Generative Sphere Packing across Dimensions and Periodic Geometries

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 sphere-packing configuration is a collection of centers with prescribed minimum pairwise separation. A periodic configuration repeats under a lattice of translations. Projection changes candidate coordinates to satisfy the chosen geometric constraints.

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 training distribution, projection, extrapolation guarantee, and downstream-improvement criterion are not specified.

Source-record limitation: Generative-model extrapolation and downstream improvement are requested without a distribution or guarantee.

Statement recorded in the supplied source.

Can one conditional generative model trained across different numbers of balls and dimensions extrapolate to larger sphere-packing instances while preserving or improving minimum separation after projection? Extend projection-corrected flow-matching methods from finite boxes to periodic packings and establish when the resulting candidates improve downstream optimization. [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

Extend generated packing configurations across sizes, dimensions, and periodic geometries.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30006438 (OWR-14299525-002). 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 MATRIX-MFO Tandem Workshop: Machine Learning and AI for Mathematics $2026$, Oberwolfach Reports, DOI 10.4171/owr/2025/43, pp. 2323--2350 https://doi.org/10.4171/owr/2025/43.
• 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 30006438 and code OWR-14299525-002 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. 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, printed pp. 2332--2333, describes a generative/local-search packing pipeline and asks about extrapolation, periodic cells, and convergence of projection-corrected sampling. The following certificates and explicit projection example are independent auxiliary calculations. No trained model, new packing record, or empirical extrapolation result is asserted.

Research attempt: Packing certificates, projection ambiguity and periodic checks

For centers $c_1,\ldots,c_n\in[0,1]^d$, $n\geq2$, balls of radius $r\geq0$ are contained in the cube and have disjoint interiors exactly when \[ r\leq c_{i\ell}\leq1-r\quad\text{for all }i,\ell, \qquad \|c_i-c_j\|^2\geq4r^2\quad(i\ne j). \] This is a finite exact certificate for rational centers and radius, using rational arithmetic. If each center is perturbed by Euclidean norm at most $\eta$, pair distances decrease by at most $2\eta$ and each wall clearance by at most $\eta$, by the triangle inequality and the coordinate bound. Thus a packing of radius $r$ remains certified after perturbation at radius $\max(0,r-\eta)$ provided the new centers stay in the cube when $r<\eta$; for the useful regime $\eta\leq r$, cube containment follows from the clearance bound. No floating-point near-contact claim is needed.

Projection onto packing constraints is nonconvex and can be ambiguous even in one dimension. Take two labeled centers in $[0,1]$ with prescribed radius $r=1/5$. The feasible set requires wall clearance $1/5$ and separation at least $2/5$. The point $(1/2,1/2)$ has two equally near feasible points, $(3/10,7/10)$ and $(7/10,3/10)$. To prove optimality, on the component $x_2-x_1\geq2/5$ the squared distance to $(1/2,1/2)$ is minimized at mean $1/2$ and gap $2/5$; the opposite component gives its reflection. Perturbing the proposed centers to $(1/2-\delta,1/2+\delta)$ selects the first component for positive small $\delta$ and the second for negative $\delta$. Thus every nearest-point selection has a discontinuity at this tie. A fixed generator repeatedly proposing the midpoint and a fixed tie rule can return the same radius-$1/5$ feasible packing forever, despite the feasible radius $1/4$ at centers $(1/4,3/4)$. Feasibility projection alone guarantees no optimization or generalization.

Indeed, for $n$ equal intervals of radius $r$ in the unit interval, ordering centers gives \[ 1\geq r+2r(n-1)+r=2nr. \] Equality is attained by centers $(2i-1)/(2n)$. Any extrapolation keeping a positive radius while increasing $n$ indefinitely must therefore fail even in dimension one.

Finite verification of a specified periodic candidate.

Let an invertible lattice matrix $A\in\mathbb R^{d\times d}$ and finitely many centers $c_i$ in one cell be specified. A periodic packing of radius $r>0$ requires \[ \|c_i-c_j+Az\|\geq2r \quad\text{for all }i,j,z\in\mathbb Z^d, \quad(i,j,z)\ne(i,i,0). \] If a violation occurs, then \[ \|z\|\leq\|A^{-1}\|_{\mathrm{op}}(2r+\|c_i-c_j\|). \] Choose a certified larger integer bound using $\|A^{-1}\|_F$ and the largest center difference. Enumerating all integer vectors in that bounded cube therefore suffices for exact checking, with rational squared norms when the data are rational. Self-translates $i=j,z\ne0$ must be included: a single center with lattice $\operatorname{diag}(1/10,10)$ already overlaps its nearest copy at radius $1/10$, although no distinct-center pair exists in its cell. The enumeration is finite but can be very expensive for ill-conditioned lattices; it is a verifier, not a generative algorithm.

Verification and adversarial audit

The perturbation statement explicitly handles the zero-radius and containment boundary. Projection optimality was checked on both labeled feasible components. Periodic verification includes self-images and uses an upper bound on inverse-lattice norm, so no distant lattice translate capable of overlap is omitted.

Audit corrections.

Distinguished certification from optimization; included self-translates in periodic constraints and the containment condition for perturbations larger than the radius.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

No convergence theorem for the source's learned nonconvex projection dynamics, cross-dimension extrapolation guarantee, or improved periodic packing is obtained. Exact candidate certification addresses only verification after a candidate has been generated.

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/52253 (consulted 2026-10-11).

Model and contributor disclosure

Codex GPT-6 completed written-proof review of this attempt at its declared scope; this is model-assisted review, not independent human proof checking. 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 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.