# Yang effective Ratner: adjudicated review

21 October 2026. Lei Yang, *Effective version of Ratner's equidistribution theorem for SL(3,R)*, Annals of Mathematics 202 (2025), 189–264.

**The published Theorems 1.2 and 1.3 are false as stated.** An explicit unimodular lattice passes their vector-and-bivector thickness tests with a strict margin while every orbit lattice retains the same nonzero vector. A fixed smooth test function then contradicts convergence to full Haar measure. The same example makes the initial retained set empty. These conclusions were checked against the official journal PDF supplied by the user; they are not merely preprint-version objections.

**A corrected effective equidistribution theorem has not been established by this review.** Strengthening the Diophantine exponent would exclude the exhibited family, but checking that change throughout the proof is a separate task. This report does not refute either expanding-curve corollary, qualitative Ratner theory, or every possible effective theorem in this setting.

## Sources and attribution

The independent native panel received the complete arXiv v6 source (19 September 2024) or specified local packets, without the external Astra analysis or coordinator calculations. Native source SHA-256: e6f3ee5af078764b1750ea6a8cdbc8c1d9577ef6d14d0bc638830f385d839db8.

The [official journal snapshot](../paper-journal.pdf) has SHA-256 fe44b57f5aa5f50884b4a126da589fe38029f1a3035f9c57bd1e325c031d9e4f. Targeted passage comparisons, including the main theorems on p. 192, are recorded in the [journal comparison](source-comparison.md) and source-evidence.json. We do not claim every journal line is identical to v6. Earlier preparation records correctly say the journal was unavailable at that earlier time; they remain unchanged.

The user supplied the explicit rational-lattice argument and attributed it to Astra. Its model ID and effort are not authenticated. The coordinator checked its mathematics and journal applicability; this is not a discovery-priority claim or a formal certificate.

## Decisive counterexample

Take

$$g=\begin{pmatrix}2&2/3&0\\0&1&0\\0&0&1/2\end{pmatrix},\qquad
\Lambda=g\mathbb Z^3.$$

For L at least 10, direct all-integer coordinate arguments give:

| Test | Exact shortest length |
|---|---:|
| a0(L) Lambda | (1/2) exp(-L/6) |
| Dual of a0(L) Lambda | (3/2) exp(-L/3) |
| a0(-L) Lambda | 2 exp(-L/3) |
| Dual of a0(-L) Lambda | exp(-L/6) |

The exterior-square test equals the dual-lattice test under the usual Hodge identification. Every length is at least (3/2) exp(-L/3). Thus the exceptional alternative is absent throughout the required window for all sufficiently large t.

But both a(t) and u(r) fix e1, and Lambda contains 2e1. All orbit lattices lie in the closed proper locus Y of lattices containing this prescribed vector. Choose a nonzero nonnegative compactly supported smooth function in X minus Y. Its orbit integral is zero, whereas its Haar integral is strictly positive. Its fixed Sobolev norm cannot reconcile that constant discrepancy with exponential decay. This refutes Theorem 1.3. The allowed moving starting point x_t=a(t)g Gamma transfers the contradiction to Theorem 1.2.

The [complete counterexample argument](counterexample.md) includes every integer branch, exact inversion and exterior-square identity, the closed-locus argument, both theorem quantifiers, and a family defeating any fixed multiplicative normalization. Finite-box checks are sanity checks only.

## The initial retained set

In Definition 6.1, journal p. 203, every point of the starting orbit still contains 2e1. The allowed choice ell=0, ell'=-2 delta1 t makes its length 2 exp(-2 delta1 t/3), eventually smaller than beta^(1/4). Every point is removed. Thus F is empty, its claimed normalized measure is undefined, and the discarded proportion is one rather than the exponentially small quantity asserted before (6.7).

This failure persists after correcting the separate reversed membership sign in Proposition 4.1. The endpoint shift changes the exponent scale: the theorem provides exp(-abs(ell' plus or minus S)/3), while the proposed nondivergence application needs exp(-S/3). An exponentially weaker endpoint bound cannot replace the latter.

## Other adjudicated findings

The [additional checks](additional-checks.md) supply proofs, examples, scope qualifications and suggested obligations.

| Location in journal | Checked finding | Scope of conclusion |
|---|---|---|
| Lemma 2.1, p. 197 | All bracket identities checked exactly | Locally supported; no certificate for later analytic arguments |
| Proposition 4.1, p. 201 | Reversed endpoint hypothesis; short fixed-vector family contradicts its uniform conclusion | Statement false literally |
| Lemma 4.2, pp. 201–202 | Identity-return counterexample at ell=1 | Stated full parameter range false under conventional coordinate/Frobenius norms; large-time replacement not refuted by this example |
| Proposition 6.2 proof, pp. 204–205 | Coefficient a0 changes to a2; false unipotent-part assertion; missing stabilizer component | False auxiliary assertions and defective supplied inference; no independent counterexample to Proposition 6.2 |
| Proposition 6.3 proof, p. 206 | Positive conjugation expands the wrong transverse block | Displayed identity false; changing sign alone does not verify the proof |
| Proposition 7.2 proof, p. 208 | Finite relatively compact neighborhoods cannot cover all noncompact X | False auxiliary covering assertion; thick-support replacement remains to be proved |
| Lemma 8.6 proof, p. 212 | Conditional probability loses its normalization denominator | Displayed inference does not follow from conditioning; final dynamical bound unresolved |
| Proposition 9.10 proof, p. 231 | beta^30 lower bound changes to beta^22 | Stronger bound not implied; later use returns to beta^30, so this may be a local transcription error |
| Lemma 11.4(1), p. 240 | Two admissible hyperbolas are closer than exp(-3s') while their parameter difference exceeds the asserted bound by beta^(-1) | Quantitative statement false, within compatible source scales |
| Lemma 11.6 proof, p. 246 | Invoked bound contains beta^(-100), which disappears in the conclusion | Cited estimate does not justify the stronger bound |
| Proposition 12.1 proof, pp. 258–260 | Weak test-function approximation does not imply an exact positive-measure decomposition; cover overlaps are not disjointized | Supplied derivation defective; alternative decomposition not ruled out |
| Corollary 1.6 proof, p. 194 | Rescaled quadratic Taylor remainder is order one | Approximation argument defective; corollary truth unresolved |

For Lemma 11.4, the checked family uses w=beta exp(s')/2, w'=w+exp(-s')/beta, k=k'=1, ell=0 and A=1. On the statement's exact clipped square the graph distance is at most (36/25) beta exp(-3s'), whereas the parameter gap is exp(-s')/beta. The full note verifies clipping, parameter bounds and compatibility with beta=exp(-epsilon^3 t). This strengthens the native proposed counterexample; the stronger construction is coordinator work, not attributed verbatim to Sol.

## Native panel and arbitration

| Assessment | Attempts | Usable final assessments |
|---|---:|---:|
| Sol high, complete source | 1 | 1 |
| Sol high, four local obligations | 4 | 4 |
| Opus 5.5 max, main theorem and curve | 2 | 0 |
| Gemini 4 Argon high, nondivergence and initial dimension | 2 | 1 |
| Independent arbitration: one Opus, two Gemini | 3 | 0 validated |

The targeted Sol response independently refutes the main statement with the identity lattice, a boundary example. Its full-source response is more conservative and does not itself assert the main theorem false. Sol and Gemini independently identify Proposition 4.1's reversed condition, and Sol independently finds the quadratic scaling issue.

All three Opus attempts exhausted their output caps without a final assessment. One initial Gemini attempt and one Gemini arbitration did likewise. The other Gemini arbitration returned substantive text but failed the typed schema because a context finding was assigned defect severity. It remains rejected; this report does not silently repair it or present it as accepted arbitration. No failed attempt supplies a mathematical verdict.

Actual native coverage is one full-source review plus eight targeted attempts over four obligations, not three independent full-paper reviews. The source index has 56 claim environments and 35 bibliography entries. Its completeness as a structural index does not establish an exhaustive proof audit. The full allegation inventory preserves all 39 full-source assessments and local findings with their separate disposition.

## External inputs and historical cutoff

The cutoff is the journal received date, 26 September 2022, not the 2024 reviewed edition date. The cited Lindenstrauss–Mohammadi Proposition 4.2 is present on p. 16 of [arXiv:2112.14562v1](https://arxiv.org/pdf/2112.14562v1), submitted 29 December 2021. Its later journal date is not an admissibility defect. Its applicability to Yang's different group, transverse dimension and scales remains unverified.

The exact Kleinbock–Margulis, Kleinbock, Bernik–Kleinbock–Margulis and quantitative closing inputs were not all independently proved or source-audited here. Bibliography entries are not certificates of their applicability. No imported theorem is needed for the elementary main counterexample. Missing definitions in a local packet are not paper errors when the complete manuscript defines them; in particular [1] is explicitly the centered unit interval.

Primary publisher, arXiv and author pages were checked again for version/correction context. No correction was found in those inspected records and targeted searches. This is not an exhaustive erratum or priority audit, and no author's response is claimed.

## Graph, costs and remaining obligations

The decorated dependency graph at the top of this panel page distinguishes refuted local statements, defective proofs, a locally supported algebra lemma, and unresolved claims. Arrows run prerequisite to consumer. Terminal nodes have outdegree zero only in the recorded partial proof graph. Red prerequisites do not make downstream statements false automatically. In particular both corollary statements remain unresolved; the curve node is red for its own defective approximation step.

All twelve attempts have reported token usage. The frozen conservative batch-rate total is **$3.8521215**, including failures, against a $10 authorization. Cache discounts may reduce actual charges; this is not an invoice. Per-attempt data are in [coverage metadata](coverage-and-cost.json). There were no retries, no new paid calls for adjudication, and no paid repair campaign.

The final outcome of this scoped review is a checked counterexample to the published main formulations and several additional precise defects. The remaining mathematical task is to formulate and prove a corrected theorem, then audit the repaired nondivergence, initial-dimension, incidence and terminal-decomposition chain. Other model concerns remain explicitly unresolved rather than being promoted to established errors.

Exact scripts/results, source quotations, edition locators, allegation inventory and graph verification are preserved alongside this report. None is a Lean certificate or an independent human grade. This publication does not establish a repaired main theorem.