[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"cn7rHMYZnD":3},"# Persistent failure of quartic SOS exactness\n\nAn explicit rational cubic in 14 variables has sphere maximum one but a\nstrictly larger degree-four moment value. The resulting quartically\nregularized polynomial fails sum-of-squares exactness for **every positive\nregularization parameter**, with its quadratic and cubic data fixed.\n\nFor the 25-term cubic $`C`$ specified in [Proof.md](Proof.md), set\n\n```math\nP_{\\sigma,a,b}(z)=\\frac{\\sigma}{4}\\|z\\|^4-\\frac a2\\|z\\|^2-bC(z),\n\\qquad \\sigma,a,b>0.\n```\n\nWith\n\n```math\nr=\\frac{3b+\\sqrt{9b^2+4a\\sigma}}{2\\sigma},\n\\qquad \\mu=-\\frac{br^3+ar^2}{4},\n\\qquad \\delta=\\frac{753357629}{500000000000},\n```\n\nthe actual global minimum is $`\\mu`$, attained at $`(r,0,\\ldots,0)`$.\nThe polynomial $`P_{\\sigma,a,b}-\\mu`$ is nonnegative and is not a sum of\nsquares of real polynomials. Every SOS lower bound $`\\gamma`$ satisfies\n\n```math\n\\gamma\\le\\mu-b\\delta r^3\u003C\\mu.\n```\n\nAt the origin the gradient vanishes and the Hessian is $`-aI_{14}`$.\nThe complete minimizer set is also classified: an isolated axis point and an\neight-dimensional sphere. The specialization $`a=b=1`$ answers the stationary, non-PSD-Hessian\nquestion in Cartis–Zhu, Remark 2.3, for the Euclidean quartic regularizer.\n\n## A unique nondegenerate minimizer\n\nA second fixed cubic, still with 25 monomials in 14 variables, is\n\n```math\n\\widehat C(z)=\\frac{1000C(z)+z_0\\|z\\|^2}{1001}.\n```\n\nReplacing $`C`$ by $`\\widehat C`$ preserves the same radius $`r`$ and\nminimum $`\\mu`$. The unique global minimizer is $`(r,0,\\ldots,0)`$,\nand its Hessian is positive definite, with diagonal blocks\n\n```math\n2a+3br,\\qquad \\frac{br}{1001}I_4,\\qquad\n\\frac{9001br}{1001}I_9.\n```\n\nNevertheless, the exact minimum shift is not SOS for any positive\nregularization parameter. Every SOS lower bound obeys\n\n```math\n\\gamma\\le\\mu-b\\widehat\\delta r^3\u003C\\mu,\n\\qquad \\widehat\\delta=\\frac{132860603}{143000000000}.\n```\n\nThe [companion proof](UniqueMinimizer.md) reuses the same rational moment\ncertificate. Positive curvature here is at the minimizer; the Hessian at\nthe stationary origin remains $`-aI_{14}`$.\n\n## Reading\n\n- [Note.md](Note.md): mathematical overview and significance.\n- [Proof.md](Proof.md): explicit construction and complete written proof.\n- [Applications.md](Applications.md): parameter scaling, minimizers and robustness.\n- [UniqueMinimizer.md](UniqueMinimizer.md): the unique nondegenerate minimizer theorem and a general selection principle.\n- [RadialConsequences.md](RadialConsequences.md): written optimized-scale bounds and flat- and positive-Hessian consequences.\n- [Sources.md](Sources.md): primary sources and exact attribution.\n- [Disclosure.md](Disclosure.md): production and review account.\n- [VERIFICATION.md](VERIFICATION.md): formal scope, checks and exact revisions.\n\nThe independent statement interface is\n[PersistentSOSChallenge.lean](PersistentSOSChallenge.lean); the proof entry\nis [PersistentSOSSolution.lean](PersistentSOSSolution.lean).\n\n## Reproduction\n\n```sh\nlake build\npython3 scripts/check_certificate.py\npython3 scripts/independent_certificate_check.py --negative-controls\npython3 scripts/check_unique_minimizer.py\npython3 scripts/check_sources.py\npython3 scripts/audit_axioms.py\n```\n\nThe formal statements, verification commands and exact-commit evidence are\nrecorded in [VERIFICATION.md](VERIFICATION.md). Submission and registration\nare separate steps. No optimal SOS value, smallest possible\ndimension, worldwide priority, or human specialist review is claimed.\n\nMIT license, copyright 2026 JD Jones.\n",1792957698371]