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Dana Paquin

Publications and source records attributed to Dana Paquin.

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What Does MMLU Actually Measure? A Psychometric Audit of Difficulty Structure in Aggregate Benchmark Scores

Although MMLU is widely adopted as a benchmark for calibrating general AI capabilities, we psychometrically demonstrate that its aggregate score primarily evaluates a model's factual retrieval capacity rather than its reasoning ability. By calibrating item difficulty for 1,000 open-weights language models over 14,042 MMLU test items using Item Response Theory, we show that evaluating both abilities via a single test is inherently flawed. Difficulty is then regressed on a deterministic, text-extractable framework of structural complexity. Applying a joint Wald test with subject-clustered covariances demonstrates that the MMLU conflates fundamentally separable constructs. The mapping from structural complexity to difficulty is not invariant across the benchmark's STEM and non-STEM partitions. This finding has practical consequences. Aggregate leaderboard ranks track non-STEM accuracy more closely than STEM accuracy, so selecting a Top-50 model on the aggregate for a reasoning-intensive deployment displaces roughly 22% of the STEM-appropriate choices. Furthermore, when controlling for the multiple-choice guessing floor natively inside the response model, we find that higher-ability models continue to degrade more steeply under increased reasoning depth. The MMLU aggregate therefore weights retrieval capacity and reasoning stability unequally, inadvertently favoring models optimized for retrieval. We release our deterministic framework as a reproducible auditing instrument and recommend disaggregated reporting.

cs.CL

Collinear Interior Lattice Points in Triangles Satisfying $B(T)\in\{4,5\}$

A positive integer $k$ is called $Bn$-collinear if at least one lattice triangle with $n$ boundary points ($B(T)=n$) and $k$ interior lattice points exists, and every such triangle has all of its interior points collinear. Building on prior work on $B(T)=3$, we completely classify the $B4$- and $B5$-collinear integers. Using canonical lattice classifications together with arithmetic properties of Alder's generalized totient function $g(k)$, we prove that the only $B4$-collinear integers are $k\in\{1,2,5\}$. Furthermore, we show that no integer is $B5$-collinear. This establishes a structural contrast: while three and four boundary lattice points exhibit some collinearity constraints, five boundary points disrupt the pattern.

math.CO

On Coprime-Preserving Transformations and Dynamic Coprime Labeling

In this paper, we introduce dynamic coprime labeling (DCL), a novel extension of coprime labeling for time-sensitive networks. In particular, we explore whether there exists a graph labeling scheme that maintains relative coprimality among adjacent vertices as the graph evolves over time. We extend the definition of coprime labeling to include an injective labeling function, a time variable, and a transformation function. A DCL on a finite simple graph is a sequence of injective vertex labelings with the property that every edge is labeled by coprime integers at each time step, and the evolution is given by a time-independent coprime-preserving transformation. We prove that a graph admits a DCL if and only if it admits a classical coprime labeling (existence equivalence). We characterize families of coprime-preserving transformations and provide proofs of the existence of DCLs for paths, wheels, cycles, and the $n$-hypercube. We also introduce two classes of coprime-preserving transformations and present an application of DCL to Carmichael's theorem. These results establish DCL as a rigorous framework for further algorithmic and applied investigations.

math.CO

Convex Lattice Polygons with $k\ge3$ Interior Points

We study the geometry of convex lattice $n$-gons with $n$ boundary lattice points and $k\geq 3$ collinear interior lattice points. We describe a process to construct a primitive lattice triangle from an edge of a convex lattice $n$-gon, hence adding one edge in a way so that the number of boundary points increases by $1$, while the number of interior points remains unchanged. We also present the necessary conditions to construct such a primitive lattice triangle, as well as an upper bound for the number of times this is possible. Finally, we apply the previous results to fully classify the positive integers for which there exists a convex $n$-gon with $k$ collinear and non-collinear interior points.

math.NT