--- title: "The sharp exponent for ordinary Hadamard powers of quartics with positive coefficients" date: 2026-09-05 version: "0.1.0-candidate" doi: 10.5281/zenodo.22346503 pdf: https://github.com/ipitchford/sharp-quartic-hadamard-powers/releases/download/v0.1.0-candidate/sharp-quartic-hadamard-powers-0.1.0-candidate.pdf repository: https://github.com/ipitchford/sharp-quartic-hadamard-powers archive: https://zenodo.org/records/22346503 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # The sharp exponent for ordinary Hadamard powers of quartics with positive coefficients ## Summary Take a degree-four polynomial whose coefficients are all positive and whose four roots are real. Now raise every coefficient to the same positive power, leaving the powers of the variable unchanged. Which exponents guarantee that the roots stay real, for every such polynomial? This unrefereed candidate gives a sharp answer. Exponent one works because it changes nothing. Immediately above one, some polynomials lose real roots. Universal preservation returns at a threshold of about **1.147720381237014**, and every larger exponent works. Above the threshold all four output roots are distinct. At the threshold, a repeated output root is possible only when all four input roots coincide. The paper supplies a complete proposed proof, not a numerical search promoted to a universal result. Exact code checks selected algebra and the threshold's decimal enclosure. External mathematical reconstruction and specialist review remain to be obtained. ## Summary for specialists Let $\mathcal R_4$ contain degree-exactly-four polynomials $f(x)=\sum_{k=0}^4 a_kx^k$ with $a_k>0$ and only real zeros, including multiplicities. Define ordinary coefficient powers by $T_pf(x)=\sum_{k=0}^4 a_k^p x^k$. The candidate proves $$ \{p>0:T_p(\mathcal R_4)\subseteq\mathcal R_4\} =\{1\}\cup[p_*,\infty), $$ where $p_*>1$ is the unique root above one of $$6^p-2\cdot4^p+2=0.$$ For $p>p_*$ every output has four distinct negative zeros. At $p=p_*$, multiple output zeros occur exactly for $f(x)=c(x+r)^4$, $c,r>0$; the normalized output factors as $$ (x+1)^2\bigl(x^2+(4^{p_*}-2)x+1\bigr). $$ The statement uses ordinary, not binomial-normalized, coefficients. It does not classify arbitrary degrees, zero-coefficient boundary inputs or general coefficient functions. The retained secondary AIM identifier is context; the originating page was not authenticated in the current audit. ## Technical account The difficult step is showing that a symmetric example controls an unsymmetrical, noncompact family. Reciprocal symmetry is **not** assumed of the input. First, Newton inequalities provide a deliberately nonsharp successful tail. A two-sided root perturbation at a universally preserving exponent yields a linear functional annihilating the input's repeated-root factor times a specified polynomial space. This restricts which multiplicities can occur when output roots collide. Next, strict lower-degree preservation excludes failures whose input roots separate into widely different scales. A last-failing-exponent argument then forces any hypothetical failure above the proposed threshold to produce a compact boundary input at a universal preserver. It does not assume that the preservation set is connected or monotone. For quartics, the boundary functional excludes all but multiplicity patterns $(4)$ and $(2,2)$. Those remaining inputs become reciprocal after positive rescaling. A reciprocal quartic reduces to a quadratic in $x+x^{-1}$, where two explicit inequalities yield the sharp threshold. Closedness is invoked only after preservation above the threshold has been established; a separate boundary argument proves the equality classification. ## Evidence, assurance and limitations The evidence is the complete written proof, a dependency map, exact auxiliary algebra, rational threshold bounds, package-integrity controls, source audits and a response to the supplied AI-assisted review. Publication checks include fresh-extraction replay, optimized-mode refusal, PDF inspection and Linux CI. Their dated receipts identify the exact scope and bytes checked. The algebra checker does not prove the scale-escape or compactness lemmas. Deliberately corrupted expressions are local controls, not whole-proof mutation tests. The assertion-based algebra checker refuses optimized Python; the separate package verifier uses explicit errors and continues to enforce hashes. The five-role editorial process is producer-coordinated internal AI review. It is not unaffiliated reproduction, formal verification, authenticated specialist review or journal peer review. Some later full texts and the original AIM page remained inaccessible. No absolute novelty, historical priority, four-star grade or measured workflow impact is claimed. ## Relationship to earlier work Wang and Zhang's 2013 paper already establishes cubic preservation, gives a quartic counterexample, and supplies a degree-dependent sufficient tail. Białas and Białas-Cież's 2017 comment concerns Hurwitz stability; the 2024 paper by Białas, Białas-Cież and Kudra also exhibits quartic failure at exponent 1.147. These are prior results, not new discoveries in this release. The proposed contribution is the exact universal quartic endpoint and its equality classification. Finite-free multiplicative convolution uses a different binomial normalization. Kudra's 2026 idealizer preprint concerns fixed linear multipliers preserving the left half-plane. Neither comparison by itself establishes or refutes this nonlinear real-rootedness theorem, and inaccessible subsidiary results remain a novelty-audit limitation. ## Who should care, and why | Audience | Potential use | Required caution | |---|---|---| | Researchers in real-rootedness and total positivity | A sharp low-degree test case for nonlinear coefficient preservers | Reconstruct the written universal argument and audit prior art | | Analysts studying boundary and extremal arguments | A multiple-root functional and separated-scale reduction | Higher-degree hypotheses and boundary patterns need fresh proofs | | Scientific-software reviewers | A compact exact replay and a clear code-to-claim map | Hashes and local controls do not certify the whole theorem | | Interested readers | An example where making an exponent slightly larger destroys a property before it returns | The guarantee concerns every admissible quartic, not every individual polynomial's failure pattern | ## Why the problem matters Real-rootedness links polynomial coefficients to strong structural inequalities. Ordinary coefficient powers look simple but need not preserve that structure for every real exponent. An exact endpoint identifies where the universal guarantee begins, and the equality case identifies its sharp boundary. The significance is a focused mathematical classification, not a claim of field-wide impact or an all-degree solution. ## How to inspect or reproduce the recorded checks Download the immutable evidence ZIP and its outer checksum. Verify the ZIP's SHA-256, extract it into a fresh directory, and follow `README.md`. The pinned Python dependencies are SymPy and mpmath. Run `verify_package.py --root . --replay`; the expected receipt is `PASS` for package integrity and scoped producer replay. Read `EVIDENCE_MAP.md` before interpreting the output. The distinct checks of the analytical argument are in the manuscript and review reports, not hidden inside that status word. Rebuilding the PDF is optional for reading the proof and requires Pandoc and pdfLaTeX; the archived PDF has its own exact hash. ## The most valuable next projects 1. Reconstruct the compactness and quartic boundary classification outside the producer workflow, and publish a scoped review or correction. 2. Complete the inaccessible-source and broader citation audit, including direct reconciliation with the originating AIM question. 3. Investigate a higher-degree critical-exponent theory. Degree five needs a threshold-relative scale argument and additional multiplicity cases; the present induction does not directly apply. 4. Formalize the analytical proof if the theorem survives specialist scrutiny. The stretch assessment identifies the higher-degree programme as potentially substantial, but it supplies neither a higher-degree theorem nor a credible six-hour four-star closure estimate. It therefore does not enlarge this release's claim. ## What is in the evidence package The package contains the manuscript PDF, accessible Markdown and TeX source; the exact algebra checker, integrity verifier and negative controls; pinned requirements and Linux CI; structured claims and a machine entry point; source, novelty, licensing and provenance records; the supplied-review response; frozen internal editorial reports; a stretch assessment; and a complete file manifest. Outer ZIP checksums and fresh-extraction replay receipts are supplied as explicit sidecars. Original prose and records are CC0; original executable code is MIT. Cited third-party material and the externally supplied review are not relicensed or bundled. ## Open directions for follow-up research - Obtain unaffiliated reconstruction of the scale-escape, last-failure and boundary arguments. - Complete inaccessible-source and citation-graph comparison, and authenticate the original AIM correspondence. - Develop a threshold-relative higher-degree escape theorem and classify new degree-five boundary patterns. - Investigate proof-assistant formalization without conflating auxiliary algebra with the complete proof. ## 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:sharp-quartic-hadamard-powers - Attempt and metric receipts: ep-attempt:sharp-quartic-hadamard-powers-assurance-publication: published / positive; scope assurance-through-publication; target A review-repaired, anonymous unrefereed quartic theorem candidate with immutable GitHub and Zenodo assets and a complete canonical Evidence Press release, or a preserved exact blocker.; active forecast 150 minutes (120-210); Fermi components Review repair, source audit and bounded stretch assessment: 1 x 30/40/60 minutes low/central/high (No supplied proof gap; literature and scope require current checks.); Five-role internal editorial gate and reproducible research package: 1 x 35/45/60 minutes low/central/high (One frozen target and deterministic controls, not whole-proof formal verification.); Immutable identities, reader-first page and provenance-bound media: 1 x 30/40/55 minutes low/central/high (Existing publication tooling and configured services, subject to live authentication.); Composite seals, CI, two deployment cycles and readback: 1 x 25/25/35 minutes low/central/high (Normal new-slug A/B then C/D release procedure.); positive-signal/closure probabilities 0.95/0.85 within 240 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 19/unknown/unknown/21/36/1; cycles positive/negative/inconclusive 1/1/0; falsification gates 2; architectures tested/rejected 2/1; result target-closed; target reached true; forecast error -131 minutes; ratio 0.127; inside interval false; positive-signal/target-closure Brier scores 0.0025/0.0225; missing telemetry activeHumanMinutes: No task-local human-work timer was available.; computeMinutes: Substantive compute was not measured separately from ordinary replay, build and publication processes; no retrospective estimate was made.; deduplicatedModelTokens: No supported fork-aware task-local token counter was available; rollout token events were not summed.; uncachedInputTokens: No supported task-local uncached-input counter was available.. 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: identified-set — Exact universal exponent set supported by a written boundary proof and scoped auxiliary arithmetic. Scope: Strictly positive ordinary monomial coefficients, exact degree four, real positive exponents and allowed repeated input roots. - Reusable methods: Structural compression (structural-compression); Exact regime stitching (regime-stitching); 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 — Tail and strict lower degrees exclude scale escape; last failure yields a universal-preserver boundary; its annihilator restricts multiplicities; reciprocal inequalities close the threshold and equality cases. Local code checks only selected identities and interval signs. Remaining risks: Written universal argument is not formally verified or externally reconstructed.; Prior-art and original AIM access remain incomplete.; No all-degree extension or numerical conditioning margin is established.. - Human judgement gates: Reconstruct the universal quantifier and noncompact boundary argument.; Check ordinary versus normalized powers and real-rootedness versus Hurwitz stability.; Preserve prior attribution, anonymous authorship and third-party rights.; Do not treat replay, a DOI or coordinated model agreement as independent validation. - Next assurance action: Obtain unaffiliated mathematical reconstruction and fuller prior-art reconciliation before stronger assurance claims. - Claim ceiling: Unrefereed quartic theorem candidate with exact auxiliary checks; no external validation, formal proof, full AIM solution, all-degree classification, historical priority, four-star grade or impact established. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — More inspectable coefficient-preserver arguments in Producer-coordinated mathematical research and publication; design none; comparator No matched conventional workflow comparator.; estimand No causal discovery-time, reliability, uptake or field-impact effect estimated.; no real-world effect evidence asserted - Parent handoffs: extends-result https://doi.org/10.1016/j.laa.2013.08.012; inherited claim: Cubic preservation, quartic failure and a degree-dependent sufficient tail are prior work.; inherited ceiling: The current note proves its own lower-degree and nonsharp-tail inputs; prior work does not establish this candidate's sharp endpoint or historical priority. ## Verification status Anonymous, AI-assisted, unrefereed theorem candidate. Five producer-coordinated internal roles recommend Accept with no P0/P1 scientific findings. This is not external peer review. The higher-degree stretch was considered but no four-star grade or higher-degree result is claimed. ## References 1. Wang, Y., and Zhang, B. (2013). Hadamard powers of polynomials with only real zeros. LAA 439, 3173–3176. 2. Białas, S., and Białas-Cież, L. (2017). Comments on 'On Hadamard powers of polynomials'. MCSS 29, article 16. 3. Białas, S., Białas-Cież, L., and Kudra, M. (2024). On the Hurwitz stability of noninteger Hadamard powers of stable polynomials. LAA 683, 111–124. 4. Fujie, K., and Ueda, Y. (2023). Law of Large Numbers for Roots of Finite Free Multiplicative Convolution of Polynomials. SIGMA 19, 004. 5. Kudra, M. (2026). The idealizer of the semigroup of stable polynomials, arXiv:2605.07628v2.