Math RefereeReturned model assessments
These are final model assessments, not adjudicated truth. Initial prompts excluded Astra and coordinator findings. Failed and rejected attempts supply no verdict.
gpt-6.1-sol — whole supplied source
The manuscript presents a coherent and potentially valuable strategy, but the supplied text does not establish its main theorem. Several auxiliary statements or displayed inferences are demonstrably false, and the central incidence and measure-decomposition arguments contain substantial unverified steps; this assessment does not assert that the main theorem itself is false.
gpt-6.1-sol — thm:main-thm
The printed theorem has a direct boundary counterexample under the standard Euclidean or coordinate-maximum lattice norms. Independently, the retained-set construction invokes a quantitative nondivergence statement that is false as printed and whose hypotheses do not match that application. The diagonal conjugacy is correct, and the structured-component stopping restriction permits the proposed terminal dimension. These checks do not validate the remaining dependencies or other editions.
gpt-6.1-sol — cor:curve
Conditional on Theorem thm:main-thm, the diagonal parameters and compactness thresholds are consistent: at time t/2 the required ell-range lies inside the displayed wider range, and X_{e^{-t/10}} supplies the required vector and exterior-square bounds. However, the stated Taylor approximation has an order-one, rather than exponentially small, remainder. The graph reduction and passage from fixed-ell estimates to simultaneous control also require justification. These are defects in the supplied argument, not counterexamples to the corollary. No external theorem is treated as verified.
gpt-6.1-sol — prop:quantitative-nondivergence
The printed proposition has an elementary counterexample under the stated uniform quantifiers. Both diagonal translates can lie outside the prescribed compact set while the entire unipotent segment retains an arbitrarily short invariant vector. Independently, the proof's final bounds are compatible with, rather than contradictory to, its hypotheses. These conclusions do not require either cited external theorem.
gpt-6.1-sol — prop:initial-bound-dimension
The supplied proof does not establish Proposition 6.2. Its retained-measure prerequisite is false as printed, the separation argument changes the coefficient being bounded, and both closing cases require an unidentified quantitative external argument. These are defects in the supplied argument, not a counterexample to Proposition 6.2 itself. Dependencies were audited as proposed inputs, not certified. Later knowledge potentially present in model weights is not admitted as evidence.
gemini-4-argon — prop:quantitative-nondivergence
Proposition 4.1 (prop:quantitative-nondivergence) is defective as stated because the hypothesis is printed with an inverted membership sign: $a_0(-s)x, a_0(s)x \not\in X_{e^{-s/3}}$ instead of $a_0(-s)x, a_0(s)x \in X_{e^{-s/3}}$. The printed statement is false and admits an elementary counterexample (taking $x = \diag(\delta,\delta,\delta^{-2})\Gamma$ with $0 < \delta < \min(\tfrac{1}{2}e^{-2s/3}, \tfrac{1}{2}\beta)$, which satisfies $a_0(\pm s)x \not\in X_{e^{-s/3}}$ while $a(s)u(r)x \not\in X_\beta$ for all $r \in [1]$). The proof itself argues by contraposition toward the intended hypothesis $a_0(-s)x, a_0(s)x \in X_{e^{-s/3}}$: assuming the quantitative non-divergence bound fails, it invokes the Kleinbock-Margulis theorem ([KM98, Theorem 5.2], [Kle08, Theorem 2.2], both published prior to the 2022-09-26 cutoff) to obtain a nonzero $\vv v \in \bigwedge^j \Z^3$ ($j=1,2$) with $\max_{r \in [1]} \|a(s)u(r)g\vv v\| \le 1$, and computes the diagonal actions $a_1(-s)\vv v' = a_0(-s)g\vv v$ (Case 1, $j=1$) and $b(-s)\vv v' = a_0(s)g\vv v$ (Case 2, $j=2$). Even for the intended hypothesis $a_0(\pm s)x \in X_{e^{-s/3}}$, there is a minor constant-factor looseness in passing from $\max_{r \in [1]}\|a(s)u(r)g\vv v\| \le 1$ to $\|a_0(\mp s)g\vv v\| < e^{-s/3}$ (since $|v'_2| \le 1$ and $|v'_2 + e^s v'_3| \le 1$ only give $|v'_3| \le 2e^{-s}$, and Euclidean vs. max norms introduce a constant factor $\le \sqrt{6}$), which can be repaired by replacing the threshold $1$ in the Kleinbock-Margulis step with a small constant $c_0 < 1/(2\sqrt{3})$.
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