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  "site": "Evidence Press",
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      "label": "Finite real-rootedness through five",
      "plainText": "For a positive alphabet rho of size n at most five, F_rho has n simple negative roots.",
      "latex": "F_{\\rho}(t)=\\sum_{k=0}^{n}s_{(3^k)}(\\rho)t^k",
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      "label": "Arbitrary-deletion strict interlacing",
      "plainText": "Deleting any chosen entry from a positive alphabet of size two through five produces a polynomial that strictly interlaces the original one.",
      "scope": "Strictly positive alphabets with 2 <= n <= 5; symmetry reduces a chosen deletion to rho = (X,y).",
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      "label": "Degree-five positive Bezout certificate",
      "plainText": "Every coefficient in each of the five leading degree-five Bezout minors is a positive integer.",
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      "plainText": "Determine whether the 3x3 Toeplitz-minor transform preserves real negative zeros in every degree.",
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      "status": "claimed-result",
      "label": "Five-support exclusion",
      "plainText": "No clean nonzero binary-decimic restriction with exactly five projective roots satisfies the frozen double-conic normal-layer equation in characteristic zero.",
      "scope": "Exactly five support points on the clean double-conic normal layer; computer-assisted exact certificates.",
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      "status": "claimed-result",
      "label": "Tangent nullcone containment",
      "plainText": "All 31 generators of the multiplicity-six binary-decimic locus lie in the radical of the tangent normal-layer ideal.",
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      "releaseTitle": "Structural reductions toward the quartic Hessian conjecture in dimension four",
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      "status": "claimed-result",
      "label": "Sixteen rational source syzygies",
      "plainText": "For the frozen fourth block there is a rational matrix T satisfying B T plus C16 equals zero.",
      "latex": "BT+C_{16}=0",
      "scope": "Sixteen selected source columns only; not the inhomogeneous fourth target.",
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      "label": "Secant fourth-target obstruction",
      "plainText": "Prove rational fourth-target membership or a full secant colon or saturation statement.",
      "scope": "Remaining secant chart; required before normal-layer closure, HC4, or JC2.",
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      "label": "Four-state all-time row-TV counterexample",
      "plainText": "For weights (1,1/2,2,1), the incomparable row distance is (2/3)(1/2)^(t-1) and the extremal distance is (1/2)^t for every integer t at least 1.",
      "latex": "d_{\\mathrm{TV}}(P^t(10,\\cdot),P^t(01,\\cdot))=\\frac23\\left(\\frac12\\right)^{t-1}>\\left(\\frac12\\right)^t=d_{\\mathrm{TV}}(P^t(00,\\cdot),P^t(11,\\cdot))",
      "status": "claimed-result",
      "scope": "The specified random-scan two-spin heat-bath chain; the scholarly object is an anonymous unrefereed candidate with producer-side replay, not an independently validated theorem.",
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      "releaseTitle": "Exact finite rigidity for Furter's R(3): all windows through n=299",
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      "label": "Furter R(3) finite window",
      "plainText": "R(3,n) holds for every integer n with 1 <= n <= 299",
      "latex": "R(3,n)\\text{ holds for every }1\\le n\\le299",
      "status": "claimed-result",
      "scope": "Exact finite principal statement of this anonymous unrefereed candidate; it does not establish universal R(3).",
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      "releaseTitle": "Exact finite rigidity for Furter's R(3): all windows through n=299",
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      "label": "Inverse-coefficient radical window",
      "plainText": "radical((g_(d,3), g_(d+1,3), g_(d+2,3)) Q[x_1,x_2,x_3]) = (x_1,x_2,x_3) for 2 <= d <= 300",
      "latex": "\\sqrt{(g_{d,3},g_{d+1,3},g_{d+2,3})\\mathbf Q[x_1,x_2,x_3]}=(x_1,x_2,x_3)\\quad(2\\le d\\le300)",
      "status": "claimed-result",
      "scope": "Equivalent finite radical formulation used by the release; producer-side exact certificates are reported for the stated range only.",
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      "releaseSlug": "full-e3-column-polydegree-conjecture",
      "releaseTitle": "A Jacobian smooth-point criterion and the full e=3 column of the Polydegree Conjecture",
      "releaseUrl": "https://evidencepress.org/releases/full-e3-column-polydegree-conjecture/",
      "releaseDoi": "10.5281/zenodo.21909085",
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      "kind": "statement",
      "label": "Full e=3 Polydegree containment",
      "plainText": "G_(d+3) is contained in closure(G_(d,4)) for every integer d >= 2",
      "latex": "\\mathcal G_{(d+3)}\\subseteq\\overline{\\mathcal G_{(d,4)}}\\quad(d\\ge2)",
      "status": "claimed-result",
      "scope": "All-degree principal statement of this anonymous unrefereed candidate; the release reports producer-side exact algebra and finite Arb certificates but no unaffiliated rerun or specialist review.",
      "sequenceTerms": [],
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      "releaseTitle": "Exact Smith Invariants and Affine Determinant Lines of Binary-Form Factorisation",
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      "kind": "identity",
      "label": "All-factor bordered Jacobian determinant",
      "plainText": "det D(m_d,g_1,...,g_(k-1)) = +/- Delta_d det(delta_b g_a)",
      "latex": "\\det D(m_{\\mathbf d},g_1,\\ldots,g_{k-1})=\\pm\\Delta_{\\mathbf d}\\det(\\delta_b g_a)",
      "status": "claimed-result",
      "scope": "Universal multi-border determinant identity asserted by the anonymous unrefereed candidate; exact replay checks examples and structural consequences but does not replace independent proof review.",
      "sequenceTerms": [],
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      "releaseTitle": "Exact Smith Invariants and Affine Determinant Lines of Binary-Form Factorisation",
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      "id": "ep-math:exact-smith-invariants-affine-determinant-lines:root-partition-degree",
      "kind": "formula",
      "label": "Labelled root-partition degree",
      "plainText": "nu = D! / product_i(d_i!) where D = sum_i d_i",
      "latex": "\\nu=\\frac{D!}{\\prod_i d_i!},\\qquad D=\\sum_i d_i",
      "status": "supporting-result",
      "scope": "Multinomial root-partition factor in the release's finite-locally-free degree formula, under its stated generic and normalisation hypotheses.",
      "sequenceTerms": [],
      "oeisId": null,
      "claimId": null,
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      "label": "Stationary-average chi-square transition scale",
      "plainText": "t_N = (log(2)/(2 delta)) N/log(N), where delta = min(2c, alpha+beta-2c)",
      "latex": "t_N=\\frac{\\log 2}{2\\delta}\\frac{N}{\\log N},\\qquad\\delta=\\min(2c,\\alpha+\\beta-2c)",
      "status": "claimed-result",
      "scope": "Open-region stationary-weighted average conditional point-start chi-square transition claimed by the anonymous unrefereed candidate; no critical-point, fixed-start, worst-case or total-variation result is asserted.",
      "sequenceTerms": [],
      "oeisId": null,
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      "releaseTitle": "Smooth-Point Certificates for Polydegree Containments: A Jacobian Interpretation of the Lewis-Perry-Straub Determinant",
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      "label": "Computed e=3 affine intersection lengths",
      "plainText": "a(d) = floor(d(d+1)/6) for 2 <= d <= 20",
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      "scope": "Exact rational and Singular computation for d=2 through d=20; this parent release explicitly leaves the uniform affine theorem open.",
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      "assuranceTaskIds": []
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      "kind": "identity",
      "label": "Multiplication-resultant Jacobian determinant",
      "plainText": "det D Phi_(r,s)(A,B) = (-1)^(s(r+1)) (r-s) Res(A,B)^2",
      "latex": "\\det D\\Phi_{r,s}=(-1)^{s(r+1)}(r-s)\\operatorname{Res}(A,B)^2",
      "status": "claimed-result",
      "scope": "All-degree determinant identity asserted by the anonymous unrefereed candidate in characteristic zero; only explicit low-degree polynomial identities received bounded computational checks.",
      "sequenceTerms": [],
      "oeisId": null,
      "claimId": null,
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      "releaseTitle": "VR2(K4) = 20: twenty vertices force two vertex-disjoint monochromatic K4s",
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      "label": "Two vertex-disjoint monochromatic K4s",
      "plainText": "VR_2(K_4) = 20",
      "latex": "\\mathrm{VR}_2(K_4)=20",
      "status": "claimed-result",
      "scope": "Principal statement of this anonymous unrefereed preprint; the upper bound inherits the z(20)=6 fixed-core certificate and encoding-soundness boundary.",
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      "releaseTitle": "z(20) = 6: resolving the first open case of Erdős problem 758",
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      "releaseDoi": "10.5281/zenodo.21647645",
      "releaseVersion": "candidate-2026-07-26",
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      "id": "ep-math:z20-equals-6:cochromatic-number-20",
      "kind": "statement",
      "label": "Cochromatic number at 20",
      "plainText": "z(20) = 6",
      "latex": "z(20)=6",
      "status": "claimed-result",
      "scope": "Principal statement of this anonymous unrefereed candidate; producer-side certificate replay is reported, while independent reproduction, end-to-end formalisation and peer review remain absent.",
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}