--- title: "Exact deletion interlacing for Fisk's Toeplitz-minor transform through five variables" date: 2026-09-01 version: "0.3.0-candidate" doi: 10.5281/zenodo.22233444 pdf: https://github.com/ipitchford/fisk-toeplitz-deletion-interlacing-through-five/releases/download/v0.3.0-candidate/paper.pdf repository: https://github.com/ipitchford/fisk-toeplitz-deletion-interlacing-through-five archive: https://zenodo.org/records/22233444 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # Exact deletion interlacing for Fisk's Toeplitz-minor transform through five variables ## Summary Steve Fisk asked whether a polynomial transformation built from $3\times3$ Toeplitz minors always keeps all of its roots real and negative. The complete question remains open. This anonymous, unrefereed candidate proves a finite part. Start with at most five positive numbers. Build the transformed polynomial from their rectangular Schur polynomials. It has only simple negative roots. Remove any one of the input numbers, and the smaller polynomial's roots sit strictly between the roots of the larger one. The finite cutoff matters. The paper proves the statement through five positive entries. It does not prove the all-degree conjecture and does not cover inputs on the zero boundary. ## The central theorem For a positive alphabet $\rho=(\rho_1,\ldots,\rho_n)$, repetitions allowed, define $$ F_\rho(t)=\sum_{k=0}^{n}s_{(3^k)}(\rho)t^k, $$ where $s_{(3^k)}$ is the Schur polynomial indexed by the $3\times k$ rectangle. The candidate theorem has two parts. 1. For $0\leq n\leq5$, $F_\rho$ has exactly $n$ simple negative roots. 2. For $2\leq n\leq5$, deleting any chosen entry of $\rho$ gives a polynomial that strictly interlaces $F_\rho$. The larger polynomial supplies the two outer roots. Because Schur polynomials are symmetric, an arbitrary chosen deletion can be written as $\rho=(X,y)$, with $y$ the deleted entry. The proof therefore needs one deletion calculation, not five separately labelled cases. ## How the mechanism works Put $f=F_{(X,y)}$ and $g=F_X$. Their signed Bezoutian gives a real symmetric matrix $B(f,g)$. Under the convention fixed in the paper, strict interlacing is equivalent to positive definiteness of this matrix once the degrees and leading signs are accounted for. Sylvester's criterion reduces positive definiteness to positivity of every leading principal minor. The package expands each required minor as a polynomial in the positive alphabet entries. If every coefficient is positive, then the minor is strictly positive throughout the positive orthant. For two, three and four entries, the package recomputes the minors directly. For five entries, Singular emits complete exponent-coefficient streams. The five streams contain respectively $$ 15,\quad 771,\quad 8{,}790,\quad 29{,}460,\quad 22{,}156 $$ positive terms, for a total of $61{,}192$. No sampled evaluation substitutes for those coefficient lists. ## What was checked and replayed - A fresh run regenerates every degree-five stream and byte-compares the deterministic sources, logs, compressed terms and certificate index. - A standard-library verifier checks hashes, coefficient signs, declared counts and order-specific modular evaluations. - Degrees two through four are recomputed directly with exact symbolic arithmetic. - Fixed Schur and Vandermonde spot checks audit the semantic bridge into the encoded polynomial family. - One mutation makes a coefficient negative; another keeps it positive but changes its value. Both must fail. - A fresh-archive route rejects unsafe or unexpected inventory and repeats the mathematical replay after extraction. - Public GitHub Actions passes the same release contract on a clean runner. - Independent downloads from GitHub and Zenodo reproduce the frozen archive, PDF and checksum hashes exactly. The supplied Major Revision review was answered point by point. A producer-coordinated five-role internal review and one exact-byte confirmation ended at `PASS_WITH_NOTES`, with no residual or new P0/P1 findings. Those are internal editorial controls, not external peer review. ## Where it stops and what remains open - Fisk's all-degree $3\times3$ Toeplitz-minor conjecture remains open. - Degree six is certified only for the first three leading Bezout minors. - Positive coefficient expansions are a sufficient route; their failure at a later degree would not by itself refute the parent conjecture. - Zero entries, degree drops, multiple roots and the limiting arguments needed on the closed boundary are not claimed. - The exhaustive small grid and seeded higher-degree search found no counterexample, but are reconnaissance rather than theorem evidence. - No unaffiliated rerun, independent reimplementation, formal proof, external specialist review or journal peer review has occurred. - The targeted literature and GitHub search does not establish novelty, historical priority or the absence of differently phrased work. ## Who should care, and why Researchers in real-rooted polynomials and total positivity get a concrete finite theorem and an executable all-degree bottleneck. Symmetric-function researchers get complete rectangular-Schur coefficient data rather than only numeric root plots. Computer-assisted mathematics reviewers get a small example in which the generator, verifier, semantic bridge, hostile mutations, archive inventory and public readback can be inspected separately. The result is also a useful handoff for anyone exploring characteristic- polynomial, matching-polynomial, network or total-positivity representations: such a representation could replace the rapidly growing coefficient expansion with a structural all-degree theorem. ## Where to inspect and replay Begin with the five-page PDF for the theorem and proof. In the archive, `certificates/degree5/` contains the complete term streams, generated Singular inputs, raw logs and verification receipt. `replay.py` is the main fresh reproduction route; `CLAIM_LEDGER.json` keeps the finite theorem, searches, failed routes and open parent conjecture separate. Run `./replay.sh` from a tagged checkout with the versions listed in `ENVIRONMENT.txt`. The GitHub Actions run is the public clean-checkout replay. `SHA256SUMS` binds the release archive and PDF, and the Zenodo DOI distributes the same bytes. ## Next research directions The nearest exact target is to finish orders four through six of the degree-six Bezout calculation, preferably with an independently authored streaming implementation. A positive result should be mined for a recurrence, network or sum-of-products structure rather than treated as a reason to extrapolate. In parallel, the sign convention and classical Bezout interlacing bridge should be reconstructed by an unaffiliated specialist and formalized independently. The universal problem should be reopened only with an all-degree proof, an exact positive rational counterexample, or a structural representation carrying a proved real-zero theorem. ## Paper, archive, and package map - **Paper:** the five-page PDF contains the finite theorem, proof bridge, certificate statement, assurance boundary and bibliography. - **Archive:** the ZIP contains source, exact degree-five streams, raw logs, verifiers, receipts, failed routes, reviews, licenses and a complete manifest. - **Repository:** the annotated `v0.3.0-candidate` tag fixes the green release source and public replay workflow. - **Zenodo:** DOI `10.5281/zenodo.22233444` archives the same ZIP, PDF and checksum sidecar as GitHub. - **Identity:** the candidate archive SHA-256 is `477118af1798d25dc0c8d0be264c5a8f8703852ccc1ed117a2e4d5a2a3e4e078`. - **Licensing:** original prose, data, certificates and review records are CC0 1.0; original code is MIT; cited third-party works retain their own terms. ## Searchable mathematical objects ### Finite real-rootedness through five - Exact searchable text: `For a positive alphabet rho of size n at most five, F_rho has n simple negative roots.` - LaTeX: `F_{\rho}(t)=\sum_{k=0}^{n}s_{(3^k)}(\rho)t^k` - Object type: statement - Relation to release: claimed-result - Scope: Strictly positive real alphabet entries, repetitions allowed, and 0 <= n <= 5. ### Arbitrary-deletion strict interlacing - Exact searchable text: `Deleting any chosen entry from a positive alphabet of size two through five produces a polynomial that strictly interlaces the original one.` - Object type: statement - Relation to release: claimed-result - Scope: Strictly positive alphabets with 2 <= n <= 5; symmetry reduces a chosen deletion to rho = (X,y). ### Degree-five positive Bezout certificate - Exact searchable text: `Every coefficient in each of the five leading degree-five Bezout minors is a positive integer.` - Object type: identity - Relation to release: computed-finite - Scope: Exactly 61,192 exponent-coefficient terms across the five degree-five minors, regenerated by Singular and checked by a standard-library verifier. ### Fisk all-degree conjecture - Exact searchable text: `Determine whether the 3x3 Toeplitz-minor transform preserves real negative zeros in every degree.` - Object type: statement - Relation to release: open-problem - Scope: The unrestricted parent problem, including degrees beyond five and its required limiting or boundary analysis. ## Open directions for follow-up research - Prove positivity of every deletion Bezout minor for arbitrary positive alphabet size, or find an exact positive rational counterexample. - Complete the degree-six leading-minor calculation beyond the first three orders and explain any resulting positive or negative structure. - Extend the finite positive-alphabet theorem to the zero boundary with a rigorous account of degree drops, multiple roots and collision limits. - Find a characteristic-polynomial, matching-polynomial, network or total-positivity representation that supplies an all-degree real-zero theorem. - Obtain an unaffiliated reconstruction, a separately authored verifier, proof-assistant formalization and external specialist assessment of correctness and closest prior art. ## 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:fisk-toeplitz-deletion-interlacing-through-five - Attempt and metric receipts: ep-attempt:fisk-toeplitz-deletion-interlacing-through-five-publication-completion: published / partial; scope publication-only; target Publish a reader-first Evidence Press release with provenance-bound fable audio, deterministic art and thumbnail, reciprocal method and work ledgers, green composite CI, guarded zero-cost deployment and exact canonical HTML, paper.json and media readback.; active forecast 95 minutes (55-160); Fermi components Claim-disciplined art, OG, thumbnail and fable audio: 1 x 15/25/40 minutes low/central/high (One new mathematical slug using established deterministic renderers and the house TTS profile.); Composite build, protocol seal and local accessibility checks: 1 x 15/25/40 minutes low/central/high (One reader-first release with reciprocal work and method records on the current schema.); Push, pull request, hosted CI and merge: 1 x 10/20/35 minutes low/central/high (One isolated site branch using the established four-job Evidence Press workflow.); Post-merge ledger reseal, guarded deploy and canonical readback: 1 x 15/25/45 minutes low/central/high (One zero-cost Cloudflare deployment with exact release, media, protocol and preservation checks.); positive-signal/closure probabilities 0.98/0.9 within 160 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 13/unknown/0/5/0/3; cycles positive/negative/inconclusive 0/0/0; falsification gates 0; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -82 minutes; ratio 0.14; 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 publication-only boundary.; deduplicatedModelTokens: No active fork-aware task counter exposed exact task-local model-token usage at or after the 15:09:32Z publication 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: executable-open-problem — A complete finite deletion-interlacing certificate through five positive alphabet entries and a sharply bounded all-degree research handoff. Scope: Exact real-rootedness and arbitrary-deletion strict interlacing for positive alphabets of size at most five, plus a degree-six first-three-minor boundary; not zero-boundary cases, all degrees, independent validation, novelty or priority. - Reusable methods: Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Counterexample- and proxy-first analysis (counterexample-proxy-first); Productive failure and stop receipts (productive-failure); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: discovery, assurance, publication - Semantic bridge: explicit — The package fixes the signed Bezout convention, maps F_(X,y) and F_X to a symmetric matrix, proves that positivity of all leading minors makes that matrix positive definite, and invokes the matching strict-interlacing criterion. Complete term streams bind the degree-five coefficient claim to the exact computation. Remaining risks: The finite proof and classical interlacing bridge have not been reconstructed by an unaffiliated specialist.; Positive coefficient expansions may fail at a higher degree even if the parent conjecture remains true by another mechanism.; Strict positivity on the open orthant does not by itself settle zero entries, degree drops or multiple-root limits.; The targeted literature and GitHub search may miss differently phrased, unindexed or unpublished prior work.. - Human judgement gates: Check the sign convention and Bezout-to-interlacing theorem as mathematics rather than inferring them from positive coefficient files.; Keep the finite theorem through five separate from the degree-six partial calculation and Fisk's unrestricted conjecture.; Treat the exact grid and seeded searches as reconnaissance rather than proof or counterexample evidence.; Keep internal review, public replay and archive availability separate from unaffiliated validation, peer review, novelty and priority. - Next assurance action: Obtain an unaffiliated reconstruction of the Bezout sign and interlacing bridge, then complete or refute the degree-six coefficient-positivity route in a separately authored stack. - Claim ceiling: An anonymous, AI-assisted, unrefereed computer-assisted finite theorem candidate through five positive alphabet entries with complete exact coefficient streams, hostile controls, producer replay, internal editorial closure and public immutable assets; not the all-degree or zero-boundary conjecture, independent validation, formal verification, external specialist or editorial peer review, historical priority or demonstrated workflow impact. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Faster or more reliable resolution and assurance of open real-rootedness conjectures in AI-assisted symmetric-function proof search, exact algebra, adversarial replay, internal review and guarded candidate publication; design none; comparator No matched conventional research, theorem-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://arxiv.org/abs/0808.1850; inherited claim: Fisk defines the higher Toeplitz-minor transform and asks whether its output preserves real negative zeros.; inherited ceiling: The source fixes the parent target and attribution but does not establish this finite theorem, its certificate, correctness, novelty or priority.; extends-result https://doi.org/10.1515/CRELLE.2011.063; inherited claim: Branden proves the corresponding two-by-two transform and develops adjacent real-rootedness-preserver context.; inherited ceiling: The published result supplies mathematical context, not the three-by-three finite certificate or all-degree conclusion. ## Verification status Anonymous, AI-assisted, unrefereed computer-assisted finite theorem candidate at internal PASS_WITH_NOTES with no residual or new P0/P1 findings in the final exact-byte confirmation. The finite theorem is a written Bezout-interlacing argument supported by complete positive coefficient streams through five positive alphabet entries. Symmetry covers deletion of any chosen entry. The exhaustive small grid and seeded degree-16-through-40 search are non-load-bearing reconnaissance, not evidence for the universal claim. Degree six has only its first three leading minors certified. Inputs with a zero entry, degree drops, collision limits needed for the unrestricted parent formulation, Fisk's all-degree conjecture, independent reconstruction, proof-assistant formalization, external specialist review, journal peer review, exhaustive novelty assessment and priority adjudication remain open or unassessed. ## References 1. Fisk, S. (2008). Questions about determinants and polynomials. arXiv:0808.1850. 2. Branden, P. (2011). Iterated sequences and the geometry of zeros. Journal fur die reine und angewandte Mathematik 658, 115-131. 3. Yoshida, R. (2013). On some questions of Fisk and Branden. Complex Variables and Elliptic Equations 58(7), 933-945. 4. American Institute of Mathematics. (2023). Theory and applications of total positivity: workshop report. 5. OpenTorus. (2026). Campaign: Fisk's 3x3 Toeplitz-minor conjecture, public status and computation audit.