Duncan Soiffer

Duncan Soiffer, Nathaniel Itty, Christopher D. Rosin et al.

A collection of hyperplanes $\mathcal{H}$ slices all edges of the $n$-dimensional hypercube $Q_n$ with vertex set $\{-1,1\}^n$ if, for every edge $e$ in the hypercube, there exists a hyperplane in $\mathcal{H}$ intersecting $e$ in its interior. Let $S(n)$ be the minimum number of hyperplanes needed to slice $Q_n$. We prove that $S(n) \leq \lceil \frac{4n}{5} \rceil$, except when $n$ is an odd multiple of $5$, in which case $S(n) \leq \frac{4n}{5} +1$. This improves upon the previously known upper bound of $S(n) \leq \lceil\frac{5n}{6} \rceil$ due to Paterson reported in 1971. We also obtain new lower bounds on the maximum number of edges in $Q_n$ that can be sliced using $k<n$ hyperplanes. We prove the improved upper bound on $S(n)$ by constructing $8$ hyperplanes slicing $Q_{10}$ aided by the recently introduced CPro1: an automatic tool that uses reasoning LLMs coupled with automated hyperparameter tuning to create search algorithms for the discovery of mathematical constructions.

Duncan Soiffer, Steven Kolawole, Virginia Smith

Cascade systems route computational requests to smaller models when possible and defer to larger models only when necessary, offering a promising approach to balance cost and quality in LLM deployment. However, they face a fundamental challenge in open-ended text generation: determining output reliability when generation quality lies on a continuous spectrum, often with multiple valid responses. To address this, we propose semantic agreement -- meaning-level consensus between ensemble outputs -- as a training-free signal for reliable deferral. We show that when diverse model outputs agree semantically, their consensus is a stronger reliability signal than token-level confidence. Evaluated from 500M to 70B-parameter models, we find that semantic cascades match or surpass target-model quality at 40% of the cost and reduce latency by up to 60%. Our method requires no model internals, works across black-box APIs, and remains robust to model updates, making it a practical baseline for real-world LLM deployment.