Press release · 1 September 2026 · version 0.3.0-candidate
Exact deletion interlacing for Fisk's Toeplitz-minor transform through five variables
Exact positive-coefficient Bezout certificates prove real negative roots and strict arbitrary-deletion interlacing for Fisk's 3x3 Toeplitz-minor transform through five positive alphabet entries; the all-degree and zero-boundary problems remain open.
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
where $s_{(3^k)}$ is the Schur polynomial indexed by the $3\times k$ rectangle.
The candidate theorem has two parts.
- For $0\leq n\leq5$, $F_\rho$ has exactly $n$ simple negative roots.
- 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
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-candidatetag fixes the green release source and public replay workflow. - Zenodo: DOI
10.5281/zenodo.22233444archives 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
Exact text is provided alongside typeset mathematics so people and automated research tools can find and compare the objects without inferring a stronger assurance state.
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.
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.
- 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.
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.
Cite
BibTeX
@misc{fisktoeplitzdeletioninterlacingthroughfive2026,
title = {Exact deletion interlacing for Fisk's Toeplitz-minor transform through five variables},
author = {Anonymous},
year = {2026},
doi = {10.5281/zenodo.22233444},
url = {https://doi.org/10.5281/zenodo.22233444},
version = {0.3.0-candidate},
howpublished = {Zenodo},
note = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/fisk-toeplitz-deletion-interlacing-through-five/}
}Also: cite.bib · paper.json · this page as Markdown