bibkey: tao2026newton authors: Terence Tao year: 2026 title: “The Newton and Maclaurin inequalities for symmetric polynomials” doi: null url: https://github.com/leanprover-community/mathlib4/blob/e3c1793d0e097d9b8d782a323e91c99c2ef0d64c/Mathlib/Analysis/MeanInequalitiesSymmetric.lean claim: “Newton inequalities for elementary symmetric functions of arbitrary real multisets; licensed source of the Crown prerequisite.” strata_touched:

• D5/S3/Analytic/RealRootedCoefficientNewton
• D5/S3/Combinatorics/Geometry/CrownOrderPolytopeLogConcavity license: Apache-2.0 triage: anchor

Newton inequalities: immutable source and license

Verified locator

URL: https://github.com/leanprover-community/mathlib4/blob/e3c1793d0e097d9b8d782a323e91c99c2ef0d64c/Mathlib/Analysis/MeanInequalitiesSymmetric.lean

The reduced Newton inequality is taken from Mathlib PR 42876 at immutable revision e3c1793d0e097d9b8d782a323e91c99c2ef0d64c. Its source SHA-256 is daab9424b8817d3e6bf62f14fe1bfb1c35cf96a47157897d46af2f64a70da2f2. It applies to arbitrary real multisets, including repeated entries and zero. The Crown proof applies it to the negated root multiset of the auxiliary scalar polynomial and converts elementary symmetric functions to coefficients by the pinned Vieta theorem. The actual geometric f-polynomial is not assumed to split.

The source’s toolchain is leanprover/lean4:v4.34.0-rc1; this repository pins v4.33.0. The port retains the upstream derivative-root and strong-induction proof. Its modifications specialize prerequisites to real numbers, localize normalization facts inside the proof, retain only the reduced Newton conclusion, and adapt imports and one natural-binomial cast to the current pin. The coefficient conversion belongs to the substantive Crown proof, with all out-of-degree cases treated there.

The licensed port is canonically frozen as a prerequisite of the Crown log-concavity theorem. The typed Proved claim belongs only to that full geometric Crown conclusion and its problem dossier. This classical Newton port is not a new open-problem resolution and earns no separate KPI. Final artifact review, required CI for this increment and dev merge remain pending; the license and retirement condition below continue to apply.

The following additional source files at the same immutable revision supply only the prerequisites needed by Newton:

All four upstream files state that they are released under Apache 2.0 as described in the upstream LICENSE. The official LICENSE fetched at this revision has SHA-256 b40930bbcf80744c86c46a12bc9da056641d722716c378f5659b9e555ef833e1; its complete text is reproduced below. The root NOTICE URL at this immutable revision returned HTTP 404 on 2026-09-20; the source headers contain no separate NOTICE reference.

Retirement condition: when this repository’s own future pinned Mathlib contains equivalent Newton and prerequisite declarations, delete the local port and replace its Crown use by direct imports and applications of those declarations. Upstream acceptance alone does not satisfy this condition.

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