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

This maintained working-paper series collects precise conjectures, corrections, proofs, partial results, and exact verification artifacts. Each release states the strongest mathematically supported conjecture and separates proved statements from computational evidence. The analysis dossier contains the corrected sharp circle L1 Poincaré–Wirtinger stability theorem, the centered Hardy–Littlewood maximal-variation reduction and quasiconcave theorem, and a certified Gaussian logarithmic-Sobolev upper bound. This update adds three audited intake results. The random monomial-unitary discrepancy claim is proved deterministically from the permutation cycles, with an almost-sure corollary under any common coupling. Holomorphic isometric embeddings of the flat plane into C × H are classified by Liouville's theorem, with the local equations and global obstructions for a general conformal metric recorded separately. A source-level audit of the 2025 GPT-5 Pro/Gemini gradient-descent episode shows that the GPT-5 proof for η ≤ 3/(2L) was correct; the later critique introduced an index mismatch. The reader also reproduces the known sharp threshold η ≤ 7/(4L). The common-half-arc selection lemma used in DOI 10.5281/zenodo.17060647 is false; the current proof replaces it with a relevant-level and disjoint-core argument. This line also supersedes DOI 10.5281/zenodo.16946199 and DOI 10.5281/zenodo.17247282. Their files, titles, credits, and metadata are preserved in the historical archive. The wider information-geometric, automorphic, motivic, Collatz, and arithmetic claims in those records are withdrawn from the maintained line unless an independent proof is supplied. The historical proof-refinement account in DOI 10.5281/zenodo.16929871 is corrected by the gradient-descent audit. The reader is accompanied by source files, exact verification scripts and outputs, a DOI lineage table, checksums, and a self-contained archive of the cited local historical sources. Independent checking and priority information are welcome.