时间线上，2018 年 Katzarkov、Kontsevich 与 Pantev 的合作让 Kontsevich 愿意再听一次，并在 2019 年莫斯科会议提出一条不必真正依赖镜像对称的路：用 four-fold 自身（而非镜像）的曲线计数来切开 Hodge 结构，逐一研究每个「原子」如何在映射到新空间时变形。关键缺口是一个描述原子如何变化的公式；疫情封控期间，他们透过 Zoom 迅速完成了分解的第一部分，但仍找不到该公式。
2023 年 7 月，Iritani 证明了一个原子变化公式（arXiv:2307.13555），团队在接下来两年把它磨到足以使用，并在 2025 年 8 月贴出证明：four-fold 总会至少保有 1 个无法被变形成「简化」四维空间所需状态的原子，因此不可参数化。这被称为数十年来分类计划最大进展之一，但也引发疑虑；Bai 组织的读书会进行了 11 次、每次 90 分钟仍觉得抓不到要点，巴黎、北京、南韩等地也出现类似小组。有人把它比作 Perelman 于 2003 年的证明：社群需要用更传统的语言重建后才会完全接受，而作者则认为阻力难免、但结果正确，且为镜像对称计划提供新的证据。
Katzarkov’s goal was to split a mirror-style curve count into “atoms,” then show a four-fold’s Hodge structure splits into matching pieces; if even one piece cannot be mapped to a “simple” 4D space, the four-fold cannot be parameterized. The original route leaned on the assumption that Kontsevich’s mirror-symmetry program holds for four-folds—widely expected to be true, but not something they could technically rely on at the time.
In 2018, work by Katzarkov, Kontsevich, and Pantev reopened the conversation, and at a 2019 Moscow conference Kontsevich outlined a workaround: use the four-fold’s own curve counts (not its mirror’s) to decompose the Hodge structure and track each atom separately under mappings to new spaces. The missing ingredient was a formula for how atoms change; during Covid lockdowns, meetings over Zoom quickly nailed the decomposition step, but the transformation formula remained out of reach.
In July 2023, Hiroshi Iritani proved an atom-variation formula (arXiv:2307.13555); over the next two years the team refined it, and in August 2025 they posted a proof that a four-fold always retains at least 1 atom that cannot be transformed to match a simplified 4D target, so four-folds are not parameterizable. It is billed as the biggest advance in decades, yet many readers remain unsure because the tools feel alien: Bai’s reading seminar has run 11 sessions of 90 minutes and still feels lost, with parallel groups in Paris, Beijing, South Korea, and elsewhere. The community compares the moment to Perelman’s 2003 proof—acceptance may require a more traditional re-derivation—while the authors expect resistance but stand by correctness and see added evidence for mirror symmetry.
