Press release · 4 September 2026 · version 0.1.0-candidate
A fixed-d Toeplitz obstruction to degree-wise coherence for FI_d
For every fixed finite d at least two in characteristic zero, a two-color Toeplitz family gives a morphism of finite-degree-presented FI_d-modules with no kernel inside that subcategory.
Summary
The category $\mathbf{FI}_d$ records injections of finite sets together with a choice of one of $d$ colors for every newly added point. A natural subcategory consists of modules that can be described using generators and relations whose object degrees have a common finite bound—even when there are infinitely many of them.
For ordinary $\mathbf{FI}=\mathbf{FI}_1$, that subcategory is abelian: kernels and cokernels stay inside it. An AIM problem asks whether the same is true for other combinatorial categories and records the expectation that it is probably false for $\mathbf{FI}_d$ when $d>1$.
This anonymous, unrefereed candidate proves the expected negative answer over every characteristic-zero field, for every fixed finite $d\geq2$. It builds one morphism between well-presented modules whose ambient kernel is not presented in finite degree. A separate Yoneda argument proves that the subcategory cannot repair the failure by choosing some different kernel object.
The mechanism has a compact picture:
Summary for specialists
Let $k$ be a field of characteristic zero and fix finite $d\geq2$. Write $\mathcal A_d$ for the full subcategory of $\mathbf{FI}_d$-modules presented in finite degree. The theorem candidate constructs a morphism
with $F,Q\in\mathcal A_d$ that has no kernel in $\mathcal A_d$.
For the principal free module $M(m)$, exact symmetric-group coinvariants give
The paper also proves the relative-free extension $\Phi(M(W))\cong R(-m)\otimes_k W_{\mathfrak S_m}$. Thus any finite-degree presentation forces $\operatorname{Tor}_1^R(k,\Phi(V))$ to be supported in a bounded set of internal degrees.
Using colors one and two, the $r$th block specializes to
Its first-syzygy module has rank one, generated up to a unit by
in internal degree $r+1$. The direct sum of all block images therefore has first Tor in arbitrarily high degrees. This excludes the ambient kernel from $\mathcal A_d$. Principal representables detect pointwise nonzero elements; Yoneda makes any putative internal kernel ambient-monic, while every finite block factors through it. The putative kernel is forced to equal the excluded ambient one.
Technical account
For a color $c$, precomposition by the one-point colored morphism gives $\alpha_c:M(1)\to M(0)$. Under $\Phi$, this becomes multiplication by $x_c$. Form the bidiagonal natural transformation
Set $D=\bigoplus_{r\geq1}D_r$, with source $E$ free in degree one and target $F$ free in degree zero. Let $K=\operatorname{im}D$ and $Q=\operatorname{coker}D$. Then $Q$ has degree-zero generators and degree-one relations, so both $F$ and $Q$ lie in $\mathcal A_d$. The ambient kernel of $F\to Q$ is $K$, generated in degree at most one.
Put $I_r=\operatorname{im}A_r$. Coprimality of $x_1$ and $x_2$ turns the row relations $x_1w_i+x_2w_{i+1}=0$ into a divisibility induction, proving the rank-one kernel formula. Hence
is minimal and $\operatorname{Tor}_1^R(k,I_r)\cong k(-(r+1))$. Since $d$ is fixed and finite, the finite-rank Koszul resolution of $k$ commutes with direct sums, giving
That is incompatible with a finite-degree presentation of $K$.
The categorical finish is essential. If $L\to F$ were a kernel internal to $\mathcal A_d$, maps from the representables $M(n)$ would show that $L\to F$ is pointwise injective, so $L\subseteq K$. Each finite block $K_r$ is finitely generated and, by local noetherianity, lies in $\mathcal A_d$; kernel universality forces every $K_r$ into $L$. Therefore $K\subseteq L$, a contradiction.
Evidence and assurance
The mathematical proof is carried by the seven-page manuscript. The package adds two exact but producer-controlled implementations: Python checks the displayed Toeplitz identity through $r=128$, and Singular computes the syzygy module independently through $r=16$. A second Python program enumerates 45 small orbit cases and 1,899 transition cases to test the $\mathbf{FI}_d$-to-polynomial encoding.
Normal and optimized Python runs execute the same explicit conditions—no verifier uses an assert that optimization could erase. Hostile controls alter sign, column order, degree shift, or color. The replay harness also rejects a known Singular hazard: an error diagnostic accompanied by exit status zero.
The final PDF was built twice to identical bytes, checked for citation-key equality, embedded subset fonts, extractable markers, a clean TeX log, no JavaScript or encryption, and inspected across all seven rendered pages. The 53-file manifest, public Ubuntu CI, GitHub release and Zenodo record expose the same release object.
The supplied review required a direct comparison with the closest published antecedent, corrected attribution, disaggregated assurance, assigned human publication accountability, and a portable build route. All were actioned; one bounded confirmation found no new critical issue. The supplied review's identity and independence are not authenticated, so it is not counted as external specialist review or an independent rerun.
Limitations and assurance boundary
Characteristic zero is load-bearing: coinvariants under finite symmetric groups are exact there. The paper makes no positive-characteristic conclusion. The example uses infinitely many free summands of uniformly bounded object degree; it does not give a finitely generated counterexample and does not conflict with local noetherianity.
Python and Singular check formulas and finite encodings, not the universal proof. Both remain within one producer-coordinated workflow. Independent reconstruction, formal verification, authenticated specialist review and journal peer review are not assessed. Linux CI demonstrates portability but the toolchain is not hermetic or fully version-locked, so environment reproducibility is partial.
A bounded literature search found no exact fixed-finite-$d$ Toeplitz proof, but that is not a novelty or priority determination. The release makes no claim to being first or previously unknown.
Relationship to earlier work
Ramos develops the $\mathbf{FI}_d$ framework and the exact characteristic-zero specialization used here, building on Sam and Snowden's Gröbner-category and local-noetherianity theory. Ramos separately proves that finite-degree-presented $\mathbf{FI}_G$-modules form an abelian category. Gan and Li prove positive results for suitable finite product categories and explicitly leave the $\mathbf{FI}_d$ case outside their method.
The closest published antecedent is Di, Li and Liang, Journal of Algebra 666 (2025), Example 3.6. It already uses a countable direct sum to combine bounded generator degree with unbounded relation degree under a sum norm. Its category has countably many object coordinates and prime-indexed supports. This candidate instead fixes finite $d$, uses two colors, obtains explicit finite-variable Toeplitz matrices from exact symmetric-group coinvariants, and closes the internal-kernel question with Yoneda. The contribution is this fixed-$d$ realization and proof, not priority for the broad mechanism.
Who should care, and why
| Audience | Potential use | Required caution |
|---|---|---|
| Representation-stability researchers | A direct answer to the fixed-$d$ characteristic-zero coherence question | Positive characteristic remains open |
| Category theorists | A concrete example where representables expose failure of an internal kernel | Ambient non-closure alone would not suffice without the Yoneda step |
| Commutative algebraists | A transparent direct sum of minimal Toeplitz resolutions with unbounded first Tor | The polynomial functor is used only as a one-way obstruction |
| Computational reviewers | Small exact Python and Singular checks with hostile controls | Both implementations are producer-controlled |
| Formalizers | A short dependency chain from coinvariants to Tor and Yoneda | Cited structural results and infinite direct sums must be formalized carefully |
| Interested readers | A vivid example of uniformly simple pieces producing a globally unbounded obstruction | Candidate publication is not field consensus |
Why the problem matters
Representation stability works by organizing sequences of symmetric-group representations as modules over combinatorial categories. Good finiteness properties make kernels and cokernels controllable, allowing homological arguments to stay inside a manageable class.
This example pinpoints a boundary. Fixed-$d$ local noetherianity controls each finitely generated block, yet presentation in finite degree permits countably many bounded-degree summands. The Toeplitz family makes the hidden relation degree grow with the block index. It shows that local control and degree-wise coherence are genuinely different phenomena and supplies a simple test pattern for other categories with polynomial specializations.
How to inspect or reproduce the checks
Use immutable tag v0.1.0-candidate or version DOI 10.5281/zenodo.22306413, not moving main.
python3 replay.pyFor component diagnosis:
python3 verification/verify_toeplitz.py --max-r 128
python3 -O verification/verify_toeplitz.py --max-r 128
Singular -q verification/verify_toeplitz.sing
python3 verification/verify_fid_coinvariants.py --max-n 5 --max-d 3
python3 -O verification/verify_fid_coinvariants.py --max-n 5 --max-d 3
python3 -m unittest discover -s tests -vThe composite command also rebuilds and preflights the manuscript and verifies the manifest. A successful run checks package integrity and encoded exact consequences; it does not independently prove the universal theorem.
The most valuable next projects
- Reconstruct the coinvariant, Tor and internal-kernel arguments under authenticated unaffiliated specialist review.
- Determine whether another exact functor or modular method yields a positive-characteristic obstruction.
- Formalize the proof, including the infinite direct-sum and Yoneda steps, in a proof assistant.
- Classify useful subclasses of finite-degree-presented $\mathbf{FI}_d$- modules that remain closed under kernels and cokernels.
- Conduct a broader specialist novelty and priority assessment centered on the relation between this construction and Di--Li--Liang Example 3.6.
What is in the evidence package
The ZIP contains the DOI-bearing PDF and LaTeX source, accessible Markdown, claim and assurance maps, Python and Singular checks, hostile controls, replay tests and receipts, source/citation/novelty audits, the full supplied review and response, producer-coordinated internal reports, one bounded confirmation, environment and licence declarations, and a complete deterministic manifest.
The frozen PDF has SHA-256 21deed75f975e38a6b6cd87fcd1a59d90bf40a4d9674761ce938d15ff9a0e0fa. The frozen ZIP has SHA-256 e915cebd8219cd9b04e6a603b142153b113c82c8675fa54f0962b26e9fac414b. The version DOI is the citation target. Any mathematical correction should be released as a versioned successor rather than silently replacing this record.
Media
The audio briefing is provided in the header above. Download the MP3 briefing · read the transcript.
Open directions for follow-up research
Also available in machine-readable form for research agents and follow-up projects.
- Determine whether an analogue of the fixed-d theorem holds in positive characteristic, where symmetric-group coinvariants are not exact.
- Obtain an authenticated unaffiliated reconstruction of the relative-free specialization, Tor obstruction and Yoneda kernel argument.
- Formalize the categorical and commutative-algebra proof in a proof assistant.
- Classify which useful subclasses of finite-degree-presented FI_d-modules remain closed under kernels and cokernels.
- Obtain external specialist review and a broader historical novelty and priority assessment, especially around the Di--Li--Liang mechanism.
Verification status
Anonymous, AI-assisted, unrefereed representation-theory theorem candidate after actioning a supplied Minor Revision review and passing producer-side replay, internal editorial review and one frozen-artifact confirmation. The result is limited to characteristic-zero fields and fixed finite d at least two. The close Di--Li--Liang antecedent is stated directly; novelty is partial and priority is unassessed.
Cite
BibTeX
@misc{fixeddtoeplitzobstruction2026,
title = {A fixed-d Toeplitz obstruction to degree-wise coherence for FI_d},
author = {Anonymous},
year = {2026},
doi = {10.5281/zenodo.22306413},
url = {https://doi.org/10.5281/zenodo.22306413},
version = {0.1.0-candidate},
howpublished = {Zenodo},
note = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/fixed-d-toeplitz-obstruction/}
}Also: cite.bib · paper.json · this page as Markdown