gpt 6 astra ultra attempt on CS-672

Research-attempt provenance

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 matroid consists of a finite ground set with a hereditary independence family satisfying exchange. It is strongly base orderable if for every two bases $B_1,B_2$ there is a bijection $\phi:B_1\to B_2$ such that for every $X\subseteq B_1$, both $(B_1\setminus X)\cup\phi(X)$ and $(B_2\setminus\phi(X))\cup X$ are bases. Its base polytope is the convex hull of base incidence vectors. The Integer Caratheodory Property requires that for every positive integer $k$ and integer $w\in kP$, one can write $w=\sum_i n_ix_i$, where the $x_i$ are affinely independent integer points of $P$, the $n_i$ are nonnegative integers, and $\sum_i n_i=k$.

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 supplied registry flags this extraction as unrecoverable. It is displayed as a literal source fragment; the definitions provide context rather than a replacement formal statement.

Source-record limitation: Source flags the statement as unrecoverable: The official report contains three separate questions, but this extraction conflates all three and repeats the first two records in this chunk while paraphrasing the third only vaguely. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance.

Related catalogue records: 30001931, 30001933, 30001934. Separate source IDs are preserved for their provenance. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim. The supplied catalogue marks this record as unrecoverable. Its extract and terminology are documented, but a complete original problem has not been reconstructed.

Statement recorded in the supplied source.

Does the intersection of two base polytopes of strongly base orderable matroids have the Integer Caratheodory Property; does the r-arborescence polytope have this property; and can matroids admit a more intuitive greedy decomposition into affinely independent integral vectors? [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 2026-08-21.

Short English statement

The supplied fragment concerns composite Repetition of Three Matroid Questions. Its complete mathematical target is not recovered.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30001932 (OWR-11451-005). 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 Combinatorial Optimization $2012$, Oberwolfach Reports, DOI 10.4171/owr/2011/53, pp. 3003--3076 https://doi.org/10.4171/owr/2011/53.
• S3 Gijswijt--Regts ICP theorem https://arxiv.org/abs/1004.4552.
• Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.

Preparation and review.

The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30001932 and code OWR-11451-005 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Related canonical identities: 30001931, 30001933, 30001934. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Literature and context check

The primary report on printed pages 3026--3027 contains three separate questions. This catalogue record conflates them, while records 30001931, 30001933 and 30001934 retain the individual formulations. Its damaged/composite provenance and ID are preserved. The following repeats complete special-case arguments from those attempts for standalone reading: bipartite matching ICP, acyclic arborescence ICP, and a counterexample to an arbitrary-tree greedy rule. They are not three new independent results and do not resolve the composite agenda.

Research attempt: Three separate decomposition routes for a composite source record

A common-base subclass: bipartite perfect matchings

Let the ground set be the allowed edges of a bipartite graph with equally many left and right vertices, with at least one vertex on each side. The empty graph instead has the singleton empty-vector polytope and its Integer Caratheodory Property is immediate. One partition matroid allows at most one edge at each left vertex, and the other at most one at each right vertex. Assume a perfect matching exists. Each matroid is strongly base orderable: for two bases, biject the unique chosen edges within each vertex block; swapping any subset of these matched pairs leaves exactly one edge in every block, in both bases. Their base polytopes are products of unit simplices, because a vector of nonnegative coordinates with block sums one is the convex average of choices, obtained by independently choosing one edge per block with those probabilities. Hence their intersection consists of nonnegative edge vectors with all row and column sums one.

Let $w$ be an integer vector in its $k$-fold dilation, $k\geq1$. Its positive-support bipartite graph has a perfect matching: for a set $U$ of left vertices, the total incident weight is $k|U|$, all lying on its neighboring right vertices, whose total available column weight is at most $k|N(U)|$. Thus $|N(U)|\geq|U|$. To recall the matching implication, a maximum matching failing to cover a left vertex yields an alternating-path search. With no augmenting path, all reached right vertices are matched to reached left vertices other than the initial unmatched one, and all neighbors of reached left vertices are reached. This would give a set with fewer neighbors than vertices, contradicting the inequality.

Choose a perfect matching incidence vector $x$ in the support and let $\lambda=\min_{e:x_e=1}w_e$. This is a positive integer. Subtraction leaves a nonnegative integer vector with every row and column sum $k-\lambda$. Repeat until the residual is zero. The positive multiplicities sum to $k$, and the original $w$ is their weighted sum of matching incidence vectors.

The chosen vectors are affinely independent, an assertion stronger than merely having an integer decomposition. At each step select an edge where the displayed minimum is attained; that edge occurs in the current matching and has zero weight in every later residual, so it occurs in no later chosen matching. In a linear dependence among the chosen vectors, take the earliest nonzero coefficient. Its selected edge has contribution only from that vector and earlier vectors; the earlier coefficients are zero and later vectors omit the edge. That coefficient must therefore be zero, a contradiction. Thus the vectors are even linearly independent, and in particular affinely independent. This proves the Integer Caratheodory Property for this common-base subclass.

An affine-independent integer decomposition of a product of simplices

Let $k\geq1$ be an integer. Consider nonnegative integer coordinates $w_{b,j}$ divided into finitely many nonempty blocks $b$, each satisfying $\sum_jw_{b,j}=k$. Form the cumulative sums $s_{b,j}=\sum_{i\leq j}w_{b,i}$ and sort all their distinct values together with zero and $k$: \[ 0=t_0<t_1<\cdots<t_h=k. \] For each interval $(t_{\ell-1},t_\ell]$, choose in each block the coordinate whose cumulative interval contains it, and let $v_\ell$ be the corresponding zero-one vector with one chosen coordinate per block. This is unambiguous even with zero coordinates, because a nonempty interval lies inside exactly one positive-length cumulative interval per block. Assign multiplicity $n_\ell=t_\ell-t_{\ell-1}$. Lengths of the relevant intervals add to $w_{b,j}$, so \[ w=\sum_{\ell=1}^h n_\ell v_\ell,\qquad n_\ell\in\mathbb Z_{>0},\qquad\sum_\ell n_\ell=k. \] To prove affine independence, suppose $\sum_\ell c_\ell v_\ell=0$ and $\sum_\ell c_\ell=0$. Each internal breakpoint $t_j$ is a cumulative sum in some block. The linear functional summing the corresponding initial coordinates of that block is one on $v_1,\ldots,v_j$ and zero on all later vectors. Applying it gives $\sum_{\ell\leq j}c_\ell=0$ for each $j<h$. Taking consecutive differences and using the total coefficient sum proves every $c_\ell=0$.

Application to acyclic directed arborescences

Let a directed acyclic graph have a first vertex $r$ in a topological order, no arcs entering $r$, and at least one incoming arc at every other vertex. Choose one incoming arc for each nonroot vertex. Following parent arcs backwards strictly decreases the order and must end at $r$, since every other vertex has a parent. Thus the chosen arcs connect every vertex to $r$ and form an outward rooted arborescence. Conversely an arborescence makes exactly these choices. The polytope is therefore a product of the incoming-arc simplices: the convex-hull equality follows by independently sampling each incoming choice from its specified probability vector. The construction above proves its Integer Caratheodory Property for every integer dilation. The graph with only $r$ has the singleton empty-vector polytope and has the property trivially.

An arbitrary tree can have a fractional maximal subtraction

Let $P$ be the spanning-tree polytope of $K_4$, $k=2$, and $w_e=1$ on all six edges. Two complementary Hamilton paths are spanning trees, so $w\in2P$. Choose the star $x$ at one vertex. On the remaining three vertices, every spanning tree has at most two internal edges. The star has none there, so a residual in $(2-\lambda)P$ must satisfy $3\leq2(2-\lambda)$, giving $\lambda\leq1/2$. At $\lambda=1/2$, the normalized residual has values $1/3$ on the three spokes and $2/3$ on the leaf-triangle edges. Average the nine trees formed by one of the three spokes and any two of the three leaf-triangle edges. Every such graph is a tree, and each spoke appears in three of them while each triangle edge appears in six. This proves feasibility and hence maximal value $1/2$ for the star. In contrast, choosing one of the Hamilton paths permits $\lambda=1$, since its complement is a tree; a chosen edge has weight one, so nonnegativity prevents any larger removal with nonnegative remaining scale. For $\lambda>2$, an unchosen edge would remain positive whereas a negative dilation of this nonnegative polytope has nonpositive coordinates, so no extra feasible values occur in this example.

Thus "choose any spanning tree and remove its largest multiple" is not a valid integer greedy rule. The original question asks for existence of a suitable tree, and this instance has such a tree; it is not a counterexample to that question.

Verification and adversarial audit

Each ICP argument proves integral multiplicities and affine independence separately. The first uses exact elimination of a positive coordinate; the second uses cumulative-breakpoint prefix functionals. The K4 example explicitly distinguishes a bad star from a good Hamilton path and hence does not refute the source's existential greedy question. The repeated material and the parent model's corroborating mathematical review are disclosed.

Audit corrections.

Separated all three original questions and labelled the repeated proofs as shared special-case analysis. No full-scope characterization or independent novelty is asserted.

External computation.

Executed check_decompositions.py: 500 deterministic pseudorandom product-simplex integer inputs passed exact reconstruction and affine-rank checks using rational Gaussian elimination. The nine-tree undirected and three-tree directed residual averages were also checked exactly with rational arithmetic. These finite tests supplement, rather than replace, the written proofs.

Scope and remaining gaps

The strongly base orderable intersection question beyond matching polytopes, the general arborescence question beyond acyclic graphs, and the general existential greedy question all remain unresolved. The source composite wording is retained rather than silently converted into one of the easier special cases.

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 calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://ems.press/content/serial-article-files/46369 (consulted 2026-10-11).

Model and contributor disclosure

The parent Codex GPT-6 model reviewed the written arguments at their declared scopes. This is model-assisted review, not independent human verification. OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 provided the mathematical corroboration credited above and performed orchestration and repository-format review; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.