Newly released models repeatedly appear near the top on the @terminalbench 2.1 leaderboard. Are these models actually on par with frontier models on terminal tasks? We took the top 20 models from that leaderboard and ran them on TB-fn. TB-fn reworks the same 89 tasks by adding steps, tightening requirements, or requiring more complex algorithms. It also removes opportunities for shortcut solutions and ensures verifiers check only the intended criteria. Models that clustered together on TB-2.1 spread across more tiers on TB-fn. That leaves GPT-5.6 Sol and Opus 5 in the lead, while some others drop significantly in performance. At the same time, models work 38% harder on average in TB-fn, leading to a redrawn Pareto frontier of performance vs. cost. The full post includes interactive figures showing how the gap for open-weight families widens. It also measures how security refusals affect scores across Anthropic, Google, and OpenAI models. Fable 5 stands out, refusing at least once on 13 of the 89 TB-fn tasks.

We’re releasing a broad range of new mathematical results produced by an internal frontier model. We’ve been consulting with the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute for Advanced Study, and we have drawn on their advice and public recommendations to inform how we release these results. github.com/openai/math

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How should we compress a binary input vector to a noisy channel while retaining as much information as possible about the channel output? Is the best strategy simply to keep part of the vector, or can coding do better? I’m excited to share a new paper with Ahmad Beirami: “The Most Informative Bit and Beyond: A Proof of the Courtade–Kumar Conjecture and Multibit Extensions.” Take a uniform binary vector X and let Y be formed by independently flipping its bits. We want to compress X into k bits while preserving as much information as possible about Y. The natural benchmark is to report k coordinates of X, which retains k[1−h₂(p)] bits of information. For k=1, this becomes the longstanding Courtade–Kumar conjecture: no one-bit function f(X) can be more informative about Y than a single coordinate of X. We prove the conjecture for every dimension and every noise level, without assuming that f is balanced. The central idea of the proof is to turn a static optimization problem into a dynamical one. Let ρ=1−2p denote the channel correlation, and consider the information gap Δ_f(ρ)=I(f(X);Yρ)−[1−h₂((1−ρ)/2)]. This measures the advantage of f over a coordinate as the channel moves from complete noise to no noise. At the noiseless endpoint, Δ_f(1)=H(f(X))−1≤0. Thus, if a function ever beat a coordinate, its information gap would have to become positive at an intermediate noise level and later return to a nonpositive value. Our proof shows that this cannot happen. Monotone rearrangement and an entropy-flow identity connect the evolution of the information gap to the edge geometry of the corresponding Boolean decision set. A sharp local entropy comparison, combined with Fourier analysis and logarithmic-Sobolev estimates for pivotal sets, then implies a global principle: once the gap becomes positive, it must continue increasing as the channel becomes cleaner. In other words, any hypothetical advantage over a coordinate would be self-reinforcing. But the noiseless endpoint forbids such an advantage. This contradiction proves the conjecture. The conceptual point is that we do not try to optimize separately at each fixed noise level. We follow the entire information trajectory as the noise changes. The entropy and geometry of a Boolean function constrain that trajectory so strongly that it cannot cross above the coordinate benchmark. We then ask the natural next question: what happens when the quantizer may output k>1 bits? Here the answer changes in a genuinely coding-theoretic way. We first show that linear maps cannot outperform coordinate selection. Nonlinear code-based quantizers can. Using the covering geometry of a shortened Hamming code, we construct an 8-bit quantizer that retains strictly more information than any eight-coordinate projection. A direct-sum construction then gives counterexamples to coordinate optimality for every k≥8. The coding is in the quantizer itself. Instead of preserving individual coordinates, it partitions the Boolean cube according to the collective geometry of a code. These structured quantization regions can remain more distinguishable after noise than the regions produced by simply selecting coordinates. What happens for k=2 through 7 remains especially intriguing. We exhaustively searched several structured code families and, for k=7, also performed Monte Carlo experiments with random codebooks. Structured codebooks consistently outperformed random ones, but none surpassed the coordinate benchmark. These findings suggest a sharp transition. We conjecture that coordinate projections remain optimal for 2≤k≤7, whereas nonlinear coding becomes advantageous from k=8 onward. We also acknowledge two concurrent and independent works resolving the one-bit conjecture: one by Zijie Chen, Amin Gohari, Adel Javanmard, Honghao Lin, Vahab Mirrokni, Chandra Nair, and David P. Woodruff; and another by Vu Khac Ky and Tuan Tran. Neither work studies the multibit extension considered here. AI disclosure: the one-bit proof was developed by generative AI under our high-level direction and review; the multibit extension was human-directed with AI assistance. We accept full responsibility for the paper. Paper: github.com/hessammahdavifar/… The source files and numerical checks are available in the same repository. Comments and feedback are very welcome.

How should we compress a binary input vector to a noisy channel while retaining as much information as possible about the channel output? Is the best strategy simply to keep part of the vector, or can coding do better? I’m excited to share a new paper with Ahmad Beirami: “The Most Informative Bit and Beyond: A Proof of the Courtade–Kumar Conjecture and Multibit Extensions.” Take a uniform binary vector X and let Y be formed by independently flipping its bits. We want to compress X into k bits while preserving as much information as possible about Y. The natural benchmark is to report k coordinates of X, which retains k[1−h₂(p)] bits of information. For k=1, this becomes the longstanding Courtade–Kumar conjecture: no one-bit function f(X) can be more informative about Y than a single coordinate of X. We prove the conjecture for every dimension and every noise level, without assuming that f is balanced. The central idea of the proof is to turn a static optimization problem into a dynamical one. Let ρ=1−2p denote the channel correlation, and consider the information gap Δ_f(ρ)=I(f(X);Yρ)−[1−h₂((1−ρ)/2)]. This measures the advantage of f over a coordinate as the channel moves from complete noise to no noise. At the noiseless endpoint, Δ_f(1)=H(f(X))−1≤0. Thus, if a function ever beat a coordinate, its information gap would have to become positive at an intermediate noise level and later return to a nonpositive value. Our proof shows that this cannot happen. Monotone rearrangement and an entropy-flow identity connect the evolution of the information gap to the edge geometry of the corresponding Boolean decision set. A sharp local entropy comparison, combined with Fourier analysis and logarithmic-Sobolev estimates for pivotal sets, then implies a global principle: once the gap becomes positive, it must continue increasing as the channel becomes cleaner. In other words, any hypothetical advantage over a coordinate would be self-reinforcing. But the noiseless endpoint forbids such an advantage. This contradiction proves the conjecture. The conceptual point is that we do not try to optimize separately at each fixed noise level. We follow the entire information trajectory as the noise changes. The entropy and geometry of a Boolean function constrain that trajectory so strongly that it cannot cross above the coordinate benchmark. We then ask the natural next question: what happens when the quantizer may output k>1 bits? Here the answer changes in a genuinely coding-theoretic way. We first show that linear maps cannot outperform coordinate selection. Nonlinear code-based quantizers can. Using the covering geometry of a shortened Hamming code, we construct an 8-bit quantizer that retains strictly more information than any eight-coordinate projection. A direct-sum construction then gives counterexamples to coordinate optimality for every k≥8. The coding is in the quantizer itself. Instead of preserving individual coordinates, it partitions the Boolean cube according to the collective geometry of a code. These structured quantization regions can remain more distinguishable after noise than the regions produced by simply selecting coordinates. What happens for k=2 through 7 remains especially intriguing. We exhaustively searched several structured code families and, for k=7, also performed Monte Carlo experiments with random codebooks. Structured codebooks consistently outperformed random ones, but none surpassed the coordinate benchmark. These findings suggest a sharp transition. We conjecture that coordinate projections remain optimal for 2≤k≤7, whereas nonlinear coding becomes advantageous from k=8 onward. We also acknowledge two concurrent and independent works resolving the one-bit conjecture: one by Zijie Chen, Amin Gohari, Adel Javanmard, Honghao Lin, Vahab Mirrokni, Chandra Nair, and David P. Woodruff; and another by Vu Khac Ky and Tuan Tran. Neither work studies the multibit extension considered here. AI disclosure: the one-bit proof was developed by generative AI under our high-level direction and review; the multibit extension was human-directed with AI assistance. We accept full responsibility for the paper. Paper: github.com/hessammahdavifar/… The source files and numerical checks are available in the same repository. Comments and feedback are very welcome.

If you are attending COLM 2026, AdvML x CoTMA workshop at COLM is hosting a @fidian sponsored social event after the workshop. We have limited spots that need approval. If interested, register for the event here luma.com/6rvuqoul. We will approve a select number of people.