Monthly Mathematical Conjectures: Corrections, Sharp Results, and Exact Verification

Abstract

This maintained working-paper series collects precise conjectures, corrections, proofs, partial results, and exact verification artifacts. The reader states the strongest mathematically supported formulation of each problem and marks the boundary between theorem, trusted-solver finite theorem, reduction, counterexample, finite diagnostic, and open work. For the discrete centered Hardy-Littlewood maximal operator, the record proves Var(Mf) at most Var(f) for every nonnegative sequence supported on three consecutive sites. The proof includes the complete unimodal radius-surgery argument. In the nonunimodal case, 0 at most b less than a at most c, the exact positive-variation deficit is the minimum of a-b and (a+c-2b)/3, and is strictly positive. Exact linear-real-arithmetic certificates establish the inequality for arbitrary nonnegative real profiles on at most eight consecutive sites. All five SMT-LIB queries, complete Z3 4.16.0 proof terms, replay receipts, and the Lean-checked three-site algebra accompany a standalone three-page proof. The general conjecture remains open. The sharp circle L1 Poincaré-Wirtinger stability theorem is supplied as a standalone four-page proof. It establishes the optimal coefficient 1/4 by a nested-core coarea argument. The archive includes the LaTeX source, an independent audit, the exact 4/5 versus 9/10 counterexample to the historical common-half-arc lemma, a 65,056-profile rational diagnostic, and a bounded Lean project. For Gaussian logarithmic-Sobolev stability, the exact-root coherent pair with t squared equal to log 3 has closest coherent states at plus or minus t/2. A 640-term entropy minorant and outward-rounded 768-bit Arb calculation prove q_N < 0.577215 and Q_N < 1.15443 pi in every dimension. The complete derivation, captured Arb balls, replay instructions, and Lean companion are provided in a standalone PDF and archive. The collection also contains the centered-maximal quasiconcave theorem and interval-Hall reduction, the deterministic cycle-discrepancy theorem for random monomial unitaries, the classification of flat holomorphic isometric embeddings into C x H, and a source-level correction of the 2025 GPT-5 Pro/Gemini gradient-descent episode together with the known sharp 7/(4L) theorem. Historical L1 and proof-refinement records retain their original files and credits and carry in-place notices linking to this stable concept DOI. The record comprises the thirteen-page reader, a directly visible Reddit-ready Markdown file, complete sources, exact verification, three result-specific proof packages, DOI lineage, manifests, and checksums. Independent checking and priority information are welcome.