with\n$0\\leq a_i\\leq d_i$; every such monomial is an intersection of divisors.\nOn powers add Poincare divisors between the copies of each individual\n$B_i$. There are no cross-factor divisors or higher mixed invariants in\nthis verified jointly generic example.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4685,
        "statement_end_line": 4700,
        "proof_start_line": null,
        "end_line": 4700,
        "latex_labels": [],
        "original_pdf_page": 61,
        "ai_pdf_page": 42
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:17.3",
      "kind": "example",
      "number": "17.3",
      "title_latex": "Repeated matching elliptic factors",
      "row": "N6",
      "assumptions_and_conclusion_latex": "Use the $k=\\Q(i)$ threefold $B$ and matching elliptic curve $E$\nfrom Example~\\ref{ex:N5weil}, and put $A=B\\times E^3$. Pairings contribute\n$(1+t+t^2+t^3)(1+9t+9t^2+t^3)$.\nA threefold volume must be paired with elliptic weight $\\mp1$.\nChoosing $a$ positive and $a+1$ negative coordinates in the three\nelliptic copies gives $2\\binom3a\\binom3{a+1}$ such invariants,\nfor $a=0,1,2$. Hence\n\\[\n P_A(t)=(1+t+t^2+t^3)(1+9t+9t^2+t^3)+6t^2+18t^3+6t^4,\n\\]\nwith dimensions $(1,10,25,38,25,10,1)$.\nLemma~\\ref{p:three-ell} supplies algebraic generators: divisors and\npullbacks of $W_k(B\\times E)$ under the $k$-linear maps $E^3\\to E$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4701,
        "statement_end_line": 4715,
        "proof_start_line": null,
        "end_line": 4715,
        "latex_labels": [],
        "original_pdf_page": 62,
        "ai_pdf_page": 42
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:18.1",
      "kind": "theorem",
      "number": "18.1",
      "title_latex": "Theorem 18.1",
      "row": "N7",
      "assumptions_and_conclusion_latex": "Every $A\\sim S_1\\times S_2\\times S_3$, with $S_i$ simple abelian\nsurfaces, satisfies the rational Hodge conjecture, as does every power.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4721,
        "statement_end_line": 4724,
        "proof_start_line": 4725,
        "end_line": 4732,
        "latex_labels": [],
        "original_pdf_page": 62,
        "ai_pdf_page": 42
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:lowdim"
      ],
      "dependency_coverage": "curated_case_main_theorem",
      "curated_transitive_external_inputs": [
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "ex:CMsurface",
      "kind": "example",
      "number": "18.2",
      "title_latex": "An explicit repeated CM surface",
      "row": "N7",
      "assumptions_and_conclusion_latex": "Let $S=J(C)$ for $C:y^2=x^5-1$. The curve has genus two, and\n$(x,y)\\mapsto(\\zeta_5x,y)$ gives CM by $F=\\Q(\\zeta_5)$ with type\n$\\{1,2\\}$. Its signed Galois orbit contains $(1,1)$ and $(1,-1)$,\nso $\\Hg(S)=U_F$ has rank two. The four distinct eigencharacters have\ncommutant $F$, proving $\\End^0(S)=F$ and simplicity.\nFor $A=S^3$, each of its two conjugate character pairs has multiplicity\nthree. Neutral monomials choose the same number of each sign, so\n\\[\n P_A(t)=(1+9t+9t^2+t^3)^2,\n \\qquad (1,18,99,164,99,18,1).\n\\]\nThey factor into degree-two pairings. Rational descent and Lefschetz\n$(1,1)$ show that the eighteen rational divisor classes generate the\nHodge ring, including the Poincare classes between repeated factors.\nHere $\\End^0(A)=M_3(F)$.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4733,
        "statement_end_line": 4749,
        "proof_start_line": null,
        "end_line": 4749,
        "latex_labels": [
          "ex:CMsurface"
        ],
        "original_pdf_page": 62,
        "ai_pdf_page": 42
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:19.1",
      "kind": "theorem",
      "number": "19.1",
      "title_latex": "Theorem 19.1",
      "row": "N8",
      "assumptions_and_conclusion_latex": "Every $A\\sim S_1\\times S_2\\times E_1\\times E_2$ in this row satisfies\nthe rational Hodge conjecture, as does every power.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4754,
        "statement_end_line": 4757,
        "proof_start_line": 4758,
        "end_line": 4763,
        "latex_labels": [],
        "original_pdf_page": 63,
        "ai_pdf_page": 43
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:lowdim"
      ],
      "dependency_coverage": "curated_case_main_theorem",
      "curated_transitive_external_inputs": [
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:19.2",
      "kind": "example",
      "number": "19.2",
      "title_latex": "A surface square and an elliptic square",
      "row": "N8",
      "assumptions_and_conclusion_latex": "Let $S$ be the explicit CM surface of Example~\\ref{ex:CMsurface} and\n$E:y^2=x^3-x$. The field $\\Q(i)$ does not embed in $\\Q(\\zeta_5)$,\nwhose only quadratic subfield is real. Thus\n$\\Hg(S\\times E)=U_{\\Q(\\zeta_5)}\\times U_{\\Q(i)}$, by\nTheorem~\\ref{p:lowdim}(2). For $A=S^2\\times E^2$, the three independent\nconjugate character pairs each have multiplicity two, and\n\\[\n P_A(t)=(1+4t+t^2)^3,\n \\qquad (1,12,51,88,51,12,1).\n\\]\nOpposite-character pairings give all invariants and therefore twelve\nrational divisor generators. This verifies a repeated CM case directly,\nas well as by the general theorem.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4764,
        "statement_end_line": 4778,
        "proof_start_line": null,
        "end_line": 4778,
        "latex_labels": [],
        "original_pdf_page": 63,
        "ai_pdf_page": 43
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:20.1",
      "kind": "theorem",
      "number": "20.1",
      "title_latex": "Theorem 20.1",
      "row": "N9",
      "assumptions_and_conclusion_latex": "Every $A\\sim S\\times E_1\\times E_2\\times E_3\\times E_4$ in this row\nsatisfies the rational Hodge conjecture, as does every power.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4783,
        "statement_end_line": 4786,
        "proof_start_line": 4787,
        "end_line": 4794,
        "latex_labels": [],
        "original_pdf_page": 63,
        "ai_pdf_page": 43
      },
      "citation_keys": [
        "MZ"
      ],
      "logical_dependencies": [
        "p:lowdim"
      ],
      "dependency_coverage": "curated_case_main_theorem",
      "curated_transitive_external_inputs": [
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:20.2",
      "kind": "example",
      "number": "20.2",
      "title_latex": "One CM surface and four equal elliptic factors",
      "row": "N9",
      "assumptions_and_conclusion_latex": "Use $S$ and $E$ from the preceding example and put $A=S\\times E^4$.\nTheir independent Hodge groups give\n\\[\n P_A(t)=(1+t)^2(1+16t+36t^2+16t^3+t^4),\n \\qquad (1,18,69,104,69,18,1).\n\\]\nTwo divisors from $S$ and sixteen from $E^4$ generate the Hodge ring;\nthe latter include graph and Poincare classes associated with the\n$\\Q(i)$-linear homomorphisms between elliptic copies.\nAll classes are algebraic intersections of these divisors.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4795,
        "statement_end_line": 4806,
        "proof_start_line": null,
        "end_line": 4806,
        "latex_labels": [],
        "original_pdf_page": 63,
        "ai_pdf_page": 43
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:21.1",
      "kind": "theorem",
      "number": "21.1",
      "title_latex": "Theorem 21.1",
      "row": "N10",
      "assumptions_and_conclusion_latex": "If a sixfold $A$ is isogenous to a product of six elliptic curves, every\nHodge class on every power of $A$ is generated by divisor classes.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4811,
        "statement_end_line": 4814,
        "proof_start_line": 4815,
        "end_line": 4846,
        "latex_labels": [],
        "original_pdf_page": 63,
        "ai_pdf_page": 43
      },
      "citation_keys": [],
      "logical_dependencies": [
        "p:fft"
      ],
      "dependency_coverage": "curated_case_main_theorem",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "result:21.2",
      "kind": "example",
      "number": "21.2",
      "title_latex": "Example 21.2",
      "row": "N10",
      "assumptions_and_conclusion_latex": "Let $E$ be $y^2=x^3-x$; then $\\End^0(E)=\\Q(i)$ and take $A=E^6$.\nThe diagonal norm-one torus has characters $\\chi,-\\chi$ on two\nsix-dimensional multiplicity spaces. In degree $2p$, an invariant selects\n$p$ vectors from each space, so\n\\[\n \\dim_\\Q\\B^p(A)=\\binom6p^2,\\qquad\n P_A(t)=1+36t+225t^2+400t^3+225t^4+36t^5+t^6.\n\\]\nThe divisor space is the $36$-dimensional space of Hermitian $6\\times6$\nmatrices over $\\Q(i)$, represented by factor polarizations and Poincare\nclasses associated with the endomorphisms $1$ and $i$ between factors.\nIts products generate every Hodge class. Here\n$\\End^0(A)=M_6(\\Q(i))$; this differs from the independent elliptic example\nbut belongs to the same numerical partition. The distinction illustrates\nwhy a dimension partition alone cannot fix Hodge-space dimensions.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4847,
        "statement_end_line": 4863,
        "proof_start_line": null,
        "end_line": 4863,
        "latex_labels": [],
        "original_pdf_page": 64,
        "ai_pdf_page": 44
      },
      "citation_keys": [],
      "logical_dependencies": [],
      "dependency_coverage": "references_indexed_only",
      "curated_transitive_external_inputs": [],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "p:conditional-sixfold",
      "kind": "pretheorem",
      "number": "P.23",
      "title_latex": "Conditional sixfold conclusion",
      "row": "Conclusion",
      "assumptions_and_conclusion_latex": "Assume the split-eightfold and CM assertions stated in\nTheorem~\\ref{p:openai-inputs}, the specified full Kuga--Satake\nalgebraicity assertion used in Theorem~\\ref{p:N3KS}, and the\nauxiliary-spin assertion (Aux) stated before\nTheorem~\\ref{p:N2auxiliary}, together with the classical external\nresults cited in the corresponding rows. Then every complex\nabelian sixfold satisfies the rational Hodge conjecture.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 4867,
        "statement_end_line": 4876,
        "proof_start_line": 4877,
        "end_line": 4890,
        "latex_labels": [
          "p:conditional-sixfold"
        ],
        "original_pdf_page": 64,
        "ai_pdf_page": 44
      },
      "citation_keys": [],
      "logical_dependencies": [
        "S1",
        "S2",
        "S3",
        "S4",
        "S5",
        "S6",
        "S7",
        "S8",
        "S9",
        "S10",
        "S11",
        "N1",
        "N2",
        "N3",
        "N4",
        "N5",
        "N6",
        "N7",
        "N8",
        "N9",
        "N10"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Aux",
        "EXT:CM",
        "EXT:Classical",
        "EXT:KS",
        "EXT:LowDim",
        "EXT:Markman6",
        "EXT:Weil8"
      ],
      "curated_transitive_conditional_inputs": [
        "EXT:Aux",
        "EXT:CM",
        "EXT:KS",
        "EXT:Weil8"
      ]
    },
    {
      "id": "TOOL:determinant-projector",
      "kind": "proof_tool",
      "number": "Tool 1",
      "row": "N2",
      "title_latex": "General quartic determinant projector",
      "assumptions": "An abelian fourfold B with a quartic CM action K and K-rank two in H^1; T is the rational sum of the four same-embedding determinant lines. No covering-family assumption.",
      "conclusion": "A rational polynomial in a general K-endomorphism on H^2, combined with the algebraic degree projector, realizes the projector onto T.",
      "assumptions_and_conclusion_latex": "There is an algebraic projector $p_T$ onto $T$ in $H^2(B)$.\nTo see this, choose a sufficiently general $k\\in K$: the four\neigenvalues $\\tau(k)^2$ can be separated from all mixed products\n$\\tau(k)\\tau'(k)$, $\\tau\\ne\\tau'$. Both sets are Galois stable,\nso a rational polynomial in $k^*$ gives the projector. Clearing\ndenominators realizes $k$ by a rational algebraic correspondence;\nthe cohomological-degree projector is algebraic on an abelian variety.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3172,
        "end_line": 3179,
        "proof_start_line": null,
        "latex_labels": [],
        "owner_label": "p:cyclic-transfer",
        "original_pdf_page": 43,
        "ai_pdf_page": 30
      },
      "logical_dependencies": [
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "TOOL:quartic-signature-group",
      "kind": "proof_tool",
      "number": "Tool 2",
      "row": "N2",
      "title_latex": "General quartic signature group dichotomy",
      "assumptions": "A weight-one polarized Hodge structure with quartic CM action, rank two and multiplicities (2,1,1,0), equivalently one pure and one balanced conjugate block. No covering equation is required.",
      "conclusion": "Either the abelian variety is CM, or its complex Hodge group is GL_2 x GL_2 with endomorphism algebra K, by the signature/cocharacter argument in this proof.",
      "assumptions_and_conclusion_latex": "We also record a group dichotomy used below. Each derived projection\nto a two-dimensional block is either trivial or $\\SL_2$, and\nGalois transitivity makes the two projections simultaneous. In the\nnontrivial case a diagonal identification is impossible: the\nadjoint Hodge circle is trivial on the pure-signature block and\nnontrivial on the balanced block. Thus either $B$ is CM, or\n$\\Hg(B)_\\C=\\GL_2^2$ and $\\End^0(B)=K$. This argument only\nuses the signatures and works for any quartic CM field with these\nsignatures. No simplicity or endomorphism-equality assumption is\nneeded for the correspondence constructed here.",
      "source": {
        "path": "latex/paper.tex",
        "start_line": 3075,
        "end_line": 3086,
        "proof_start_line": null,
        "latex_labels": [],
        "owner_label": "p:cyclic-transfer",
        "original_pdf_page": 42,
        "ai_pdf_page": 30
      },
      "logical_dependencies": [
        "EXT:Classical"
      ],
      "dependency_coverage": "curated_core",
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "S1",
      "kind": "case_contract",
      "title": "Type I(1)",
      "assumptions": "A complex abelian sixfold in row S1, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Full symplectic image; exclude the 2 x 6 tensor representation using the Hodge cocharacter and rational normal factors.",
      "strategy_zh": "完整辛群；以 Hodge 余特征及有理正规因子排除 2×6 张量表示。",
      "powers_scope": "all_powers",
      "method": "representation_argument",
      "logical_dependencies": [
        "p:small",
        "p:fft",
        "t:S1"
      ],
      "declared_conditional_inputs": [],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:S1"
        ],
        "start_line": 989,
        "original_pdf_page": 14,
        "ai_pdf_page": 12
      },
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "S2",
      "kind": "case_contract",
      "title": "Type I(2)",
      "assumptions": "A complex abelian sixfold in row S2, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Small symplectic constituents and commutant/Goursat separation give the full product; apply pairing invariant theory.",
      "strategy_zh": "小维辛表示与交换子／Goursat 分离给出完整乘积群，再用配对不变量。",
      "powers_scope": "all_powers",
      "method": "classical_application",
      "logical_dependencies": [
        "result:2.1",
        "p:small",
        "p:goursat",
        "p:fft"
      ],
      "declared_conditional_inputs": [],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:S2"
        ],
        "start_line": 1028,
        "original_pdf_page": 14,
        "ai_pdf_page": 13
      },
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "S3",
      "kind": "case_contract",
      "title": "Type I(3)",
      "assumptions": "A complex abelian sixfold in row S3, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "The same full-product argument, with degree-three real multiplication.",
      "strategy_zh": "三次实乘情形使用同一完整乘积群论证。",
      "powers_scope": "all_powers",
      "method": "classical_application",
      "logical_dependencies": [
        "result:3.1",
        "p:small",
        "p:goursat",
        "p:fft"
      ],
      "declared_conditional_inputs": [],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:S3"
        ],
        "start_line": 1066,
        "original_pdf_page": 15,
        "ai_pdf_page": 13
      },
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "S4",
      "kind": "case_contract",
      "title": "Type I(6)",
      "assumptions": "A complex abelian sixfold in row S4, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Rank-two symplectic blocks, separated by the commutant, give divisor generators.",
      "strategy_zh": "交换子分离二维辛块，得到除子生成元。",
      "powers_scope": "all_powers",
      "method": "classical_application",
      "logical_dependencies": [
        "result:4.1",
        "p:small",
        "p:goursat",
        "p:fft"
      ],
      "declared_conditional_inputs": [],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:S4"
        ],
        "start_line": 1104,
        "original_pdf_page": 15,
        "ai_pdf_page": 14
      },
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "S5",
      "kind": "case_contract",
      "title": "Type II(1)",
      "assumptions": "A complex abelian sixfold in row S5, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Quaternionic multiplicity space has full Sp_6 image; pairings generate every power.",
      "strategy_zh": "四元数重数空间具有完整 Sp_6 像；配对生成各次幂。",
      "powers_scope": "all_powers",
      "method": "classical_application",
      "logical_dependencies": [
        "result:5.1",
        "p:small",
        "p:fft"
      ],
      "declared_conditional_inputs": [],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:S5"
        ],
        "start_line": 1142,
        "original_pdf_page": 16,
        "ai_pdf_page": 14
      },
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "S6",
      "kind": "case_contract",
      "title": "Type II(3)",
      "assumptions": "A complex abelian sixfold in row S6, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Three rank-two quaternionic multiplicity blocks give the full symplectic product.",
      "strategy_zh": "三个二维四元数重数块给出完整辛群乘积。",
      "powers_scope": "all_powers",
      "method": "classical_application",
      "logical_dependencies": [
        "result:6.1",
        "p:small",
        "p:goursat",
        "p:fft"
      ],
      "declared_conditional_inputs": [],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:S6"
        ],
        "start_line": 1182,
        "original_pdf_page": 17,
        "ai_pdf_page": 14
      },
      "curated_transitive_external_inputs": [
        "EXT:Classical"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "S7",
      "kind": "case_contract",
      "title": "Type III(1)",
      "assumptions": "A complex abelian sixfold in row S7, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Find an imaginary-quadratic Weil plane, then use quaternionic endomorphisms and sum maps to generate exceptional volume tensors.",
      "strategy_zh": "先取虚二次 Weil 平面，再以四元数端同态与求和映射生成例外体积张量。",
      "powers_scope": "all_powers",
      "method": "exceptional_generators",
      "logical_dependencies": [
        "t:S7structure",
        "t:S7complete",
        "p:all-sixfold-weil"
      ],
      "declared_conditional_inputs": [
        "EXT:Weil8"
      ],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:S7"
        ],
        "start_line": 1216,
        "original_pdf_page": 17,
        "ai_pdf_page": 15
      },
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:Weil8"
      ],
      "curated_transitive_conditional_inputs": [
        "EXT:Weil8"
      ]
    },
    {
      "id": "S8",
      "kind": "case_contract",
      "title": "Type IV(1,1)",
      "assumptions": "A complex abelian sixfold in row S8, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Unbalanced signatures give a full unitary group; balanced signature leaves a Weil plane, then polarization of generators gives powers.",
      "strategy_zh": "非平衡签名给完整酉群；平衡签名留下 Weil 平面，再以生成元极化处理各次幂。",
      "powers_scope": "all_powers",
      "method": "unitary_dichotomy",
      "logical_dependencies": [
        "t:S8",
        "p:all-sixfold-weil",
        "p:polarization"
      ],
      "declared_conditional_inputs": [
        "EXT:Weil8"
      ],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:S8"
        ],
        "start_line": 1454,
        "original_pdf_page": 20,
        "ai_pdf_page": 17
      },
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:Weil8"
      ],
      "curated_transitive_conditional_inputs": [
        "EXT:Weil8"
      ]
    },
    {
      "id": "S9",
      "kind": "case_contract",
      "title": "Type IV(2,1)",
      "assumptions": "A complex abelian sixfold in row S9, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Determine the determinant torus; correct the rational discriminant by a positive real-quadratic endomorphism.",
      "strategy_zh": "确定行列式环面；用全正实二次端同态调整有理判别式。",
      "powers_scope": "all_powers",
      "method": "polarization_construction",
      "logical_dependencies": [
        "result:9.1",
        "p:smallunitary",
        "p:quartic-scale",
        "p:coupled-det"
      ],
      "declared_conditional_inputs": [],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:S9"
        ],
        "start_line": 1531,
        "original_pdf_page": 21,
        "ai_pdf_page": 17
      },
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "S10",
      "kind": "case_contract",
      "title": "Type IV(3,1)",
      "assumptions": "A complex abelian sixfold in row S10, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Determine the determinant torus; its exceptional Weil plane is handled by stabilization and contraction.",
      "strategy_zh": "确定行列式环面；用稳定化与收缩处理例外 Weil 平面。",
      "powers_scope": "all_powers",
      "method": "determinant_invariants",
      "logical_dependencies": [
        "result:10.1",
        "p:smallunitary",
        "p:all-sixfold-weil",
        "p:coupled-det"
      ],
      "declared_conditional_inputs": [
        "EXT:Weil8"
      ],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:S10"
        ],
        "start_line": 1702,
        "original_pdf_page": 24,
        "ai_pdf_page": 19
      },
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:Weil8"
      ],
      "curated_transitive_conditional_inputs": [
        "EXT:Weil8"
      ]
    },
    {
      "id": "S11",
      "kind": "case_contract",
      "title": "Type IV(6,1)",
      "assumptions": "A complex abelian sixfold in row S11, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Exclude Hodge rank four by a two-dimensional relation space; split the rank-five Weil form using inverse derivatives and interpolation.",
      "strategy_zh": "以二维关系空间排除 Hodge 秩四；用逆导数与插值构造秩五情形的分裂 Weil 形式。",
      "powers_scope": "all_powers",
      "method": "rank_and_trace_construction",
      "logical_dependencies": [
        "p:CMsixrank",
        "p:CMsplit",
        "p:CMrankfive",
        "p:CMcomplete"
      ],
      "declared_conditional_inputs": [],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:S11"
        ],
        "start_line": 2002,
        "original_pdf_page": 28,
        "ai_pdf_page": 21
      },
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "N1",
      "kind": "case_contract",
      "title": "5+1",
      "assumptions": "A complex abelian sixfold in row N1, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "A prime-dimensional factor leaves at most one mixed Weil plane; odd-rank polarization adjustment makes it split.",
      "strategy_zh": "素数维因子至多留下一个混合 Weil 平面；奇秩极化调整使之分裂。",
      "powers_scope": "sixfold_only_as_stated",
      "method": "mixed_weil_reduction",
      "logical_dependencies": [
        "result:12.1",
        "p:prime-ell",
        "p:oddscale"
      ],
      "declared_conditional_inputs": [],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:N1"
        ],
        "start_line": 2491,
        "original_pdf_page": 34,
        "ai_pdf_page": 25
      },
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "N2",
      "kind": "case_contract",
      "title": "4+2",
      "assumptions": "A complex abelian sixfold in row N2, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Separate independent strata. The residual quartic mixed model reduces to one cycle; Aux constructs it and a CM correspondence transfers it.",
      "strategy_zh": "分离独立分层；剩余四次混合模型归约为一个循环，由 Aux 构造后经 CM 对应转移。",
      "powers_scope": "sixfold_full_row_and_residual_all_powers",
      "method": "auxiliary_spin_construction",
      "logical_dependencies": [
        "p:N2strata",
        "p:N2nonCMstructure",
        "p:N2CMstructure",
        "p:quartic-mixed",
        "p:mixed-allpowers",
        "p:N2CM-new",
        "p:N2auxiliary"
      ],
      "declared_conditional_inputs": [
        "EXT:Aux",
        "EXT:CM"
      ],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:N2"
        ],
        "start_line": 2543,
        "original_pdf_page": 35,
        "ai_pdf_page": 26
      },
      "curated_transitive_external_inputs": [
        "EXT:Aux",
        "EXT:CM",
        "EXT:Classical",
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": [
        "EXT:Aux",
        "EXT:CM"
      ]
    },
    {
      "id": "N3",
      "kind": "case_contract",
      "title": "4+1+1",
      "assumptions": "A complex abelian sixfold in row N3, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Treat independent and repeated elliptic factors first; use a K3 realization, full Clifford matrix extraction and contraction in the residual model.",
      "strategy_zh": "先处理独立与重复椭圆因子；剩余模型用 K3 实现、完整 Clifford 矩阵提取及收缩。",
      "powers_scope": "sixfold_full_row_and_residual_all_powers",
      "method": "complete_KS_construction",
      "logical_dependencies": [
        "p:N3strata",
        "p:N3repeated",
        "p:N3residual",
        "p:N3onecycle",
        "p:mixed-allpowers",
        "p:N3CM-new",
        "p:N3KS"
      ],
      "declared_conditional_inputs": [
        "EXT:KS",
        "EXT:CM"
      ],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:N3"
        ],
        "start_line": 3696,
        "original_pdf_page": 49,
        "ai_pdf_page": 34
      },
      "curated_transitive_external_inputs": [
        "EXT:CM",
        "EXT:Classical",
        "EXT:KS",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": [
        "EXT:CM",
        "EXT:KS"
      ]
    },
    {
      "id": "N4",
      "kind": "case_contract",
      "title": "3+3",
      "assumptions": "A complex abelian sixfold in row N4, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Test the actual Hodge cocharacters of common quotients; the remaining determinant relation gives an odd-rank split Weil plane.",
      "strategy_zh": "检验共同商的实际 Hodge 余特征；剩余行列式关系给出可由奇秩调整分裂的 Weil 平面。",
      "powers_scope": "sixfold_only_as_stated",
      "method": "joint_Hodge_circle_test",
      "logical_dependencies": [
        "p:CMthreepair",
        "p:N4strata",
        "p:N4complete",
        "p:oddscale"
      ],
      "declared_conditional_inputs": [],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:N4"
        ],
        "start_line": 4321,
        "original_pdf_page": 57,
        "ai_pdf_page": 39
      },
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "N5",
      "kind": "case_contract",
      "title": "3+2+1",
      "assumptions": "A complex abelian sixfold in row N5, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Separate the surface, then use the threefold-elliptic fourfold Weil plane.",
      "strategy_zh": "分离曲面因子，再用三维簇与椭圆曲线组成的四维 Weil 平面。",
      "powers_scope": "sixfold_only_as_stated",
      "method": "low_dimensional_reduction",
      "logical_dependencies": [
        "result:16.1",
        "p:surface-three",
        "p:three-ell"
      ],
      "declared_conditional_inputs": [],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:N5"
        ],
        "start_line": 4621,
        "original_pdf_page": 61,
        "ai_pdf_page": 41
      },
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "N6",
      "kind": "case_contract",
      "title": "3+1+1+1",
      "assumptions": "A complex abelian sixfold in row N6, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Group elliptic isogeny classes; sum-map pullbacks of the fourfold Weil plane generate the possible mixed classes.",
      "strategy_zh": "按同源类整理椭圆因子；以四维 Weil 平面的求和映射拉回生成可能的混合类。",
      "powers_scope": "sixfold_only_as_stated",
      "method": "low_dimensional_reduction",
      "logical_dependencies": [
        "result:17.1",
        "p:three-ell"
      ],
      "declared_conditional_inputs": [],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:N6"
        ],
        "start_line": 4670,
        "original_pdf_page": 61,
        "ai_pdf_page": 42
      },
      "curated_transitive_external_inputs": [
        "EXT:Classical",
        "EXT:LowDim",
        "EXT:Markman6"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "N7",
      "kind": "case_contract",
      "title": "2+2+2",
      "assumptions": "A complex abelian sixfold in row N7, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Apply the product theorem for abelian varieties of dimension at most two.",
      "strategy_zh": "直接应用各因子维数至多二的阿贝尔簇乘积定理。",
      "powers_scope": "all_powers",
      "method": "direct_external_application",
      "logical_dependencies": [
        "result:18.1",
        "p:lowdim"
      ],
      "declared_conditional_inputs": [],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:N7"
        ],
        "start_line": 4716,
        "original_pdf_page": 62,
        "ai_pdf_page": 42
      },
      "curated_transitive_external_inputs": [
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "N8",
      "kind": "case_contract",
      "title": "2+2+1+1",
      "assumptions": "A complex abelian sixfold in row N8, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Apply the same low-dimensional product theorem, including repeated and CM factors.",
      "strategy_zh": "应用同一低维乘积定理，允许重复及 CM 因子。",
      "powers_scope": "all_powers",
      "method": "direct_external_application",
      "logical_dependencies": [
        "result:19.1",
        "p:lowdim"
      ],
      "declared_conditional_inputs": [],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:N8"
        ],
        "start_line": 4750,
        "original_pdf_page": 62,
        "ai_pdf_page": 43
      },
      "curated_transitive_external_inputs": [
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "N9",
      "kind": "case_contract",
      "title": "2+1+1+1+1",
      "assumptions": "A complex abelian sixfold in row N9, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Apply the same low-dimensional product theorem.",
      "strategy_zh": "应用同一低维乘积定理。",
      "powers_scope": "all_powers",
      "method": "direct_external_application",
      "logical_dependencies": [
        "result:20.1",
        "p:lowdim"
      ],
      "declared_conditional_inputs": [],
      "dependency_coverage": "curated_case_contract",
      "source": {
        "path": "latex/paper.tex",
        "latex_labels": [
          "sec:N9"
        ],
        "start_line": 4779,
        "original_pdf_page": 63,
        "ai_pdf_page": 43
      },
      "curated_transitive_external_inputs": [
        "EXT:LowDim"
      ],
      "curated_transitive_conditional_inputs": []
    },
    {
      "id": "N10",
      "kind": "case_contract",
      "title": "1+1+1+1+1+1",
      "assumptions": "A complex abelian sixfold in row N10, with the simple-factor convention and all stated strata included.",
      "conclusion": "Rational Hodge conjecture on the sixfold relative to the stated external inputs.",
      "strategy": "Collect elliptic isogeny classes; symplectic and opposite-character pairings generate the invariants.",
      "strategy_zh": "按同源类整理椭圆因子；辛配对及相反角色配对生成不变量。",
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      "title": "CM abelian Hodge assertion",
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}