gpt 6 astra ultra attempt on CS-724

SONC geometry: a complete sparse quadratic model and closure issues

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

A polynomial cone is closed under nonnegative linear combinations. SONC denotes sums of nonnegative circuit polynomials. A dual cone consists of linear functionals nonnegative on every element of the primal cone; moment functionals are defined by polynomial integration.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The linear program, non-auxiliary variables, and the distinction between its fibers and projections are not provided. No exact missing relaxation is reconstructed.

Source-record limitation: The supplied statement names a SONC linear-programming relaxation but gives neither that LP nor its variables, and fixing non-auxiliary variables could mean a fiber or a projection in different spaces. Without the source-specific construction, the requested primal/dual relation cannot be stated faithfully.

Statement recorded in the supplied source.

For the SONC-cone linear-programming relaxation, what is the relation between the primal polyhedron obtained by fixing the non-auxiliary variables and the SONC cone, and what is the relation between the dual SONC cone and sparse moment cones? [S2]

Source status.

The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Compare the primal and dual geometry of the cited SONC relaxation.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30004439 (OWR-17475-005). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Real Algebraic Geometry with a View Toward Hyperbolic Programming and Free Probability $2021$, Oberwolfach Reports, DOI 10.4171/owr/2020/12, pp. 639--712 https://doi.org/10.4171/owr/2020/12.
• S3 Dressler--Iliman--de Wolff--Theobald, S-cone framework https://www.math.uni-frankfurt.de/~theobald/publications/s-cone.pdf.
• Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.

Verbatim supplied catalogue question

For the SONC-cone linear-programming relaxation, what is the relation between the primal polyhedron obtained by fixing the non-auxiliary variables and the SONC cone, and what is the relation between the dual SONC cone and sparse moment cones?

The actual source relaxation

Dressler's report, printed pages 694--696, first uses exponential sums on real space and equivalently polynomials on the positive orthant. It describes fixing coefficient variables in a lifted dual-cone formulation, leaving an LP in auxiliary variables. It does not print the full LP or the resulting primal inequalities. Thus an LP fiber cannot be identified here with the entire coefficient cone. We compute the primal cone, dual cone, and moment issue completely for support $A=\{0,1,2\}$ on $t>0$, then test two proposed generalizations. This model does not substitute a reconstructed LP for the missing one.

The full three-coefficient calculation

Write $f(t)=a+bt+ct^2$. Nonnegativity for all $t>0$ is equivalent to \[ C=\{(a,b,c):a,c\ge0,\ b\ge-2\sqrt{ac}\}. \] Necessity of $a,c\ge0$ follows by taking $t\downarrow0$ and $t\to\infty$. If $b\ge0$ these conditions suffice. If $b<0$ and $a=0$ or $c=0$, the polynomial is negative for sufficiently small or large positive $t$, respectively. For $a,c>0$, its minimum on the positive axis is at $t=-b/(2c)$, with value $a-b^2/(4c)$, by completing the square. This proves the stated condition in every case. When $b<0$, the polynomial is itself the one-dimensional nonnegative circuit with endpoints 0 and 2 and interior exponent 1. When $b\ge0$ it is a sum of nonnegative monomials. Conversely every such circuit or monomial is nonnegative. Thus on this support the SONC cone, with the positive-orthant convention, equals $C$.

Use the ordinary coefficient pairing $\langle(a,b,c),(u,v,w)\rangle=au+bv+cw$. The dual is \[ C^*=\{(u,v,w):u,v,w\ge0,\ v^2\le uw\}. \] Testing the three nonnegative coordinate rays shows $u,v,w\ge0$. For fixed $a,c\ge0$ the smallest pairing over allowed $b$ is $au+cw-2v\sqrt{ac}$. If $v^2\le uw$, this is nonnegative because $au+cw\ge2\sqrt{acuw}\ge2v\sqrt{ac}$, where the first inequality is $(\sqrt{au}-\sqrt{cw})^2\ge0$. Conversely, if $u,w>0$, choose $a=w,c=u$ to get $2uw-2v\sqrt{uw}\ge0$, forcing $v\le\sqrt{uw}$. If $u=0$ and $v>0$, choose $c=1$ and let $a$ grow; the expression $w-2v\sqrt a$ becomes negative. The case $w=0$ is symmetric. This proves the formula including all zero boundaries.

Dual membership need not be an actual positive-axis moment

A finite positive measure $\mu$ on $(0,\infty)$ with finite second moment has vector $(\int1\,d\mu,\int t\,d\mu,\int t^2\,d\mu)$ in $C^*$: the middle coordinate is nonnegative, and Cauchy--Schwarz follows by integrating $(t-s)^2\ge0$ against the product measure, giving $2uw-2v^2\ge0$. But $(1,0,1)\in C^*$ is not such a vector. If $\int t\,d\mu=0$, then each set $\{t\ge1/r\}$ has measure zero, since its contribution to the integral is at least its measure divided by $r$. Their union is $(0,\infty)$, so $\mu=0$, contradicting mass one.

It is a limit of genuine two-atom moment vectors. For $0<e<1$ take \[ \mu_e=(1-e^2)\delta_e+e^2\delta_{1/e}. \] Its moments are $(1,2e-e^3,1+e^2-e^4)$ and converge to $(1,0,1)$. This shows that even in a case where SONC equals all nonnegative polynomials on the support, the dual need not equal the unclosed cone of actual finite moments. The boundary of the domain and escape of small mass to infinity both matter.

Why a closure is the natural general relation.

For any finite real function family with evaluation vectors $v(t)$, let $M$ be their finite nonnegative conic combinations and let $P$ be the coefficient cone of functions nonnegative everywhere on the domain. Then $P=M^*$ directly from the definition of the pairing, and $P^*=\overline M$. Here is a finite-dimensional proof of the second assertion. Inclusion $\overline M\subseteq P^*$ follows from continuity. If $z\notin\overline M$, choose a closest point $y$ in the closed cone $\overline M$; existence follows by restricting to a sufficiently large closed ball and compactness. Minimizing squared distance along the segment from $y$ to any $v\in\overline M$ gives $(z-y)\cdot(v-y)\le0$. Taking $v=0$ and $v=2y$ shows $(z-y)\cdot y=0$. Thus $h=y-z$ satisfies $h\cdot v\ge0$ for all $v\in\overline M$, while $h\cdot z=-\|y-z\|^2<0$. Hence $h\in P$ and $z\notin P^*$, as required. For a general SONC subcone $S\subseteq P$, this gives only $\overline M=P^*\subseteq S^*$; equality needs a further argument.

Why polyhedral fibers do not make the cone polyhedral

For fixed $a,c\ge0$, the allowed $b$ form the half-line $b\ge-2\sqrt{ac}$, a polyhedron in one dimension. Nevertheless $C$ is not polyhedral. Intersect it with $a+c=1$ and $b\le0$, and use coordinates $x=2a-1$, $y=b$. The section is exactly the lower half disk $x^2+y^2\le1$, $y\le0$. Its curved boundary contains infinitely many points of the open lower semicircle. If this set were given by finitely many linear inequalities, every boundary point would make at least one nontrivial inequality tight: otherwise finitely many strict margins would give a small open ball inside the set. Each corresponding line meets the circle in at most two points, by substituting its linear equation into the quadratic circle equation. Finitely many lines cannot cover the semicircle, a contradiction.

Nor can $C$ be the exact linear projection of a finite-dimensional polyhedron. Eliminating one variable from finitely many linear inequalities preserves polyhedrality: write all nonzero-coefficient inequalities as lower or upper bounds on that variable, keep the zero-coefficient ones, and require each lower bound to be at most each upper bound. These conditions are sufficient by choosing a value between the largest lower and smallest upper bounds, with one-sided cases unrestricted on the other side. Repetition eliminates every auxiliary variable. An exact projected LP description would therefore make $C$ polyhedral, contrary to the preceding section. Fixing coefficients before taking an LP dual is compatible with this fact; it is not an exact global linear extended formulation of $C$.

Model-assisted audit and remaining scope

The audit checked the positive-axis domain, pairing normalization, zero coefficients, actual versus limiting moments, and fiber versus projection. The three-coefficient calculations are known elementary geometry, not a new characterization of arbitrary SONC supports. The primary S-cone paper was opened as a contextual reference; no unread general theorem from it is invoked. No numerical computation or theorem prover was used. The missing source LP prevents identifying its particular primal polyhedron, and strictness or equality for general sparse dual cones remains unproved here.

Process and contributions

Mathematics and model-assisted adversarial self-audit: OpenAI gpt-6-astra, ultra reasoning, as dispatched for this research batch; exact serving revision not exposed. Codex (GPT-6; exact serving revision not exposed) materially assisted with orchestration and format review. No human mathematical authorship, mathematical edits, or independent human proof checking is asserted. The coordinating Codex (GPT-6; exact serving revision not exposed) also provided a separate model-assisted written-proof review of this manuscript. This is not independent human verification. Contribution prompt A was applied to the full definition and supplied source row, then prompt B was applied adversarially to the derived results. The item-specific audit above records the scope checks and rejected extrapolations. Local assembly and structural checks do not establish mathematical correctness. The source catalogue status is preserved in TOP_PROBLEM and is separate from this attempt status.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The proved material is a rederivation of known or auxiliary facts rather than a substantiated new advance on the original target. The exact source LP and its primal polyhedron are not supplied. The full quadratic-support calculation and general inclusion with the closed moment cone do not characterize arbitrary SONC supports or the requested relaxation.

References

The complete original Sources subsection above is retained, including catalogue attribution and licence notices. The additional list records primary sources actually consulted for this attempt on 11 October 2026; uncited source links reproduced above were not all rechecked.

• Dressler, Global optimization via the dual SONC cone and linear programming, OWR 12/2020, pp. 694--696; full formulation, LP description, and questions 2--3 inspected in the cached primary report. https://doi.org/10.4171/owr/2020/12.
• Primary S-cone framework paper opened for context; none of its general theorems is used as an unproved result in the calculations above. https://www.math.uni-frankfurt.de/~theobald/publications/s-cone.pdf.

Final status

Unresolved in this attempt. The exact source LP and its primal polyhedron are not supplied. The full quadratic-support calculation and general inclusion with the closed moment cone do not characterize arbitrary SONC supports or the requested relaxation.