A bounded index?

The claims, in plain words

Take a contraction f: X → Z with connected fibers, where minus the canonical divisor K is ample over Z: a Fano contraction. A complement near a point of Z is an effective boundary B that makes K + B trivial there: N(K + B) is an integral Cartier divisor linearly equivalent to zero, and N is the index. On the projective line over a point, two points with coefficient one give index one, but that pair is only log canonical; a klt complement needs every coefficient below one, like four points with coefficient one half, index two.

Shokurov introduced complements in his work on threefold flips and conjectured that their index can be bounded in each dimension. Birkar proved bounded log canonical complements, a key step toward his boundedness theorem for Fano varieties. Klt complements are harder, because bad singularities push the index up: the cone over a rational normal curve of degree n has a vertex with log discrepancy 2/n, and its klt complements need index at least n/2. So one fixes ε and allows only ε-lc varieties, whose log discrepancies are all at least ε. Chen reduced the conjecture to a local threshold statement and proved it over bases of dimension at most two.

Result 066 is two OpenAI preprints, both dated September 25, 2026. The first (93 pages) claims: for each dimension d and rational ε there is an index N(d, ε) such that every ε-lc Fano contraction of dimension d, over any algebraically closed field of characteristic zero, has a klt complement of that index near every point of the base. Over ℂ this gives the finite-rational-coefficient form of Shokurov’s bounded-klt-complement conjecture, as Chen stated it. The second (106 pages) claims the Cartier-divisor conjecture of Birkar and Shokurov: for an ε-lc Fano type contraction with nef −(K + B) and a positive-dimensional base, through any base point there is a Cartier divisor whose pullback has log canonical threshold bounded below in terms of the dimension and ε alone. Over ℂ, Chen showed the two conjectures are equivalent; the papers reach them by independent routes.

What they don’t claim

The complements are klt, not ε-lc (the stronger version in Chen’s Remark 1.2 is not asserted), and the coefficient version needs a finite set of rational coefficients. The index depends on the dimension and ε; without a fixed ε it cannot be bounded, as the cones show.

Where it stands

Posted as preprints

OpenAI, both dated September 25, 2026, released October 6

Peer reviewed

Not yet

Checked in Lean

Main theorems: no; OpenAI’s Lean check covers one supporting compactness lemma of the companion paper, and uncatalogued files add more of its local algebra (we have not run them)

Last checked October 19, 2026. Neither preprint has been revised. The two cite each other as independent companions. The Cartier paper adapts two lemmas from the ordinary-double-point gap paper of episode 037, without using its theorem as a black box, and discusses a local phase assertion from episode 036 without relying on it. Inputs include Birkar’s boundedness theorems and complements, Chen’s reduction, minimal model theory, Ambro’s canonical bundle formula, normalized volumes (Li, Blum, Xu, Zhuang) and Takagi’s comparison of multiplier and test ideals. Both papers are long, and each adds new technical steps that experts must check. We found no public assessment by specialists. OpenAI’s catalogued Lean check covers that compactness lemma: 13 files and 1,620 lines, excluding Mathlib. Its files also contain 10 uncatalogued (not linked from the collection’s catalog) Lean files, 1,788 lines, with more of the companion’s local algebra from its Section E: initial exponents of power series, valuation bounds and bounds on chart retractions. An umbrella file ties them together (23 files, 3,408 lines). We found no unproved steps in them and have not run them. The main theorems are not formalized. Correction (October 15, 2026): the video says the only Lean check is the compactness lemma; uncatalogued files add more supporting lemmas, not the main theorems.

How the proof goes

1. A threshold. Through Chen’s equivalence, the first paper needs a uniform lower bound for the log canonical threshold of the pullback of a base point’s maximal ideal, in the birational case with no boundary. For the cones this threshold is exactly 2/n, never below ε for ε-lc members.
2. By contradiction. A sequence breaking the bound has divisors whose order along the ideal is huge compared with their log discrepancy. Small log discrepancy at a real valuation is not enough, because rescaling rescales it, so the proof must also control the normalization.
3. Volumes and degenerations. Anticanonical sections are encoded in an affine cone; normalized volume is minimized under a log canonical constraint, after an exact-penalization step; stable degeneration and a torus of maximal rank give a graded model, and counts of independent sections control the valuations.
4. Traces. An auxiliary reduction to positive characteristic uses Cartier trace maps, compared with multiplier ideals through Takagi’s theorem, to produce sections of controlled degree and value. Values are rounded to integers layer by layer until the initial terms form a full transcendence basis; then a bounded multiple of the valuation is a whole-number multiple of a prime divisor whose log discrepancy is below ε, a contradiction.
5. The companion route. The Cartier paper uses characters on affine cones and a pinning obstruction instead.

The numbers in the video

• A short Python script checks the least klt complement index 2 on the projective line and, for the cones P(1,1,n), the vertex log discrepancy 2/n from the Hirzebruch surface F_n, the threshold 2/n of the vertex’s maximal ideal, and the least klt complement index max(2, ⌈n/2⌉), at most max(2, ⌈1/ε⌉) for ε-lc members.

Read the source

• Bounded klt complements for Fano contractions (PDF), September 25, 2026 (93 pages).
• Uniform Cartier sections for Fano type contractions (PDF), September 25, 2026 (106 pages).
• OpenAI’s note on the scope of the Lean check.
• Chen, on bounded klt complements and local thresholds (arXiv).
• The collection’s catalog, where this is result 066.

The 2 papers in this result

OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.

• Bounded klt complements for Fano contractions main paper
• Uniform Cartier sections for Fano type contractions companion

Transcript

Spot a mistake?

This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.