--- title: "A Degree-Independent Rhombus Criterion in Three Variables, and the Higher-Dimensional Boundary" date: 2026-08-31 version: "0.2.0-candidate" doi: 10.5281/zenodo.22211016 pdf: https://github.com/ipitchford/degree-independent-rhombus-criterion/releases/download/v0.2.0-candidate/degree-independent-rhombus-criterion-v0.2.0-candidate.pdf repository: https://github.com/ipitchford/degree-independent-rhombus-criterion archive: https://zenodo.org/records/22211016 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # A Degree-Independent Rhombus Criterion in Three Variables, and the Higher-Dimensional Boundary ## Summary Rhombus inequalities compare four neighbouring coefficients in a triangular array. A theorem of Petter Brändén says that, for a positive homogeneous polynomial in three variables, sufficiently strong rhombus inequalities force real stability. His published sufficient factor grows with the degree. This anonymous, unrefereed candidate argues that the fixed factor $3$ works in every degree. It also shows why the higher-dimensional part of the problem cannot be answered by simply reusing the same words: three natural extensions lead to three different boundaries, and one meaningful normalized formulation remains open in four, five and six variables. ## Why the problem matters Real-stable polynomials connect complex zero geometry to combinatorics, probability and discrete convexity. A degree-independent local criterion would turn a growing global stability test into a uniform family of small coefficient comparisons. The dimensional boundary matters just as much. “Rhombus log-concavity” has a canonical triangular meaning in three variables, but no single automatic meaning in higher dimension. A useful answer must state which extension is being tested before claiming either success or impossibility. ## The exact theorem and higher-dimensional boundary Let $$ P(x,y,z)=\sum_{\alpha_1+\alpha_2+\alpha_3=d} a_\alpha x^{\alpha_1}y^{\alpha_2}z^{\alpha_3}, \qquad a_\alpha>0. $$ The central candidate theorem says: if every elementary hive-rhombus quotient, with Brändén's numerator and denominator orientation, is at least $3$, then $P$ is real-stable. The number $3$ is independent of $d$ and is not claimed to be optimal. The paper then separates three higher-dimensional readings. 1. **Coordinate-face control.** For every $m\ge 4$ and every prescribed finite factor $q$, there is a positive homogeneous quadratic whose coordinate-face rhombus quotients are all at least $q$ but which is not stable. 2. **Uniform strict exchange.** One precisely defined symmetrized one-exchange multiplicative margin greater than $1$ is inconsistent on positive full-support quadratics once four distinct indices occur. 3. **Normalized $M$-concavity.** This formulation is meaningful, but finite positive families converging to the known Fano obstruction rule out an unrestricted theorem from seven variables onward. The normalized cases $m=4,5,6$ are not classified here. ## How the proof works The ternary proof writes the logarithm of each coefficient as a fixed quadratic baseline plus an ordinary hive. The baseline absorbs exactly one unit across each elementary rhombus, while the residual hive extends to a concave function on the simplex. To compare neighbouring coefficient rows of the univariate slice $P(1,1,z)$, products are separated by the parity of two indices. Each parity class has an injective midpoint parametrization, so no degree-sized multiplicity factor is introduced. The remaining penalties form two convergent theta-type sums. At $q=3$ their elementary bounds fit strictly under the coefficients in one quarter of a square: $$ C_0(3)<\frac14, \qquad C_1(3)<\frac12. $$ This yields Hutchinson's strict coefficient inequality for every interior row. The published Brändén slice-and-boundary argument then converts the univariate real-rootedness statement into ternary real stability. The higher-dimensional statements use different mechanisms: a small-eigenvalue quadratic family defeats coordinate-face control; the three pairings of four indices contradict a strict uniform exchange margin; and a finite-positive limit transfers the prior-art Fano non-stability obstruction to the normalized setting from seven variables onward. ## What was checked and replayed The release package performs deterministic producer-side checks rather than claiming that finite computation proves the universal theorem. - Python in normal and optimized modes replays the midpoint identities, theta bounds, exact rational counterexample instances, Fano distance checks and semantic mutations. - C++ exactly enumerates 880 declared $M$-convex supports before running a bounded long-double grid scan. `NO WITNESS` from that scan is explicitly non-probative. - Digest-pinned, network-disabled containers rebuild the 12-page PDF and reproduce exact fresh Python, PDF and architecture-specific C++ bytes. - Concordance checks bind load-bearing statements across TeX, Markdown, DOCX and PDF. Inventory tests reject an extra file, a corrupted byte, an altered receipt and an altered generated artifact. - Public GitHub Actions replayed the frozen commit successfully, and the ZIP, PDF and checksum sidecar downloaded from GitHub and Zenodo match local bytes. ## Evidence and assurance boundary The written manuscript is the evidence for the universal mathematical claims. The finite programs audit identities, indexing, representative exact instances and bounded searches. They do not replace the proof. The supplied review was actioned point by point. A producer-coordinated five-role review and bounded confirmation reached `PASS_WITH_NOTES`. Those are internal editorial records, not authenticated unaffiliated specialist review. Public availability, deterministic replay, independent rerun, independent reimplementation, formal verification, specialist review, editorial peer review, novelty and priority remain distinct assurance dimensions. ## Limitations and what remains open - The normalized $M$-concavity cases with four, five and six variables remain open. - The constant $3$ is sufficient in the candidate proof but is not claimed optimal. - The three higher-dimensional formulations do not exhaust every possible quantitative stability condition. - The Fano non-stability obstruction is prior work; the release's scoped step is the finite-positive family and limit corollary. - No authenticated unaffiliated reconstruction, proof-assistant formalization, external specialist review, journal peer review or historical-priority adjudication is attached. - A bounded novelty search cannot establish that the result is new or first. ## Who should care The release is aimed at researchers in stable and Lorentzian polynomials, matroid half-plane properties, hives, tropical and discrete convexity, and negative-dependence theory. It may also be useful to reviewers interested in a compact example of how one ambiguous higher-dimensional question can be split into separately falsifiable formulations. ## Where to inspect and replay Start with the PDF for the complete arguments. In the archive, `SOURCE_BRIDGE.md` maps every imported theorem to its exact use, `CLAIMS.json` separates the five claim types, and `OPEN_PROBLEMS.md` gives an executable handoff for the unresolved normalized dimensions. Run `run_all.sh` from the repository root with Docker to reproduce the digest-pinned build and stored receipts. The GitHub Actions run linked in the assurance panel is the public clean-checkout replay. `MANIFEST.sha256` is the complete 59-file package inventory, while the release-level `SHA256SUMS` binds the downloadable ZIP and PDF. ## Next work The first mathematical priority is an unaffiliated reconstruction of the parity pairing and stability bridge. In parallel, the normalized $m=4,5,6$ cases can be attacked through the support enumeration and coefficient-search interfaces in `OPEN_PROBLEMS.md`. A proof-assistant development should separate the elementary theta estimates from the imported stability-preserving steps, making the source bridge mechanically explicit. ## Paper, archive, and package map - **Paper:** the canonical 12-page PDF contains the complete proofs and bibliography. - **Archive:** the ZIP contains source, accessible derivatives, structured claims, source maps, review records, verification programs, receipts, licenses and the complete manifest. - **Repository:** the annotated tag fixes the reviewed source and public CI workflow. - **Zenodo:** the version DOI archives the same ZIP, PDF and checksum sidecar. - **Licensing:** original prose and data are CC0-1.0; original code, tests and workflows are MIT; third-party works are cited but not redistributed. ## Open directions for follow-up research - Classify the normalized M-concavity formulation for m=4,5,6, beginning with the executable support and coefficient targets in OPEN_PROBLEMS.md. - Determine the optimal absolute ternary rhombus factor, or prove that 3 is sharp. - Find a non-vacuous higher-dimensional quantitative condition that is stronger than coordinate-face control but does not impose the inconsistent strict exchange axiom. - Independently reconstruct the parity-pairing proof and all higher-dimensional obstruction families in a materially separate stack. - Formalize the ternary stability bridge, theta estimates and symbolic obstruction arguments in a proof assistant. - Obtain an unaffiliated specialist assessment of correctness, closest prior art and historical priority. ## 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:degree-independent-rhombus-criterion - Attempt and metric receipts: ep-attempt:degree-independent-rhombus-criterion-publication: published / partial; 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 candidate's dimensional or assurance boundaries.; active forecast 95 minutes (60-145); Fermi components Reader-first page and machine record: 1 x 15/25/35 minutes low/central/high (One 12-page candidate with four distinct dimensional formulations and a load-bearing open m=4,5,6 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 claim-specific triangular-rhombus visual that does not imply all-dimensional closure.); Operating records and composite A/B and C/D seals: 1 x 15/20/30 minutes low/central/high (One new prospective work link and method-cluster assignment with append-only ledger reconciliation.); Hosted CI, guarded deployment and exact readback: 1 x 15/25/45 minutes low/central/high (One canonical URL requiring merged-main verification, zero-cost deployment and bounded custom-domain convergence checks.); positive-signal/closure probabilities 0.97/0.86 within 145 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 30/unknown/0/7/0/0; cycles positive/negative/inconclusive 0/0/0; falsification gates 0; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -65 minutes; ratio 0.32; inside interval false; positive-signal/target-closure Brier scores 0.0009/0.0196; 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 16:10:16Z publication-ledger boundary, so tokens are not reconstructed from conversation context.; uncachedInputTokens: The runtime does not expose an uncached-input token counter.; appended measurement corrections release-candidate resource snapshot, terminal resource outcome and assurance timing: Retain computeMinutes 0, reworkMinutes 0 and the pending-canonical clause in the immutable release-candidate measurement and assurance snapshot, and retain the same zero/zero totals in the terminal metrics outcome for internal consistency. Record the later observed three-compute-minute and five-rework-minute supplemental totals only in this appended correction, while the appended published status, public-release milestone and canonical evidence references establish the later availability state. The mathematical claim and assurance dimensions are unchanged. (reason: The first canonical deployment made the 16:10 release-candidate snapshot public with zero compute minutes, zero rework minutes and canonical Evidence Press publication still pending. The terminal closeout subsequently observed three compute minutes and five rework minutes and established canonical availability. Replacing the already-public provisional fields would erase the publication-time observation.). 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: other — A degree-independent ternary sufficiency theorem candidate paired with explicit higher-dimensional counterexample, impossibility and open-boundary objects. Scope: The factor-3 positive ternary criterion in every degree, failure of finite coordinate-face control for m at least 4, inconsistency of one strict full-support quadratic exchange rule, and a normalized Fano-derived obstruction from m=7; not the normalized m=4,5,6 cases, the optimal ternary constant, every higher-dimensional quantitative criterion, novelty, priority or external validation. - Reusable methods: Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Counterexample- and proxy-first analysis (counterexample-proxy-first); 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 fixes Branden's rhombus orientation, normalizes by the quadratic baseline, records the parity-pairing bijection and theta bounds, states the stability bridge, and defines each higher-dimensional formulation before proving its distinct boundary. CLAIMS.json and OPEN_PROBLEMS.md keep written universal claims, finite replay and unresolved dimensions separate. Remaining risks: The universal ternary proof and Fano limit bridge have only producer-coordinated review, not unaffiliated reconstruction.; The coordinate-face, strict-exchange and normalized formulations do not exhaust all meaningful higher-dimensional quantitative criteria.; The normalized m=4,5,6 cases remain open and a bounded m=4 cubic scan is non-probative.; The targeted novelty search may miss differently phrased, unindexed or informal prior work.. - Human judgement gates: Check the parity pairing, theta-series domination and imported one-variable stability bridge as mathematics rather than inferring the theorem from finite replay.; Keep the three higher-dimensional formulations separate and do not describe their obstructions as one complete impossibility theorem.; Keep the prior-art Fano obstruction separate from the finite-positive limiting construction used here.; Keep the unresolved normalized dimensions four through six prominent in summaries and media.; Treat internal review, deterministic replay and public availability as distinct from unaffiliated validation, peer review, novelty and priority. - Next assurance action: Obtain an authenticated unaffiliated reconstruction of the parity-pairing proof and symbolic families, then have a stable-polynomial specialist assess the source bridge and normalized m=4,5,6 programme. - Claim ceiling: An anonymous, AI-assisted, unrefereed factor-3 ternary theorem candidate with explicit higher-dimensional coordinate-face, strict-exchange and Fano-derived boundaries, written proofs, public immutable assets, producer replay and internal PASS_WITH_NOTES; not a resolution of normalized dimensions four through six, a complete all-dimensional criterion, 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 resolution and assurance of open questions in stable-polynomial and discrete-convexity theory in AI-assisted proof construction, symbolic obstruction search, 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://web.archive.org/web/20240208020313id_/http://aimpl.org/totalpos/1/; inherited claim: AIM Problem 1.1 asks whether the degree dependence in Branden's ternary sufficient rhombus condition can be removed and asks what happens for more than three variables.; inherited ceiling: The archived problem fixes the target and attribution but does not establish this candidate's theorem, definitions, obstruction families, correctness, novelty or priority.; extends-result https://doi.org/10.1137/090758738; inherited claim: Branden's Theorem 14 supplies a degree-dependent sufficient ternary factor and the hive-to-univariate stability architecture used by the candidate.; inherited ceiling: The published theorem supplies the predecessor and bridge, not the factor-3 improvement or any higher-dimensional conclusion in this release. ## Verification status Anonymous, AI-assisted and unrefereed partial mathematics candidate at internal PASS_WITH_NOTES. The degree-independent ternary theorem and higher-dimensional families are written arguments; the finite scripts audit identities, indexing and declared bounded searches but do not prove the universal statements. The Fano non-stability obstruction is prior work. The normalized dimensions four through six remain open, the constant 3 is not claimed optimal, and the bounded novelty search does not establish priority. ## References 1. American Institute of Mathematics. Theory and applications of total positivity, Problem 1.1. 2. Branden, P. (2010). Discrete concavity and the half-plane property. SIAM Journal on Discrete Mathematics 24(3), 921-933. 3. Branden, P. (2007). Polynomials with the half-plane property and matroid theory. Advances in Mathematics 216(1), 302-320. 4. Hutchinson, J. I. (1923). On a remarkable class of entire functions. Transactions of the American Mathematical Society 25(3), 325-332. 5. Borcea, J., & Branden, P. (2009). The Lee-Yang and Polya-Schur Programs I. Inventiones Mathematicae 177(3), 541-569. 6. Branden, P., & Huh, J. (2020). Lorentzian polynomials. Annals of Mathematics 192(3), 821-891. 7. Kummer, M., & Sawall, D. (2025). Three results related to the half-plane property of matroids. Algebraic Combinatorics 8(1), 1-15.