--- title: "A fixed-d Toeplitz obstruction to degree-wise coherence for FI_d" date: 2026-09-04 version: "0.1.0-candidate" doi: 10.5281/zenodo.22306413 pdf: https://github.com/ipitchford/fixed-d-toeplitz-obstruction/releases/download/v0.1.0-candidate/fixed-d-toeplitz-obstruction-v0.1.0-candidate.pdf repository: https://github.com/ipitchford/fixed-d-toeplitz-obstruction archive: https://zenodo.org/records/22306413 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # A fixed-d Toeplitz obstruction to degree-wise coherence for FI_d ## Summary The category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: $$ \text{two colors} \longrightarrow \text{Toeplitz blocks} \longrightarrow \text{unbounded first syzygies} \longrightarrow \text{no internal kernel}. $$ ## 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 $$ \pi:F\longrightarrow Q $$ 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 $$ \Phi(M(m))\cong R(-m),\qquad R=k[x_1,\ldots,x_d]. $$ 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 $$ A_r:R(-1)^{r+1}\longrightarrow R^r, \qquad (A_rw)_i=x_1w_i+x_2w_{i+1}. $$ Its first-syzygy module has rank one, generated up to a unit by $$ \bigl((-x_2)^r,\ x_1(-x_2)^{r-1},\ \ldots,\ x_1^r\bigr), $$ 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 $$ D_r= \begin{pmatrix} \alpha_1&\alpha_2&0&\cdots&0\\ 0&\alpha_1&\alpha_2&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&0\\ 0&\cdots&0&\alpha_1&\alpha_2 \end{pmatrix} :M(1)^{r+1}\to M(0)^r. $$ 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 $$ 0\longrightarrow R(-(r+1))\longrightarrow R(-1)^{r+1} \longrightarrow I_r\longrightarrow0 $$ 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 $$ \operatorname{Tor}_1^R(k,\Phi(K)) \cong\bigoplus_{r\geq1} k(-(r+1)). $$ 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`. ```sh python3 replay.py ``` For component diagnosis: ```sh 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 -v ``` The 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 1. Reconstruct the coinvariant, Tor and internal-kernel arguments under authenticated unaffiliated specialist review. 2. Determine whether another exact functor or modular method yields a positive-characteristic obstruction. 3. Formalize the proof, including the infinite direct-sum and Yoneda steps, in a proof assistant. 4. Classify useful subclasses of finite-degree-presented $\mathbf{FI}_d$- modules that remain closed under kernels and cokernels. 5. 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. ## Open directions for follow-up research - 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. ## 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:fixed-d-toeplitz-obstruction - Attempt and metric receipts: ep-attempt:fixed-d-toeplitz-obstruction-assurance-publication: published / positive; scope assurance-through-publication; target Publish the exact reviewed fixed-d FI_d theorem candidate with deterministic replay, immutable GitHub and Zenodo identities, reader-first Evidence Press page, provenance-bound media, reciprocal operating records, green hosted CI and canonical readback.; active forecast 120 minutes (80-180); Fermi components Review response, proof precision and source comparison: 1 x 20/30/45 minutes low/central/high (One minor-revision report with mandatory prior-art, attribution, assurance and portability actions.); Deterministic replay, cross-platform repair and immutable identity: 1 x 20/35/55 minutes low/central/high (One exact Python and Singular package, seven-page XeLaTeX manuscript, GitHub prerelease and Zenodo DOI.); Reader-first page, provenance-bound media and reciprocal records: 1 x 20/30/45 minutes low/central/high (One new representation-theory release using established deterministic site generators and the house fable TTS profile.); Composite site CI, merge, guarded deployment and canonical readback: 1 x 20/25/35 minutes low/central/high (One append-only Evidence Press publication on the existing zero-cost Cloudflare route.); positive-signal/closure probabilities 0.98/0.9 within 180 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 60/unknown/30/16/0/10; cycles positive/negative/inconclusive 0/0/0; falsification gates 6; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -60 minutes; ratio 0.5; inside interval false; positive-signal/target-closure Brier scores 0.0004/0.01; missing telemetry activeHumanMinutes: No instrument captured human direction or review time inside the prospective assurance-through-publication boundary.; uncachedInputTokens: The goal counter exposes aggregate task tokens but does not separate uncached input tokens.; appended measurement corrections measurement.computeMinutes -> metrics.outcome.computeMinutes: Retain the immutable 15-minute release-candidate snapshot and record the conservative 30-minute terminal total only in metrics.outcome.computeMinutes. The terminal total comprises 1,549 completed hosted job-seconds across five research and two site runs, rounded upward with observed local replay and composite-build wall time. (reason: The first public release-candidate snapshot recorded 15 compute minutes before the final tag-triggered research replay, two four-job Evidence Press CI runs and the guarded production build had completed; silently replacing that already-public observation would erase the measurement sequence.). 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: obstruction — A fixed-finite-d two-color Toeplitz family whose unbounded first-syzygy degrees obstruct closure under kernels for modules presented in finite degree. Scope: FI_d-modules over characteristic-zero fields for fixed finite d at least two; not positive characteristic, finite generation, external verification, novelty or priority. - Reusable methods: 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 paper maps colored FI_d transitions to polynomial variables using exact symmetric-group coinvariants, maps the block-diagonal natural transformation to Toeplitz matrices, converts unbounded first Tor into failure of finite-degree presentation, and uses representables plus Yoneda to rule out a different internal kernel. Remaining risks: The universal proof and cited FI_d specialization have not been reconstructed by an authenticated unaffiliated specialist.; Python and Singular check formulas and finite encodings but do not prove the universal categorical theorem.; The bounded literature search may miss an implicit or differently phrased fixed-d construction.; The proof does not extend through the same exactness step in positive characteristic.. - Human judgement gates: Keep finite generation distinct from presentation in finite degree with infinitely many uniformly bounded summands.; Check that characteristic zero is used precisely for exactness of every symmetric-group coinvariant functor.; Verify that unbounded syzygy degree belongs to the polynomial specializations Phi(K_r), not directly to K_r.; Audit the Yoneda step so failure of an ambient kernel is not mistaken for failure of an internal kernel without proof.; Keep producer replay, internal review, independent reconstruction, formal verification, specialist review, novelty and priority separate. - Next assurance action: Obtain an authenticated unaffiliated specialist reconstruction, then formalize the proof and investigate the positive-characteristic obstruction boundary. - Claim ceiling: An anonymous, AI-assisted, unrefereed fixed-d characteristic-zero theorem candidate with a complete written proof, immutable public assets, producer replay, hostile controls and internal editorial closure; not independent validation, formal verification, authenticated specialist review, editorial peer review, exhaustive novelty, historical priority or demonstrated workflow impact. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Faster or more reliable resolution and assurance of categorical coherence obstructions in AI-assisted structural reduction, exact replay, internal review and guarded candidate publication; design none; comparator No matched conventional research, specialist-review 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/3/; inherited claim: AIM item 3.1 asks whether the FI finite-degree coherence theorem persists for other categories and records the expectation that FI_d for d greater than one is probably negative.; inherited ceiling: The problem source states the question and expectation but supplies no proof of this theorem, construction, novelty or priority.; extends-result https://doi.org/10.1090/proc/13618; inherited claim: Ramos develops FI_d representation stability and records the exact characteristic-zero coinvariant specialization used here.; inherited ceiling: The cited result supplies definitions, local-noetherian context and the exact functor, not the fixed-d Toeplitz obstruction or internal-kernel theorem.; extends-result https://doi.org/10.1016/j.jalgebra.2024.11.020; inherited claim: Di, Li and Liang Example 3.6 gives a countable-direct-sum failure with bounded generator degree and unbounded relation degree for a sum norm on an N-infinity-type category.; inherited ceiling: The published antecedent supplies the broad mechanism, not this fixed-finite-d two-color Toeplitz specialization or its explicit internal Yoneda kernel closure. ## 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. ## References 1. American Institute of Mathematics. Representation stability: Coherence, AIM Problem Lists item 3.1 (2016). 2. Ramos, E. (2017). Generalized representation stability and FI_d-modules. Proceedings of the American Mathematical Society 145, 4647–4660. 3. Ramos, E. (2017). On the degree-wise coherence of FI_G-modules. New York Journal of Mathematics 23, 873–895. 4. Sam, S. V., and Snowden, A. (2017). Gröbner methods for representations of combinatorial categories. Journal of the American Mathematical Society 30, 159–203. 5. Gan, W. L., and Li, L. (2020). Castelnuovo–Mumford regularity of representations of certain product categories. Journal of Algebra 555, 246–264. 6. Di, Z., Li, L., and Liang, L. (2025). Representations of N-infinity-type combinatorial categories. Journal of Algebra 666, 47–81.