Factorial spikes separate componentwise Noetherianity from FI-Noetherianity

Summary

Noetherianity says that ascending chains of ideals eventually stop. For an FI-algebra there are two different places to ask for that stability:

• inside one fixed finite component; and
• across the whole compatible system of finite sets and injections.

This anonymous, unrefereed candidate separates them explicitly. Over any field $k$, let

be the standard polynomial FI-algebra. For every $r\ge2$, introduce the factorial spike

Let $A$ be the sub-FI-algebra generated by every translate of every $g_r$. Then every fixed component $A(S)$ is an affine, finitely presented Noetherian $k$-algebra. Nevertheless, the FI-ideals generated through successive widths form a strict chain

Thus componentwise Noetherianity does not imply FI-Noetherianity for unrestricted sub-FI-algebras of $V$. The proof uses only monomials and integer exponents, so it is independent of characteristic.

The restriction “unrestricted” matters: this $A$ is not finitely generated as an FI-algebra. A new generator orbit appears in every width.

The idea in widths two, three and four

The first spikes are

Each spike has one peak variable and $r-1$ light variables. Relabelling may move the peak and choose a different light set, but it does not change the pattern of exponents.

At a fixed width $n$, only the generators with $2\le r\le n$ can occur. There are finitely many of them. Across the FI-system, however, there is no largest width: $g_{n+1}$ is genuinely new.

The factorials are chosen to make that last statement exact. To build a width-$R$ spike from smaller spikes, every factor must put its peak on the same target variable. Their light-variable sets must partition the other $R-1$ variables. This constrains both the number of light variables and the peak exponent.

The exact result: factorisation criterion

More generally, replace $r!$ by integers $a_r\ge2$ and write

The native width-$R$ monomial $g_R(a)$ is a product of translates of lower-width generators exactly when there are widths $s_1,\ldots,s_q<R$ such that

The first equation counts the light variables. The second matches the peak exponent.

For factorial exponents, every $s_j<R$ satisfies $s_j!\le(R-1)!$, while the first equation implies $q\le R-1$. Hence

The peak equation cannot hold. Therefore $g_R$ cannot be assembled from lower-width generator orbits.

Why the FI-ideal chain is strict

Let $J_R$ be the FI-ideal generated by $g_2,\ldots,g_R$. Certainly $J_R\subseteq J_{R+1}$.

To prove strictness, inspect the native width-$R$ spike $g_R$. The algebra $A([R])$ is a monomial subalgebra. A monomial belongs to an ideal generated by monomials exactly when it is one of those generators times another monomial of the subalgebra.

If $g_R$ belonged to $J_{R-1}([R])$, it would therefore contain a translated lower-width spike as a monomial factor. The remaining multiplier can itself be expanded into defining generator monomials. No width-$R$ factor can occur, because such a factor already has the full total degree of $g_R$ and the ideal generator has positive degree. We would obtain a complete factorisation of $g_R$ into smaller spikes, contradicting the factorial inequality.

Thus

which proves every inclusion is strict.

Why every fixed component is Noetherian

Fix a finite set $S$ with $|S|=n$. A translated width-$r$ generator is determined by its peak variable and its unordered set of $r-1$ light variables. There are

such monomials. Summing over $2\le r\le n$ gives

distinct displayed generators.

Therefore $A(S)$ is a finitely generated $k$-algebra. Hilbert's basis theorem makes it Noetherian. It is also finitely presented: map a polynomial ring on the finite displayed generating set onto $A(S)$; the kernel is finitely generated because the source polynomial ring is Noetherian.

This is the separation in one sentence: each finite width sees only finitely many spikes, but the FI-system sees a new spike orbit forever.

The module clause, carefully scoped

The regular FI-module ${}_AA$ is free of rank one over $A$, hence finitely presented. The union

is an FI-submodule of $A$. It is not finitely generated: any finite generating set would lie in some $J_R$, while $g_{R+1}$ lies in $J\setminus J_R$.

So a finitely presented module over the constructed sub-FI-algebra has a non-finitely-generated FI-submodule.

This is a formal consequence under one literal interpretation of the source question. It is not:

• a non-free finitely presented module;
• a module over the ambient polynomial FI-algebra $V$; or
• evidence against the positive theorem for finitely generated modules over this specific width-one polynomial FI-algebra.

Those stronger readings should be treated as separate problems.

What the candidate does not prove

The construction does not settle any variant requiring the sub-FI-algebra itself to be finitely generated or finitely presented as an FI-algebra. In fact, the same factorial obstruction proves that no finite collection of its elements can generate all widths.

It also does not:

• produce a non-free finitely presented module;
• produce a non-Noetherian finitely presented module over $V$;
• classify Noetherian sub-FI-algebras of width-one polynomial FI-algebras;
• establish that the factorial-spike construction is historically new;
• supply independent reconstruction, formal verification, specialist review or editorial peer review.

Evidence, controls and limits

The immutable package contains the eight-page paper, aligned Markdown, the universal proof, exact Python replay, four hostile controls, source and citation records, a bounded novelty audit, the supplied review, five internal editorial reports, one frozen-target confirmation, licences, environment information and a complete manifest.

The checker has two finite routes:

1. a mask dynamic program implementing the structural partition criterion;
2. a generic ambient exponent-vector search through width seven.

They agree on the tested widths. Four deliberately damaged cases must also be rejected: an additive peak sequence that really factors, overlapping light sets, a factor peaking on a target light variable, and incomplete orbit enumeration.

These computations test the encoding. They do not replace the universal proof. Both routes are producer-authored and do not amount to independent reimplementation.

The supplied review found no fatal mathematical defect and requested scoped repairs. A five-role internal panel then found one blocking page-count and reproducibility inconsistency; it was repaired and one exact RC2 confirmation reported no remaining P0, P1 or P2 finding. This is producer-coordinated editorial evidence, not authenticated external FI-algebra specialist review.

The live AIMPL item returned HTTP 502 during the audit. Its wording was recovered from the current UnsolvedMath v1.6.0 record. That source-recovery boundary remains explicit.

Relationship to earlier work

Nagel and Römer provide the FI-algebra and FI-module framework used here. They prove positive Noetherianity for finitely generated modules over the standard polynomial FI-algebra and give adjacent examples and positive monomial-subalgebra regimes. Those results explain why the present module scope and the failure of FI-finite generation are load-bearing.

Draisma, Eggermont, Farooq and Meier develop positive component-counting and topological Noetherianity results for symmetric wide-matrix settings. Maraj and Nagel study shift-invariant algebras, and Morrow and Nagel develop equivariant free-resolution algorithms in positive coefficient-algebra regimes. These are relevant context, not direct prior statements of this factorial-spike separation.

The UnsolvedMath v1.6.0 record itself already contains a machine-generated factorial-spike attempt. This package is an attributed reconstruction and strengthening of that attempt, not independent discovery.

Targeted searches found no scholarly publication of this exact construction. That is bounded negative search evidence, not proof of novelty or priority.

Who should care, and why

How to inspect or reproduce the checks

Use immutable tag v0.1.0-candidate or version DOI 10.5281/zenodo.22206978, not moving main.

python3 -m unittest -v test_verify.py
python3 -O -m unittest -v test_verify.py
python3 verify.py --max-width 9 --receipt /tmp/factorial-spikes-replay.json
cmp /tmp/factorial-spikes-replay.json REPLAY_RECEIPT.json
python3 test_release_metadata.py
python3 tools/pdf_tex_preflight.py paper.pdf
shasum -a 256 -c MANIFEST.sha256

A successful run confirms the encoded finite consequences, hostile controls, PDF metadata and package integrity. It does not prove the universal theorem, establish source intention or confer peer review.

The most valuable next projects

1. Seek a finitely generated or finitely presented sub-FI-algebra separation.
2. Clarify and address a stronger non-free finitely presented module version.
3. Reconstruct the proof in a materially separate implementation and notation.
4. Formalize the universal monomial argument in a proof assistant.
5. Obtain authenticated FI-algebra specialist review and broader novelty search.

What is in the evidence package

The ZIP contains the DOI-bearing PDF and source, aligned Markdown, exact replay and receipt, four hostile controls, source and citation records, editorial reports and response, licences, environment declaration, checksums, CI workflow and complete manifest.

The frozen ZIP is 295,581 bytes with SHA-256 3783c891018422fa6db5647b8172cdaa88c4c1a22feeb35439a127aa2946c14b. The version DOI is the citation target. Any mathematical correction should be released as a versioned successor rather than silently replacing this candidate.

@misc{factorialspikesfinoetherianity2026,
  title        = {Factorial spikes separate componentwise Noetherianity from FI-Noetherianity},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.22206978},
  url          = {https://doi.org/10.5281/zenodo.22206978},
  version      = {0.1.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/factorial-spikes-fi-noetherianity/}
}