---
title: "Factorial spikes separate componentwise Noetherianity from FI-Noetherianity"
date: 2026-08-31
version: "0.1.0-candidate"
doi: 10.5281/zenodo.22206978
pdf: https://github.com/ipitchford/factorial-spikes-fi-noetherianity/releases/download/v0.1.0-candidate/paper.pdf
repository: https://github.com/ipitchford/factorial-spikes-fi-noetherianity
archive: https://zenodo.org/records/22206978
license: CC0-1.0
status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)
---

# 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

$$
V(S)=k[x_s\mid s\in S]
$$

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

$$
g_r=x_1x_2\cdots x_{r-1}x_r^{r!}.
$$

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

$$
J_2\subsetneq J_3\subsetneq J_4\subsetneq\cdots.
$$

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

$$
g_2=x_1x_2^2,qquad
g_3=x_1x_2x_3^6,qquad
g_4=x_1x_2x_3x_4^{24}.
$$

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

$$
g_r(a)=x_1\cdots x_{r-1}x_r^{a_r}.
$$

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

$$
\sum_{j=1}^{q}(s_j-1)=R-1
\qquad\text{and}\qquad
\sum_{j=1}^{q}a_{s_j}=a_R.
$$

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

$$
\sum_j s_j!
\le (R-1)(R-1)!
<R!.
$$

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

$$
g_R\in J_R([R])\setminus J_{R-1}([R]),
$$

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

$$
n\binom{n-1}{r-1}
$$

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

$$
n(2^{n-1}-1)
$$

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

$$
J=\bigcup_{R\ge2}J_R
$$

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

| Audience | Potential use | Required caution |
|---|---|---|
| FI-algebra researchers | A compact separation between finite-component and functor-level Noetherianity | The algebra is not FI-finitely generated |
| Representation-stability researchers | A stress test for which global finiteness hypotheses positive theorems really need | The literal module corollary may be weaker than the intended question |
| Commutative algebraists | A monomial example where every component is affine but the compatible ideal system does not stabilize | Ordinary Noetherianity is not failing inside any fixed component |
| Computational reviewers | Two exact finite routes and four semantic mutations | Reimplement independently rather than importing producer predicates |
| Formalizers | A short universal proof using FI-functoriality, monomial membership and integer inequalities | Source conventions and the module interface must be formalized explicitly |
| Interested readers | A clear example of local finiteness failing to control a growing system | Candidate publication is not field consensus |

## How to inspect or reproduce the checks

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

```sh
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.




## Open directions for follow-up research

- Find a non-Noetherian sub-FI-algebra of the standard width-one polynomial FI-algebra that is finitely generated, or prove that none exists under a natural finite-presentation hypothesis.
- Construct a non-free finitely presented module over a finitely presented non-Noetherian sub-FI-algebra, if the intended module clause requires more than the regular module.
- Independently reconstruct the exact lower-width factorisation criterion and strict FI-ideal chain in a materially separate stack.
- Formalize the FI-algebra construction, monomial-ideal lemma, factorial obstruction and regular-module consequence in a proof assistant.
- Obtain authenticated FI-algebra specialist review and a broader MathSciNet or zbMATH novelty and priority assessment.

## Research process, metrics and reusable methods

This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work.

- Work ID: ep-work:factorial-spikes-fi-noetherianity
- Attempt and metric receipts: ep-attempt:factorial-spikes-fi-noetherianity-publication: published / positive; scope publication-only; target Publish a reader-first Evidence Press release with provenance-bound media, append-only operating records, two composite seals, hosted CI, guarded zero-cost deployment and exact canonical readback without broadening the literal unrestricted-algebra result or its assurance status.; active forecast 110 minutes (65-165); Fermi components Reader-first page and machine record: 1 x 15/25/35 minutes low/central/high (One eight-page FI-algebra candidate with a delicate componentwise-versus-FI and module-scope boundary.); Deterministic art, byte-bound audio, Open Graph image and thumbnail: 1 x 15/25/35 minutes low/central/high (One release using established generators but requiring a new factorial-spike visual that must not imply FI-finite generation.); Operating records and composite A/B and C/D seals: 1 x 15/25/40 minutes low/central/high (One new prospective work link and method cluster with append-only ledger reconciliation.); Hosted CI, guarded deployment and exact readback: 1 x 20/35/55 minutes low/central/high (One canonical URL requiring merged-main verification, zero-cost deployment and bounded custom-domain convergence checks.); positive-signal/closure probabilities 0.97/0.84 within 165 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 33/unknown/2/7/0/2; cycles positive/negative/inconclusive 0/0/0; falsification gates 0; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -77 minutes; ratio 0.3; inside interval false; positive-signal/target-closure Brier scores 0.0009/0.0256; missing telemetry activeHumanMinutes: No instrument captured human direction or review time at the prospective publication-only boundary.; deduplicatedModelTokens: No active fork-aware goal counter exposed exact task-local model-token usage at or after the 11:46:57Z publication-ledger boundary, so tokens are not reconstructed from conversation context.; uncachedInputTokens: The runtime does not expose an uncached-input token counter.. Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json
- Intended aims: science
- Artifact roles: research-output, evidence-assessment, method-demonstration, communication
- Decision object: counterexample — An explicit characteristic-free sub-FI-algebra with affine Noetherian fixed components and a strict infinite ascending chain of FI-ideals. Scope: The unrestricted sub-FI-algebra A generated by factorial spikes and the regular A-module consequence; not an FI-finitely-generated algebra, a non-free module, a module over V, a classification theorem, a novelty finding or external validation.
- Reusable methods: Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json
- Targeted clocks: assurance, publication, translation
- Semantic bridge: explicit — The package records the recovered AIM wording, FI convention, factorial-spike generators, exact native-width factorisation equations, monomial-ideal membership step, strict FI-ideal witnesses, component count, module interpretation and replay predicates. This exposes every transition from the source request to the separation claim. Remaining risks: The live AIMPL item returned HTTP 502 during the audit, so exact current wording was recovered through the UnsolvedMath v1.6.0 record rather than direct readback.; Both finite checker routes are producer-authored and do not independently establish the universal proof.; The literal module interpretation may be weaker than the source author's intended challenge.; The bounded novelty search cannot establish historical priority..
- Human judgement gates: Keep componentwise Noetherianity separate from Noetherianity of the FI-system.; State prominently that A is not finitely generated as an FI-algebra.; Scope the finitely presented module consequence over A and do not call it non-free or a module over V.; Treat finite replay and hostile controls as regression evidence for the encoding, not as proof of the universal theorem.; Keep the supplied review, internal editorial confirmation, authenticated specialist review, formal verification and peer review distinct.; Make no first-discovery or historical-priority claim from a bounded search.
- Next assurance action: Obtain an authenticated unaffiliated FI-algebra reconstruction of the strict-chain proof and clarify whether a finitely generated or non-free module variant is the intended stronger target.
- Claim ceiling: An anonymous, AI-assisted, unrefereed theorem candidate separating componentwise Noetherianity from FI-Noetherianity by a characteristic-free factorial-spike construction, with a carefully delimited regular-module consequence, complete written proof, public immutable assets, producer replay and internal review; not an FI-finitely-generated counterexample, a non-free module, historical-priority finding, unaffiliated validation, formal verification, authenticated external specialist review or editorial peer review.
- Aim-scoped impact evidence:
  - science: NO_IMPACT_EVIDENCE — Faster or more reliable construction and assurance of FI-algebra examples addressing open problem-list requests in AI-assisted proof reconstruction, source correspondence, adversarial replay, internal review and guarded candidate publication; design none; comparator No matched conventional research or publication workflow was registered.; estimand No effect on discovery time, active human effort, compute, correction rate, proof quality, independent-assurance time, uptake, citation or field outcomes was estimated.; no real-world effect evidence asserted
- Parent handoffs: depends-on-claim https://aimpl.org/repnstability/1/; inherited claim: The AIM problem asks for a concrete non-Noetherian sub-FI-algebra of the standard polynomial FI-algebra, asks about finitely presented modules and asks whether characteristic matters.; inherited ceiling: The source request fixes the target wording but does not establish this construction, its correctness, intended module scope, novelty or priority.; extends-result https://doi.org/10.1016/j.jalgebra.2019.06.029; inherited claim: Nagel and Römer provide the FI-algebra and FI-module conventions, positive Noetherianity for finitely generated modules over the standard polynomial FI-algebra and adjacent monomial-subalgebra results.; inherited ceiling: Their theorems supply framework and nearby positive regimes, not the present unrestricted non-finitely-generated subalgebra or a novelty finding for it.



## Verification status

Anonymous, AI-assisted, unrefereed FI-algebra theorem candidate after actioning a supplied full review, completing a five-role producer-coordinated editorial gate and passing one bounded confirmation on an exact frozen archive. Reviewer identity, specialist credentials and unaffiliated status were not authenticated. The literal unrestricted subalgebra and characteristic clauses receive a complete candidate proof; the regular-module corollary is deliberately scoped over A. The construction is not FI-finitely generated, and stronger finite-generation variants remain open.

## References

1. American Institute of Mathematics. Representation stability, section 1, item 1.2, sub-FI-algebra problem. <https://aimpl.org/repnstability/1/>
2. Nagel, U., & Römer, T. (2019). FI- and OI-modules with varying coefficients. Journal of Algebra 535, 286-322. <https://doi.org/10.1016/j.jalgebra.2019.06.029>
3. Draisma, J., Eggermont, R. H., & Farooq, A. (2022). Components of symmetric wide-matrix varieties. Journal für die reine und angewandte Mathematik 793, 143-184. <https://doi.org/10.1515/crelle-2022-0064>
4. Draisma, J., Eggermont, R. H., Farooq, A., & Meier, L. (2025). Image closures of symmetric wide-matrix varieties. Journal of Algebra 668, 190-207. <https://doi.org/10.1016/j.jalgebra.2024.12.028>
5. Maraj, A., & Nagel, U. (2023). Shift invariant algebras, Segre products and regular languages. Journal of Algebra 631, 236-266. <https://doi.org/10.1016/j.jalgebra.2023.04.016>
6. Morrow, M., & Nagel, U. (2025). Equivariant free resolutions of sequences of symmetric modules. arXiv:2507.11650. <https://arxiv.org/abs/2507.11650>