gpt 6 astra ultra attempt on CS-95

Research-attempt provenance

Definitions and mathematical statement

Let $S\subset\mathbb R^3$ be a finite set with no four coplanar points. A tetrahedralization of $S$ is a face-to-face subdivision of $\operatorname{conv}(S)$ into nondegenerate tetrahedra whose vertex set is exactly $S$. In particular, every input point is used and no Steiner vertices are added.

A $2\leftrightarrow3$ bistellar flip replaces two tetrahedra sharing a triangular face by three sharing an edge, or reverses this operation. The five involved vertices must form a convex triangular bipyramid, and both local subdivisions must have the same support and boundary. Only replacements that remain geometric tetrahedralizations are legal.

Form a graph $\mathcal F(S)$ with one node per tetrahedralization and an edge for each legal flip. The question is whether $\mathcal F(S)$ is connected for every such $S$. Equivalently, can any two tetrahedralizations of the same point set be related by a finite sequence of these moves?

The point coordinates stay fixed throughout. Vertex-inserting $1\to4$ and vertex-deleting $4\to1$ moves are excluded. Connectivity of abstract triangulations, or connectivity restricted to a special class such as regular triangulations, does not settle the general geometric statement.

Short English statement

Can any two tetrahedralizations of the same points in three-dimensional space be connected by legal two-to-three and three-to-two flips, without moving, inserting, or deleting vertices?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5500028 (AMR-054-0028), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problems Project - Computational Geometry, Problem 28. Original source indexed by AMR-054-0028. https://topp.openproblem.net/p28.

Literature and context check

TOPP Problem 28 was consulted. It distinguishes geometric general-position tetrahedralizations from topological examples and from higher-dimensional disconnected flip graphs. The supplied prior prose suggesting a known general geometric three-dimensional disconnection is not supported by this consulted page. No such assertion is used in the attempt.

Research attempt: A nonflippable locally non-Delaunay face

A natural route to connectivity is to fix a preferred triangulation and repair every locally bad face. The following five-point calculation shows why the elementary two-to-three face repair is not always available. Set \[ A=(0,0,0),\ B=(1,0,0),\ C=(0,1,0),\ D=(1/2,1/2,1),\ E=(7/10,7/10,-1/10). \] The tetrahedra $ABCD$ and $ABCE$ meet in $ABC$ with tips on opposite sides. The segment $DE$ meets $z=0$ at $(15/22,15/22,0)$, outside $ABC$ since its first two coordinates sum to $15/11>1$. Hence their union is not convex, and replacing this pair by the three tetrahedra around $DE$ is not a legal two-to-three move. To justify nonconvexity directly, this point lies on the segment between two points of the union; the union meets $z=0$ exactly in $ABC$.

Nevertheless $ABC$ fails the local empty-sphere test. The sphere through $A,B,C,D$ has center $(1/2,1/2,1/4)$ and squared radius $9/16$. The squared distance of $E$ to its center is $2(1/5)^2+(-7/20)^2=81/400<9/16$. Thus a local non-Delaunay face need not be repairable by flipping that face.

This example occurs inside a full tetrahedralization, not just an isolated nonconvex pair. The unique affine dependence can be written \[ \tfrac12 B+\tfrac12 C=\tfrac4{15}A+\tfrac1{15}D+\tfrac23E. \] All coefficients are positive. In particular the five points are in convex position, and every four are affinely independent: direct determinants for $ABCD,ABCE,ABDE,ACDE,BCDE$ are nonzero (the first four are immediate, and the dependence with all five coefficients nonzero then implies the fifth). The three tetrahedra \[ ABCD,\quad ABCE,\quad BCDE \] triangulate their convex hull around edge $BC$. For a detailed coverage check, start from any convex combination of the five points and add a multiple of the displayed affine dependence until one of the coefficients of $A,D,E$ first becomes zero. The resulting nonnegative combination lies in one of these three tetrahedra. Two different stopping choices can coincide only when at least two of those coefficients vanish, which is a shared face or lower-dimensional face; uniqueness of the one-dimensional affine dependence proves disjoint interiors.

There is a legal three-to-two flip around $BC$, yielding $ABDE$ and $ACDE$. Therefore this is not a disconnected flip graph or an isolated node. It refutes only the proposed face-by-face greedy argument and exhibits exactly the extra edge move that the argument must consider. A global descent proof would still need to guarantee an available improving move at every undesired triangulation.

Verification and adversarial audit

The segment intersection, sphere center, squared distances and positive affine dependence were checked by exact rational arithmetic in the displayed formulas. All five four-point subsets are noncoplanar. The existence of the reverse edge flip was included to actively falsify any attempted isolation claim.

Audit corrections.

The nonflippable face was embedded in a valid full tetrahedralization and its available 3-to-2 flip was exhibited; local obstruction is not global disconnection.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

No global potential argument guaranteeing escape from every tetrahedralization has been proved, and no counterexample to connectivity has been constructed. The first gap is an improving legal move or a different connecting mechanism when an offending face itself is nonflippable.

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://topp.openproblem.net/p28 (consulted 2026-10-11).

Model and contributor disclosure

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 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. This is a research attempt with an explicit failed route or restricted calculation, not a full solution, not a current-status certification, and not a claim that the elementary material is new. The catalogue's original source-status label is preserved independently of this attempt status.