Unbounded generator degrees in moment ideals via graph faces

Summary

Can the equations describing independent-variable moments always be built from equations of some fixed maximum degree? This candidate gives a constructive negative answer when both the number of variables and the moment order may grow.

For any proposed degree limit, the proof constructs a full moment model with a relation that lower-degree equations cannot generate. The key is to embed an even cycle so that the remaining model coordinates cannot provide a shortcut. The release also embeds general graph configurations and gives an example where the moment variety is intrinsically nonnormal.

Summary for specialists

With zeroth moments normalized to one, let $I_{n,d}$ be the kernel of the monomial map $m_\alpha\mapsto\prod_{i:\alpha_i>0}\mu_{i,\alpha_i}$ for $|\alpha|=d$. For every $r\geq3$, put

$$n_r=r^2,\qquad d_r=\frac{(r+1)^{r^2-2r+2}-1}{r}.$$

The written proof shows that $I_{n_r,d_r}$ has an indispensable degree-$r$ binomial and cannot be generated by arbitrary polynomials of degree below $r$. This addresses the full-model generator-degree conjecture in Section 4 of Alexandr–Kileel–Sturmfels (2023), not the separate fixed-partition coordinate models.

Every finite simple graph with an edge admits an exposed-face realization preserving its graded toric ideal. A seven-vertex graph yields a nonnormal homogeneous moment coordinate ring and an intrinsically nonnormal projective moment variety at $n=20$, $d=628292358730$.

Technical account

The construction encodes edge selection using binary vertex variables and nonedge slack variables. Packing the equality constraints into separate integer digits gives a fixed moment order without carries. Each position may then use only its zero level or one specified positive level.

A nonnegative supporting functional vanishes exactly on these permitted columns. That exposed-face property is essential: setting every other moment variable to zero preserves the kernel. A difficult coordinate submodel alone would not prove the result for the full model.

For an even cycle, the target parameter image has exactly two preimages: the monomials of its two perfect matchings. They are coprime and have degree $r$. Any nonzero lower-degree contribution to their difference would need a common monomial factor, but none exists. A second argument uses the principal face ideal and its retraction.

For nonnormality, the two-odd-cycle graph obstruction supplies a semigroup hole. The proof shows that this hole persists after adding arbitrarily many copies of a chosen edge, so it survives localization on a projective chart. Failure of integral closure of the homogeneous ring alone would not justify this stronger conclusion.

Evidence, assurance and limitations

The all-parameter results rest on a self-contained written proof. Finite checks corroborate examples; they do not certify the uniform theorem. Constructive checks cover $r=3,\ldots,8$, while a separately written meet-in-the-middle implementation covers $r=3,4,5$. Thirteen distinct semantic mutations are rejected. Full certificates agree in normal and optimized Python, and a regression test checks that frozen-release replay changes no files.

The publication process includes internal, producer-coordinated editorial assessment. It does not establish unaffiliated reproduction, external specialist review, journal peer review or formal verification. The supplied acceptance review reports extra checks, but its linked audit archive was not supplied here; those reports are not promoted into inspected independent evidence.

The parameters grow rapidly. The release does not prove fixed-$n$ unboundedness, optimal parameter growth, sharp largest-generator degrees, bounds for set-theoretic defining equations, general mixture results or practical hardness of statistical algorithms. Historical priority remains unestablished after a bounded source search.

The theorem concerns formal algebraic moment parameters. It does not classify feasible real probability distributions or positivity constraints, nor assert that every point of an embedded graph face is probabilistically realizable. The existence construction is not a practical fitting procedure, runtime lower bound or sample-complexity result.

Relationship to earlier work

The originating conjecture concerns full fixed-total moment ideals. Classical ingredients include cycle toric relations, fiber-based indispensability, exposed-face transfer, digit packing and the graph odd-cycle normality obstruction. Hypergraph universality is also established in other settings. The contribution claimed here is their exact realization in this moment configuration, together with the resulting lower bound and projective nonnormality consequence—not invention of those ingredients.

Who should care, and why

Why the problem matters

Low-degree equations often make algebraic models easier to describe and calculate with. A uniform bound would constrain their possible algebraic complexity. The construction shows why no such bound can hold across this full family. Its graph embedding also connects equation complexity to a separate geometric property: normality.

How to inspect or reproduce the recorded checks

Read the manuscript's definitions and Lemmas 2–3 before inspecting the cycle example. The small example now includes an explicit mask-to-edge key. Sections 3.2 and 3.3 give two lower-bound arguments; Theorem 6 supplies the full projective-chart argument.

After extracting the release ZIP, run python3 make_review_bundle.py --checks-only. This uses temporary storage, prints its receipt and leaves release files unchanged. Run python3 test_release.py and python3 -O test_release.py for the preservation regression. The exact checks use only Python's standard library. Use a scratch copy for a full PDF rebuild, which intentionally rewrites generated files.

The most valuable next projects

Reduce the number of moment positions or the enormous encoded order; investigate unboundedness at fixed dimension; and identify further graph-ring properties that survive the embedding. Each is a new research problem, not an implication already established by the release. A proof-assistant treatment of the encoding and face retraction would provide a distinct assurance contribution.

What is in the evidence package

The archive contains the full manuscript in PDF, Markdown and LaTeX, exact checkers and certificates, a machine-readable claim index, replay regression tests, internal audits and editorial reports, a review-response matrix, source and novelty records, component licences and a checksummed inventory. The separate archive replay receipt binds the exact ZIP bytes without a circular self-hash. Audio and art explain the result; they are not additional mathematical evidence.

@misc{momentidealsunboundedgeneratordegrees2026,
  title        = {Unbounded generator degrees in moment ideals via graph faces},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.22680540},
  url          = {https://doi.org/10.5281/zenodo.22680540},
  version      = {0.1.1-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/moment-ideals-unbounded-generator-degrees/}
}