CHIMERA: Compact Synthetic Data for Generalizable LLM Reasoning

Xinyu Zhu; Yihao Feng; Yanchao Sun; Xianzhi Du; Pingzhi Li; Olli Saarikivi; Yun Zhu; Yu Meng

cs.CLMar 1, 2026

CHIMERA: Compact Synthetic Data for Generalizable LLM Reasoning

Authors: Xinyu Zhu, Yihao Feng, Yanchao Sun, Xianzhi Du, Pingzhi Li, Olli Saarikivi, Yun Zhu, Yu Meng

Organizations: Department of Computer Science, University of Virginia, Charlottesville, VA, USA · Apple, Cupertino, CA, USA · Department of Computer Science, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA

Abstract

Large Language Models (LLMs) have recently exhibited remarkable reasoning capabilities, largely enabled by supervised fine-tuning (SFT)- and reinforcement learning (RL)-based post-training on high-quality reasoning data. However, reproducing and extending these capabilities in open and scalable settings is hindered by three fundamental data-centric challenges: (1) the cold-start problem, arising from the lack of seed datasets with detailed, long Chain-of-Thought (CoT) trajectories needed to initialize reasoning policies; (2) limited domain coverage, as most existing open-source reasoning datasets are concentrated in mathematics, with limited coverage of broader scientific disciplines; and (3) the annotation bottleneck, where the difficulty of frontier-level reasoning tasks makes reliable human annotation prohibitively expensive or infeasible. To address these challenges, we introduce CHIMERA, a compact synthetic reasoning dataset comprising 9K samples for generalizable cross-domain reasoning. CHIMERA is constructed with three key properties: (1) it provides rich, long CoT reasoning trajectories synthesized by state-of-the-art reasoning models; (2) it has broad and structured coverage, spanning 8 major scientific disciplines and over 1K fine-grained topics organized via a model-generated hierarchical taxonomy; and (3) it employs a fully automated, scalable evaluation pipeline that uses strong reasoning models to cross-validate both problem validity and answer correctness. We use CHIMERA to post-train a 4B Qwen3 model. Despite the dataset's modest size, the resulting model achieves strong performance on a suite of challenging reasoning benchmarks, including GPQA-Diamond, AIME 24/25/26, HMMT 25, and Humanity's Last Exam, approaching or matching the reasoning performance of substantially larger models such as DeepSeek-R1 and Qwen3-235B.

Figures & tables

Figure 1: Data synthesis pipeline overview: stage 1 expands a small set of seed subjects into thousands of fine-grained topics; stage 2 creates well-defined problems with concise, verifiable answers based on these topics; stage 3 generates detailed reasoning trajectories and labels their correctness.

Dataset	# Problems	# Subjects	# Topics	Prompt	Solution	Answer	Solution
				Length	Length	Format	Annotator
Human-Curated Reasoning Datasets
GSM8K	$7{,}473$	$1$	–	$45.1$	$51.7$	Numeric	Human
MATH	$7{,}500$	$1$	–	$33.0$	$89.5$	Free-form	Human
NuminaMath-CoT	$859{,}494$	$1$	–	$44.0$	$205.3$	Free-form	Human
Synthetic Reasoning Datasets

Table 1: Statistics of commonly used reasoning datasets. Chimera features substantially longer problem statements and more detailed solutions than existing datasets (lengths are measured in words), enabling complex and long-horizon reasoning for training modern LLMs.

Figure 2: Distribution of problem subjects in Chimera . The left panel illustrates broad disciplinary coverage, with mathematics accounting for

48.3\%

of the dataset, followed by computer science, chemistry and physics. The right panel decomposes the mathematics subset into fine-grained subfields. This distribution reflects the Chimera ’s emphasis on disciplinary breadth and topic diversity.

Large Scale ( $>$ 70B)
Model	# Params	GPQA-D	AIME24	AIME25	AIME26	HMMT Feb 25	HMMT Nov 25	HLE
DeepSeek-R1	671B	71.5	79.8	70.0	–	41.7	–	8.5
DeepSeek-R1-0528	671B	81.0	91.4	87.5	–	79.4	–	17.7
Qwen3-235B-A22B	235B	71.1	85.7	81.5	–	62.5	–	11.8
Qwen3-235B-A22B-Thinking-2507	235B	81.1	–	92.3	–	83.9	–	18.2
o3-mini (medium)	–	76.8	79.6	74.8	–	–	–	10.3

Table 2: Main results on reasoning benchmarks. Models are categorized into Standard Scale (

\leq

70B) and Large Scale (

>

70B). Notably, fine-tuning the Qwen3-4B base model on Chimera yields performance competitive with substantially larger models ( e.g. , DeepSeek-R1, Qwen3-235B-A22B), highlighting the strong data efficiency of our dataset.

Panel of Figure 3 — Figure 3: Pass@ $k$ results on GPQA-Diamond and HLE. Fine-tuning on Chimera consistently improves pass@ $k$ , indicating enhanced reasoning coverage and improved solution discovery under increased sampling.

Benchmark	Base	SFT	SFT + RL
GPQA-D	65.8	68.8	70.1
AIME24	81.6	86.5	86.9
AIME25	81.0	79.8	80.7
AIME26	80.8	80.3	82.7
HMMT Feb 25	59.2	63.1	65.7
HMMT Nov 25	57.3	66.3	67.0

Table 3: Reasoning benchmark performance of the base model, the SFT model trained on Chimera , and the subsequent RL model. SFT alone accounts for the majority of performance gains across benchmarks, with RL providing additional improvements.

Figure 4: Accuracy of Qwen3-4B-Thinking-2507 on existing synthetic datasets and Chimera . The base model achieves near-saturation performance on prior datasets, whereas Chimera remains substantially more challenging.

Figure 5: Average problem quality scores across sources, evaluated by o4-mini and gemini-2.5-pro . Under both evaluators, LLM-generated problems receive higher average scores than human-curated problems in this ranking protocol.

Benchmark	8-gram	13-gram
GPQA-Diamond	$0$	$0$
HLE	$7.89\times 10^{-6}$	$0$

Table 4:

n

-gram Jaccard overlap between Chimera and test benchmarks.

Subject: Physics
Topic: Topological Quantum Field Theory
Problem
Consider the 2+1-dimensional untwisted Dijkgraaf–Witten finite gauge TQFT with gauge group $G=Z_{n}$ (the cyclic group of order $n$ ). In this theory, the partition function on a closed oriented 3-manifold $M$ is defined by $Z(M)=\|\mathrm{Hom}(\pi_{1}(M),Z_{n})\|/n.$ Let $L(p,q)$ be a lens space, and use the standard fact that $\pi_{1}(L(p,q))\cong Z_{p}$ (a cyclic group of order $p$ ), independent of $q$ . Compute $Z(L(p,q))$ as a function of $n$ and $p$ . Here $\gcd(a,b)$ denotes the greatest common divisor.
Answer:
$\boxed{Z(L(p,q))=\frac{\gcd(n,p)}{n}}$
Detailed Solution

Table 5: A data sample from Chimera under the physics subject.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Usage	Prompt Template
Subject Expansion	I have a broad subject. Generate a list of more specific topics that fall under this subject. Each topic should be unique and different from the other topics. The topics should be an important field of study within this subject. Here is the broad subject: {subject}. Be as specific and diverse as possible when listing topics. For example, subject “Probability Theory” could have topics like “Probability Distributions”, “Random Variable”, “Conditional Probability”, “Bayesian Statistics”, etc. Output should be in the format of a list of topics: <topic1>\n<topic2>\n<topic3>\n …
Problem Generation	Given a topic, generate a new problem that falls under this topic. The problem should be at a PhD level or higher. Here is the topic: {topic}. Your answer must strictly follow the format and criteria below, without any markdown formatting: Problem: A clearly stated problem that requires advanced knowledge, techniques, or reasoning related to the topic. The problem must have a single, unambiguous, and objectively verifiable answer (e.g., a specific numeric value, closed-form expression, proof, or well-defined result). Clearly define all variables, parameters, and assumptions to avoid ambiguity. Reasoning: A detailed, logical reasoning process that shows how to solve the problem step by step, ensuring each step is verifiable and reproducible. Solution: A complete and rigorous solution derived from the reasoning steps above, including all necessary calculations or proofs. Answer: The final, concise answer to the problem. It must be clearly checkable and easy to verify as correct.
Solution Synthesis	<\|im_start\|>\| user\n{problem} <\|im_end\|> \n
Problem Validator	Given a topic and a problem with the answer, verify whether the problem and answer are valid and satisfy the requirements. Topic: {topic} Problem: {problem} Answer: {answer} Check the following: - The problem matches the topic and is at a PhD level or higher. - The statement is clear, self-contained, and unambiguous. - There is exactly one objectively verifiable answer. - The final answer is correct. Your answer must strictly follow the format and criteria below, without any markdown formatting: Verdict: VALID or INVALID Reason: one short sentence explaining the decision
Correctness Verifier	Judge whether the following predicted_solution to question is correct or not based on the precise and unambiguous correct_answer below. question: {question} correct_answer: {correct_answer} predicted_solution: {predicted_solution} Your judgement must be in the format and criteria specified below: extracted_final_answer: The final exact answer extracted from the predicted_solution. Put the extracted answer as ’None’ if there is no exact, final answer to extract from the predicted solution. reasoning: Explain why the extracted_final_answer is correct or incorrect based on correct_answer, focusing only on if there are meaningful differences between correct_answer and the extracted_final_answer. Do not comment on any background to the problem, do not attempt to solve the problem, do not argue for any answer different than correct_answer, focus only on whether the answers match. correct: Answer ’yes’ if extracted_final_answer matches the correct_answer given above, or is within a small margin of error for numerical problems. Answer ’no’ otherwise, i.e. if there if there is any inconsistency, ambiguity, non-equivalency, or if the extracted answer is incorrect. confidence: The extracted confidence score between 0% and 100% from predicted_solution. Put 100 if there is no confidence score available.

Appendix

Table 6: Prompts used for the data synthesis pipline.

Subject	Topic
Mathmematics	‘3-manifolds and hyperbolic structures’, ‘4-manifolds and smooth structures’, ‘Abelian varieties, Jacobians, and Pryms’, ‘Absoluteness and reflection principles’, ‘Actuarial science and risk theory’, ‘Additive combinatorics’, ‘Additive combinatorics and sum-product phenomena’, ‘Additive number theory and sumset problems’, ‘Adic and perfectoid spaces’, ‘Algebra’, ‘Algebraic coding theory’, ‘Algebraic combinatorics’, ‘Algebraic cycles and Chow groups’, ‘Algebraic geometry and moduli’, ‘Algebraic geometry and tropical combinatorics’, ‘Algebraic number theory’, ‘Algebraic spaces and stacks’, ‘Algebraic topology’, ‘Algebraic topology and homotopy theory’, ‘Algorithmic class field theory’, ‘Algorithms and randomized algorithms’, ‘Analysis’, ‘Analytic combinatorics and singularity analysis’, ‘Analytic number theory’, ‘Applications to analysis and Banach spaces’, ‘Applied combinatorics in networks and data’, ‘Applied probability’, ‘Applied topology and persistent homology’, ‘Approximate counting and correlation decay’, ‘Approximation algorithms for graph problems’, ‘Approximation theory and splines’, ‘Arithmetic combinatorics’, ‘Arithmetic geometry and Arakelov theory’, ‘Arithmetic geometry and Diophantine geometry’, ‘Arithmetic statistics’, ‘Arithmetic statistics and Cohen–Lenstra heuristics’, ‘Association schemes and distance-regular graphs’, ‘Automated reasoning and theorem proving’, ‘Automated theorem proving for mathematics’, ‘Automorphic forms and the Langlands program’, ‘Automorphism groups and Cayley graphs’, ‘Axiomatic frameworks and foundations of mathematics’, ‘Axiomatic set theory (ZFC and alternatives)’, ‘BSD conjecture and L-series of elliptic curves’, ‘Backward SDE and FBSDE’, ‘Bandits and online learning’, ‘Bayesian nonparametrics and Dirichlet processes’, ‘Bayesian statistics and Bayesian nonparametrics’, ‘Bijective combinatorics’, ‘Bioinformatics and phylogenetics’, ‘Biophysics and neuroscience models’, ‘Biostatistics and clinical trials’, ‘Bipartite and multipartite graphs’, ‘Birational geometry and minimal model program’, ‘Blockchain primitives and number-theoretic protocols’, ‘Borel equivalence relations and classification theory’, ‘Branching processes and coalescent theory’, ‘Brauer groups and central simple algebras’, ‘Bridgeland stability conditions’, ‘Brill–Noether theory and syzygies’, ‘Brownian motion and Gaussian processes’, ‘CAT(0) and Alexandrov spaces’, ‘Calabi–Yau manifolds and special holonomy’, ‘Calabi–Yau varieties and K3 surfaces’, ‘Calculus of variations’, ‘Calibrations and special Lagrangians’, ‘Cardinal characteristics of the continuum’, ‘Category theory and topos theory’, ‘Causal inference under uncertainty’, ‘Chabauty–Coleman methods and p-adic integration’, ‘Chaos and bifurcation theory’, ‘Character sheaves and Springer theory’, ‘Cheeger inequalities and isoperimetry on graphs’, ‘Chip-firing games and sandpile groups’, ‘Chordal, interval, and comparability graphs’, ‘Chromatic invariants and Tutte polynomials’, ‘Class field theory’, ‘Classical algebraic geometry (varieties and maps)’, ‘Classifying spaces and characteristic classes’, ‘Clique-width and rank-width’, ‘CoHAs and categorification in topology’, ‘Coarse and controlled topology’, ‘Cobordism theory and Thom spectra’, ‘Coding theory and combinatorial designs’, ‘Coding theory and compressed sensing’, ‘Coding theory and number-theoretic constructions’, ‘Combinatorial commutative algebra’, ‘Combinatorial optimization’, ‘Combinatorial representation theory’, ‘Combinatorial set theory’, ‘Combinatorics for ML and graph neural networks’, ‘Combinatorics in statistical physics (Ising, Potts, dimers)’, ‘Combinatorics of polynomials and real stable polynomials’, ‘Commutative algebra’, ‘Comparison geometry and Ricci curvature’, ‘Complex and Kähler geometry’, ‘Computability and recursion theory’, ‘Computational algebra and Gröbner bases’, ‘Computational algebraic geometry and Gröbner bases’, ‘Computational and algorithmic number theory’, ‘Computational and applied mathematics’, ‘Computational geometry’, ‘Concentration of measure and isoperimetry’, ‘Conformal and CR geometry’, ‘Conformal and topological quantum field theories’, ‘Conformal prediction and uncertainty quantification in ML’, ‘Constructible universe L and fine structure’, ‘Constructive and intuitionistic mathematics’, ‘Contact geometry and Reeb dynamics’, ‘Container method and hypergraph containers’, ‘Continuum theory and Peano continua’, ‘Control and observability on networks’, ‘Control theory and systems theory’, ‘Convex geometry and Brunn–Minkowski theory’, ‘Core model theory and extender models’, ‘Coupling methods and contractivity’, ‘Covering spaces and fundamental group’, ‘Cryptography and algebraic cryptography’, ‘Cryptography and post-quantum cryptography’, ‘Data science and manifold learning’, ‘Decorated character varieties and higher Teichmüller theory’, ‘Deep learning theory and generalization’, ‘Deligne–Mumford and Artin stacks’, ‘Derived categories of coherent sheaves’, ‘Descriptive set theory’, ‘Descriptive set theory (Borel and projective hierarchies)’, ‘Descriptive set-theoretic dynamics’, ‘Design theory and block designs’, ‘Design theory and finite geometries’, ‘Determinacy axioms (AD, ADL(R))’, ‘Differential geometry’, ‘Diophantine approximation and transcendence’, ‘Diophantine geometry and arithmetic geometry’, ‘Discrete Morse theory’, ‘Discrete and computational geometry’, ‘Discrete conformal geometry and circle packings’, ‘Discrete geometry and incidence geometry’, ‘Discrete geometry and polyhedral combinatorics’, ‘Discrete mathematics’, ‘Disordered systems and spin glasses’
	‘Distance-regular graphs and association schemes’, ‘Distributed and local graph algorithms’, ‘Distribution of rational points and unlikely intersections’, ‘Donaldson and Seiberg–Witten theory’, ‘Donaldson–Thomas and Pandharipande–Thomas invariants’, ‘Econophysics and financial mathematics’, ‘Edge coloring and total coloring’, ‘Edge expansion, vertex expansion, and conductance’, ‘Effective descriptive set theory’, ‘Electrical networks and effective resistance’, ‘Elementary number theory’, ‘Elliptic curve methods and isogeny computations’, ‘Elliptic curves and abelian varieties’, ‘Embeddings and metric distortion’, ‘Emerging directions’, ‘Empirical processes and Donsker theorems’, ‘Empirical processes and concentration bounds’, ‘Enumerative and extremal combinatorics’, ‘Enumerative combinatorics and generating functions’, ‘Enumerative geometry and Hurwitz theory’, ‘Environmental and climate risk modeling’, ‘Epidemic models and branching processes’, ‘Epidemics and contagion on networks’, ‘Epidemiology and infectious disease modeling’, ‘Erdos distinct distances and Szemerédi–Trotter’, ‘Erdos–Ko–Rado and intersecting families’, ‘Ergodic methods on homogeneous spaces’, ‘Ergodic theory and dynamical systems’, ‘Ergodic theory and mixing’, ‘Euclidean and classical geometry’, ‘Exchangeability and de Finetti theorems’, ‘Expander constructions and pseudorandomness’, ‘Expander constructions via number theory’, ‘Expanders and expander constructions’, ‘Exponential sums and the circle method’, ‘Extremal combinatorics’, ‘Extreme value theory and heavy tails’, ‘Extreme value theory and regular variation’, ‘Face numbers, f-vectors, and h-vectors’, ‘Facility location and scheduling’, ‘Fairness and privacy in algorithms’, ‘Fairness and robustness via probabilistic modeling’, ‘Faltings’ theorem and Mordell-type problems’, ‘Fano varieties and rationality problems’, ‘Federated Bayesian learning and distributed inference’, ‘Fibre bundles and principal bundles’, ‘Field theory and Galois theory’, ‘Filtering and state estimation (Kalman, particle filters)’, ‘Financial mathematics and derivatives pricing’, ‘Fine-grained lower bounds for combinatorial problems’, ‘Finite element and spectral methods’, ‘Finite geometries and incidence structures’, ‘Floer homology and pseudo-holomorphic curves’, ‘Flows, cuts, and circulations’, ‘Flows, cuts, and sparsifiers’, ‘Foliations, laminations, and measured laminations’, ‘Forcing and iterated forcing’, ‘Formal methods in cyber-physical systems’, ‘Fractal geometry’, ‘Fractional coloring and circular coloring’, ‘Free probability and noncommutative probability’, ‘Free probability in applications’, ‘Functional CLTs and invariance principles’, ‘Functional analysis and operator theory’, ‘Functional inequalities (Poincaré, log-Sobolev, transport)’, ‘Galois representations and deformation theory’, ‘Galois theory of number fields’, ‘Game theory and mechanism design’, ‘Gaussian processes and kernel methods’, ‘General relativity and geometric PDE’, ‘Generalized metric spaces and compactness’, ‘Generators, Dirichlet forms, and potential theory’, ‘Generic absoluteness and forcing axioms (MA, PFA, MM)’, ‘Geometric Langlands program’, ‘Geometric analysis’, ‘Geometric analysis and PDE on manifolds’, ‘Geometric deep learning and graph neural networks’, ‘Geometric graphs and unit disk graphs’, ‘Geometric group theory’, ‘Geometric group theory and quasi-isometries’, ‘Geometric measure theory’, ‘Geometric measure theory and currents’, ‘Geometric set cover and epsilon-nets’, ‘Geometric topology’, ‘Geometric topology interfaces’, ‘Geometrization and 3-manifold geometry’, ‘Geometry and topology’, ‘Geometry of Banach spaces’, ‘Geometry of numbers and lattice point problems’, ‘Gibbs measures, spin systems, and phase transitions’, ‘Graph coloring and list coloring’, ‘Graph compression and summarization’, ‘Graph databases and knowledge graphs’, ‘Graph decomposition and minors structure’, ‘Graph drawing and visualization’, ‘Graph embeddings and metric embeddings’, ‘Graph homomorphisms and constraint satisfaction’, ‘Graph isomorphism and canonical labeling’, ‘Graph limits and graphons’, ‘Graph minors and Robertson–Seymour theory’, ‘Graph polynomials (Tutte, chromatic, interlace)’, ‘Graph products and powers’, ‘Graph representation learning and GNNs’, ‘Graph theory and spectral graph theory’, ‘Graph theory and structural graph theory’, ‘Greedoid theory’, ‘Gromov–Witten theory and quantum cohomology’, ‘Group theory’, ‘Hamiltonian systems and symplectic dynamics’, ‘Hardness of approximation and PCP connections’, ‘Harmonic analysis and Fourier analysis’, ‘Heegaard Floer homology and contact invariants’, ‘Heights, Arakelov geometry, and Northcott property’, ‘High-dimensional probability and concentration inequalities’, ‘High-dimensional probability for data science’, ‘Hilbert schemes and Quot schemes’, ‘Hodge theory’, ‘Hodge theory and period mappings’, ‘Hodge theory and variations of Hodge structure’, ‘Holographic and Holant frameworks’, ‘Homological algebra’, ‘Homological mirror symmetry’, ‘Homology, cohomology, and cohomology operations’, ‘Homotopical algebra and model categories’, ‘Homotopy theory and Postnikov towers’, ‘Homotopy type theory in proof assistants’, ‘Hopf algebras and quantum groups’, ‘Hydrology and extreme events’, ‘Hyperbolic geometry and Kleinian groups’, ‘Hypergraph theory’, ‘Hypergraph theory and set systems’, ‘Hypergraphs and uniformity’, ‘Ideals on omega and forcing with ideals’, ‘Imaging science and tomography’, ‘Index theory and Atiyah–Singer theorem’, ‘Infinity categories and higher topos theory’, ‘Information geometry and statistical manifolds’, ‘Information geometry for learning’, ‘Information theory and entropy methods’, ‘Information-theoretic inequalities and entropy methods’, ‘Inner model theory and mice’, ‘Insurance mathematics and solvency modeling’, ‘Integrable probability and last passage percolation’, ‘Integrable systems’, ‘Integral geometry and geometric probability’, ‘Integral polyhedra and total unimodularity’
	‘Interacting particle systems and hydrodynamic limits’, ‘Interacting particle systems and mean-field models’, ‘Interactive proof assistants and formalization of mathematics’, ‘Interdisciplinary and applied areas’, ‘Intersection theory and Poincaré duality’, ‘Intersection theory and Riemann–Roch’, ‘Invariant theory’, ‘Inverse problems and regularization’, ‘Iwasawa theory’, ‘K-theory (algebraic and topological)’, ‘K-theory (topological and algebraic)’, ‘KPZ universality and integrable probability’, ‘Kazhdan–Lusztig theory and Hecke algebras’, ‘Knot theory and link invariants’, ‘Knot theory and low-dimensional topology’, ‘L-functions and functional equations’, ‘L-functions and zeta functions’, ‘Langlands program (classical and geometric)’, ‘Large cardinal axioms’, ‘Large cardinals and elementary embeddings’, ‘Large deviations and Gartner–Ellis theorem’, ‘Latin squares and orthogonal arrays’, ‘Lattice algorithms and LLL’, ‘Levy processes and jump diffusions’, ‘Lie groups and Lie algebras’, ‘Linear systems and base loci’, ‘Liouville quantum gravity and random geometry’, ‘Local fields and ramification’, ‘Local limit theorems and Berry–Esseen bounds’, ‘Logarithmic geometry’, ‘Low-dimensional topology’, ‘Manifold topology and smoothing theory’, ‘Manin conjecture and Peyre constants’, ‘Mapping class groups and surfaces’, ‘Markov chain Monte Carlo and mixing analysis’, ‘Markov chains and Markov processes’, ‘Markov processes and semigroup theory’, ‘Martingales and optional stopping’, ‘Matching theory and matroid intersection’, ‘Matching, matroid parity, and factors’, ‘Mathematical biology and epidemiology’, ‘Mathematical physics’, ‘Matroid theory and oriented matroids’, ‘Mean curvature flow and Ricci flow’, ‘Mean field games’, ‘Mean-field models and McKean–Vlasov equations’, ‘Measurable cardinals and ultrafilters’, ‘Measure theory and integration’, ‘Measure-theoretic probability and integration’, ‘Measure-theoretic probability and stochastic processes’, ‘Metric geometry and Gromov hyperbolicity’, ‘Microlocal analysis and pseudodifferential operators’, ‘Minimal surfaces and geometric flows’, ‘Minimal surfaces and mean curvature flow’, ‘Mirror symmetry and Fukaya categories’, ‘Mirror symmetry and SYZ’, ‘Mixed Hodge structures and degenerations’, ‘Model theory’, ‘Model theory in number theory and o-minimality’, ‘Moderate deviations and LIL’, ‘Modular forms and automorphic forms’, ‘Moduli of curves and stable maps’, ‘Moduli problems and geometric invariant theory’, ‘Monte Carlo and Markov chain mixing’, ‘Monte Carlo methods and variance reduction’, ‘Motives and motivic cohomology’, ‘Multiplier ideals and singular metrics’, ‘Network design and survivable networks’, ‘Network design, reliability, and survivability’, ‘Network inference and community detection’, ‘Network science and complex systems’, ‘Network science and multilayer networks’, ‘Neural differential equations and neural operators’, ‘Non-Euclidean geometry (hyperbolic and spherical)’, ‘Noncommutative algebra’, ‘Noncommutative algebraic geometry’, ‘Noncommutative topology and C-algebraic topology’, ‘Nonlinear analysis’, ‘Nonstandard analysis’, ‘Number theory’, ‘Numerical analysis and scientific computing’, ‘Numerical linear algebra’, ‘O-minimality and definable sets in AG’, ‘Obstruction theory’, ‘Operations research and mathematical programming’, ‘Operations research under uncertainty’, ‘Operator algebras (C-algebras, von Neumann algebras)’, ‘Optimal experimental design and active learning’, ‘Optimal transport and Wasserstein geometry’, ‘Optimal transport and probability metrics’, ‘Optimal transport for probabilistic modeling’, ‘Optimization (convex, nonconvex, integer, semidefinite)’, ‘Orbifolds and stacks’, ‘Order dimension and comparability graphs’, ‘PCF theory and singular cardinals’, ‘Parameterized algorithms and kernels on graphs’, ‘Parameterized complexity and kernelization’, ‘Partial differential equations (elliptic, parabolic, hyperbolic)’, ‘Partition calculus and infinite combinatorics’, ‘Percolation and interacting particle systems’, ‘Percolation and phase transitions’, ‘Percolation and phase transitions on networks’, ‘Perfect graphs and the strong perfect graph theorem’, ‘Perfectoid spaces and p-adic Hodge theory’, ‘Permutation patterns and sorting networks’, ‘Perpetuities and fixed-point equations in distribution’, ‘Perverse sheaves and D-modules’, ‘Perverse sheaves and microlocal sheaf theory’, ‘Philosophy and history of mathematics’, ‘Physics-informed machine learning and scientific ML’, ‘Planar graphs and graphs on surfaces’, ‘Point-set topology and separation axioms’, ‘Poisson geometry’, ‘Polyhedral combinatorics’, ‘Polyhedral geometry and polytopes’, ‘Polytopes, zonotopes, and associahedra’, ‘Posets, lattices, and Möbius inversion’, ‘Positivity of line bundles and vanishing theorems’, ‘Post-quantum cryptography and isogeny-based crypto’, ‘Primality testing and integer factorization’, ‘Prime number theory and zero-density estimates’, ‘Privacy-preserving analytics and differential privacy’, ‘Probabilistic combinatorics’, ‘Probabilistic combinatorics and random structures’, ‘Probabilistic numerics’, ‘Probabilistic potential theory’, ‘Probabilistic programming and approximate inference’, ‘Probability and statistics’, ‘Projective and affine geometry’, ‘Proof theory’, ‘Propagation of chaos and chaos decompositions’, ‘Property testing and sublinear algorithms’, ‘Property testing for graph properties’, ‘Pseudo-Riemannian and Lorentzian geometry’, ‘Pseudorandomness and extractor constructions’, ‘Quadratic forms and modular curves’, ‘Quantum field theory (constructive, algebraic, axiomatic)’, ‘Quantum gravity and string theory’, ‘Quantum information theory and quantum error correction’, ‘Quantum mechanics and spectral analysis’, ‘Quantum unique ergodicity and arithmetic quantum chaos’, ‘Queueing theory and network queues’, ‘Quiver varieties and symplectic resolutions’, ‘Ramsey theory’, ‘Ramsey theory and van der Waerden-type problems’, ‘Random Schrödinger operators’, ‘Random Schrödinger operators and Anderson localization’, ‘Random fields and Gaussian free fields’, ‘Random geometry and SLE in applications’
	‘Random graphs and complex networks’, ‘Random graphs and hypergraphs’, ‘Random graphs and percolation theory’, ‘Random graphs and stochastic block models’, ‘Random matrices and free probability’, ‘Random matrices and spectral statistics’, ‘Random matrices and universality’, ‘Random matrix theory and L-function statistics’, ‘Random multiplicative functions’, ‘Random multiplicative functions and pretentious methods’, ‘Random permutations and descents’, ‘Random tilings and dimer models’, ‘Random walk in random environment’, ‘Random walks and diffusion on graphs’, ‘Random walks on groups and heat kernel bounds’, ‘Random walks, mixing times, and cover times’, ‘Random, Cohen, and Sacks forcing’, ‘Rational connectedness and Campana program’, ‘Rational homotopy theory’, ‘Rational points on higher-dimensional varieties’, ‘Real and complex analysis’, ‘Reinforcement learning with probabilistic models’, ‘Reliability engineering and maintenance’, ‘Reliability theory and survival analysis’, ‘Representation theory’, ‘Reverse mathematics’, ‘Ricci flow and comparison geometry’, ‘Riemannian and pseudo-Riemannian geometry’, ‘Riemannian geometry’, ‘Rigid analytic and Berkovich spaces’, ‘Ring theory and module theory’, ‘Risk-sensitive and robust decision-making’, ‘Rough paths and regularity structures’, ‘Sato–Tate equidistribution and automorphic Sato–Tate’, ‘Schemes and morphisms of schemes’, ‘Schmidt subspace theorem and Roth’s theorem’, ‘Schramm–Loewner evolution and conformal invariance’, ‘Schubert calculus and Coxeter combinatorics’, ‘Semimartingales and stochastic integration’, ‘Sequential Monte Carlo and particle methods’, ‘Set theory and foundations’, ‘Set-theoretic topology and selection principles’, ‘Shape theory and ANR spaces’, ‘Sheaf theory and cohomology’, ‘Shimura varieties and automorphic sheaves’, ‘Shimura varieties and modularity’, ‘Sieve methods and additive number theory’, ‘Sieve methods and distribution in arithmetic progressions’, ‘Signal processing and compressed sensing’, ‘Simplicial complexes and collapsibility’, ‘Singularity theory and resolutions’, ‘Spatial statistics and geostatistics’, ‘Spatial-temporal modeling and point processes’, ‘Spectral geometry and isoperimetric inequalities’, ‘Spectral graph theory and Laplacians’, ‘Spectral graph theory and expanders’, ‘Spectral hypergraph theory’, ‘Spectral sequences and derived functors’, ‘Spectral sparsification’, ‘Spectral theory’, ‘Sperner theory and lattice of subsets’, ‘Spin geometry and Dirac operators’, ‘Square and diamond principles’, ‘Stable homotopy and spectra’, ‘Stable homotopy category and spectra’, ‘Stable laws and domains of attraction’, ‘Stable pairs and wall crossing’, ‘Stanley–Reisner theory and face rings’, ‘Stationary sets and club guessing’, ‘Statistical decision theory and inference’, ‘Statistical mechanics and phase transitions’, ‘Statistical mechanics and probabilistic combinatorics’, ‘Steiner systems and finite projective planes’, ‘Stein’s method and normal approximation’, ‘Stochastic PDE and random fields’, ‘Stochastic PDE in applications’, ‘Stochastic analysis on manifolds’, ‘Stochastic calculus (Itô and Stratonovich)’, ‘Stochastic calculus and SDE’, ‘Stochastic control and optimal stopping’, ‘Stochastic control, HJB, and viscosity solutions’, ‘Stochastic differential equations and flows’, ‘Stochastic networks and communication systems’, ‘Stochastic optimization and stochastic gradient methods’, ‘Stochastic partial differential equations’, ‘Stratified spaces and intersection homology’, ‘Streaming algorithms for combinatorial problems’, ‘Structural graph theory’, ‘Sub-Riemannian and Finsler geometry’, ‘Sublinear and streaming graph algorithms’, ‘Submodular optimization’, ‘Sumset inequalities and Freiman-type results’, ‘Surface classification and higher-dimensional birational geometry’, ‘Suslin trees and Aronszajn trees’, ‘Symplectic and contact geometry’, ‘Symplectic and contact topology’, ‘Symplectic geometry and Hamiltonian dynamics’, ‘Systems biology and network biology’, ‘TQFT and modular tensor categories’, ‘Teichmüller theory and moduli of curves’, ‘Teichmüller theory and moduli spaces’, ‘Temporal and dynamic graphs’, ‘Theoretical computer science and complexity’, ‘Theta functions and moduli of vector bundles’, ‘Threshold phenomena and sharp thresholds’, ‘Tilings, packings, and sphere packing’, ‘Time series analysis and stochastic forecasting’, ‘Time-frequency analysis and wavelets’, ‘Topological combinatorics’, ‘Topological data analysis’, ‘Topological data analysis and persistent homology’, ‘Topological dynamics and minimal sets’, ‘Topological graph theory and crossing number’, ‘Topological groups and Lie groups’, ‘Topological methods in combinatorics’, ‘Topological methods in data analysis’, ‘Topological quantum computing’, ‘Topological robotics and configuration spaces’, ‘Topology (point-set, geometric, and algebraic)’, ‘Toric and tropical geometry’, ‘Toric varieties and fans’, ‘Tournament and oriented graphs’, ‘Transcendence and Diophantine approximation’, ‘Treewidth, clique-width, and decompositions’, ‘Treewidth, pathwidth, and decompositions’, ‘Tropical geometry and tropical intersection’, ‘Type theory and homotopy type theory’, ‘Uncertainty quantification’, ‘VC theory and uniform laws of large numbers’, ‘Variational inequalities and complementarity problems’, ‘Waring’s problem and Vinogradov methods’, ‘Weak convergence and Skorokhod topology’, ‘Young tableaux and Schur functions’, ‘Zero-free regions and the Riemann hypothesis’, ‘Zeta functions of graphs’, ‘p-adic Hodge theory and Fontaine modules’, ‘p-adic analysis and Hida families’, ‘p-adic number theory and local fields’, ‘q-series, partitions, and symmetric functions’
Computer Science	‘AI for systems and systems for AI’, ‘AR, VR, and XR interfaces’, ‘Accessibility and inclusive design’, ‘Active learning and optimal data selection’, ‘Algorithms and data structures’, ‘Animation and physics-based simulation’, ‘Approximation algorithms and PTAS’, ‘AutoML and hyperparameter optimization’, ‘Bayesian deep learning and uncertainty estimation’, ‘Bilevel optimization and hyperparameter tuning’, ‘Binary analysis and software hardening’, ‘Blockchain and distributed ledger technologies’, ‘Category theory in CS’, ‘Causal inference and causal discovery in CS’, ‘Causal representation learning’, ‘Causality in ML and causal discovery’, ‘Cloud computing and virtualization’, ‘Computability and automata theory’, ‘Computational biology and bioinformatics’, ‘Computational chemistry and materials informatics’, ‘Computational complexity theory’, ‘Computational economics and market design’, ‘Computational imaging and computational photography’, ‘Computational neuroscience’, ‘Computer architecture and hardware-software co-design’, ‘Computer graphics and rendering’, ‘Computer networks and programmable data planes’, ‘Computer vision and multimodal learning’, ‘Computer vision and pattern recognition’, ‘Computer-supported cooperative work’, ‘Convolutional, recurrent, and attention-based networks’, ‘Cryptography and cryptographic protocols’, ‘Cyber-physical systems and real-time computing’, ‘Data mining and knowledge discovery’, ‘Data-centric AI and data governance’, ‘Data-centric AI, data governance, and quality’, ‘Database systems and data engineering’, ‘Datacenter architectures and disaggregated systems’, ‘Deep learning and representation learning’, ‘Deep learning architectures’, ‘Differential privacy and federated learning’, ‘Differential privacy and secure computation’, ‘Digital currencies and cryptoeconomics’, ‘Digital twins and cyber-physical security’, ‘Distributed systems and consensus’, ‘Domain generalization and distribution shift’, ‘Edge AI and on-device learning’, ‘Edge computing and IoT’, ‘Education technologies and computing education research’, ‘Embedded systems and robotics platforms’, ‘Energy-aware and green computing’, ‘Energy-based models and score matching’, ‘Fairness, accountability, and ethics’, ‘Fairness, accountability, and transparency in AI’, ‘Federated and decentralized learning’, ‘Formal languages and parsing’, ‘Formal methods and model checking’, ‘Foundation models and large language models’, ‘GPU and accelerator computing’, ‘Generative chemistry and materials discovery’, ‘Generative modeling’, ‘Geometric and topological data analysis’, ‘Geometry processing and CAD’, ‘Graph algorithms and network algorithms’, ‘Graph and relational learning for science’, ‘Graph neural networks and geometric deep learning’, ‘Hierarchical and multi-agent RL’, ‘Human-AI interaction and interactive ML’, ‘Human-computer interaction and UX’, ‘Implicit neural representations and neural fields’, ‘In-memory and approximate computing’, ‘Information retrieval and search’, ‘Information-theoretic learning bounds’, ‘Instruction tuning, RLHF, and RLAIF’, ‘Interpretability and explainability’, ‘Inverse RL and imitation learning’, ‘Knowledge representation and reasoning’, ‘Lambda calculus and functional programming theory’, ‘Large language models and foundation models’, ‘Learning PDE solvers and neural operators’, ‘Logic in computer science’, ‘MLOps and ML systems’, ‘MLOps, deployment, and monitoring’, ‘Machine learning and data-driven methods’, ‘Mechanistic interpretability of neural networks’, ‘Meta-learning and few-shot learning’, ‘Model compression, pruning, quantization, distillation’, ‘Multi-task and continual learning’, ‘Multiagent systems and game-theoretic AI’, ‘Natural language processing and LLMs’, ‘Natural language processing and computational linguistics’, ‘Networking for AI workloads’, ‘Neural architecture search and AutoML’, ‘Neurosymbolic AI and program synthesis’, ‘Nonconvex optimization and landscape analysis’, ‘Offline and batch RL’, ‘Online algorithms and competitive analysis’, ‘Online learning and bandits’, ‘Operating systems and kernels’, ‘Optimization for ML’, ‘PAC learning and VC theory’, ‘Parallel and high-performance computing’, ‘Parameterized and fine-grained complexity’, ‘Perception, SLAM, and control in robotics’, ‘Physics-informed ML and operator learning’, ‘Planning, search, and constraint programming’, ‘Policy gradients and actor-critic methods’, ‘Post-Moore architectures and neuromorphic computing’, ‘Post-quantum cryptography’, ‘Privacy engineering and compliance’, ‘Probabilistic modeling and graphical models’, ‘Probabilistic programming and inference engines’, ‘Program analysis and synthesis’, ‘Program synthesis and neuro-symbolic systems’, ‘Programming languages and compilers’, ‘Proof assistants and program verification’, ‘Protein structure prediction and generative biology’, ‘Quantum programming languages and compilers’, ‘Randomized algorithms and probabilistic methods’, ‘Recommender systems and personalization’, ‘Reinforcement learning (model-free and model-based)’, ‘Reinforcement learning and decision-making’, ‘Responsible and safe AI’, ‘Retrieval-augmented generation and tool use’, ‘Robotics and autonomous systems’, ‘Robustness and adversarial ML’, ‘Runtime systems and garbage collection’, ‘Safe and risk-sensitive RL’, ‘Scientific computing and numerical methods’, ‘Scientific machine learning’, ‘Second-order and adaptive optimization’, ‘Secure hardware and side-channel defenses’, ‘Security and privacy’, ‘Self-supervised and contrastive learning’, ‘Semi-supervised and weakly supervised learning’, ‘Serverless computing and microservices’, ‘Smart contracts and formal verification’, ‘Social computing and computational social science’, ‘Software engineering and software architecture’, ‘Software testing, fuzzing, and reliability’, ‘Software-defined networking and NFV’, ‘Speech and audio processing’
	‘Speech, audio, and music ML’, ‘Stability and compression-based generalization’, ‘Stochastic gradient methods and variance reduction’, ‘Storage systems and filesystems’, ‘Supervised learning (classification and regression)’, ‘Surrogate modeling and uncertainty quantification’, ‘Systems and network security’, ‘Theoretical ML and generalization’, ‘Theoretical ML and statistical learning theory’, ‘Time series forecasting and anomaly detection’, ‘Transaction processing and concurrency control’, ‘Transfer learning and domain adaptation’, ‘Transformers and sequence modeling’, ‘Trustworthy AI and robustness’, ‘Type systems and program semantics’, ‘Ubiquitous and wearable computing’, ‘Unsupervised learning (clustering, density estimation)’, ‘Usable security and human factors’, ‘VAEs, GANs, normalizing flows, and diffusion models’, ‘Visualization and visual analytics’, ‘Web and internet protocols’, ‘Wireless, mobile, and 5G/6G networks’, ‘World models and model-based planning’

Appendix

Table 7: Subjects and their expanded topic sets used for creating Chimera .

Example from Chimera
Subject: Math
Topic: Sieve methods and distribution in arithmetic progressions
Problem:
Let $q\geq 1$ be fixed, and let $a$ be an integer coprime to $q$ . For $x\geq e^{e}$ , fix $u\in[2,3]$ , and set $z=\frac{x^{1/u}}{(\log x)^{A}},\hskip 18.49988ptA\geq 5.$ Let $\mathcal{P}$ denote the set of primes not dividing $q$ . Define the sifted count in the arithmetic progression $S(x;q,a;z)=\#\Bigl\{n\leq x:n\equiv a\pmod{q},\ \text{and every prime divisor }p\mid n\text{ satisfies }p\geq z\Bigr\}.$ Prove that, as $x\to\infty$ , uniformly in $a$ with $(a,q)=1$ (and for fixed $q$ ), the following two-sided sieve bounds hold: $\frac{x}{\phi(q)}V(z)f(u)+O\!\left(\frac{x}{\phi(q)}\frac{V(z)}{\log z}\right)-O(z^{u})\leq S(x;q,a;z)\leq\frac{x}{\phi(q)}V(z)F(u)+O\!\left(\frac{x}{\phi(q)}\frac{V(z)}{\log z}\right)+O(z^{u}),$ where • $V(z)=\prod_{\begin{subarray}{c}p<z\\ p\nmid q\end{subarray}}\left(1-\frac{1}{p}\right),$ • for $u\in[2,3]$ , $f(u)=\frac{2e^{\gamma}\log(u-1)}{u},\hskip 18.49988ptF(u)=\frac{2e^{\gamma}\bigl(1+\log(u-1)\bigr)}{u},$ • and $\gamma$ is the Euler–Mascheroni constant. Moreover, deduce the asymptotic form $S(x;q,a;z)=\frac{x}{\phi(q)}\frac{e^{-\gamma}}{\log z}\left(\prod_{p\mid q}\left(1-\frac{1}{p}\right)^{-1}\right)\left(\Phi(u)+O\!\left(\frac{1}{\log z}\right)\right)+O(z^{u}),$ with $\Phi(u)$ satisfying $f(u)\leq\Phi(u)\leq F(u),$ and in particular $S(x;q,a;z)=\frac{x}{\phi(q)\log z}\left(\prod_{p\mid q}\left(1-\frac{1}{p}\right)^{-1}\right)\left(e^{-\gamma}\Phi(u)+o(1)\right)+O(z^{u}).$
Answer:
For fixed $q\geq 1$ , with $(a,q)=1$ , $u\in[2,3]$ , and $z=\frac{x^{1/u}}{(\log x)^{A}},\hskip 18.49988ptA\geq 5,$ as $x\to\infty$ one has $\frac{x}{\phi(q)}V(z)f(u)+O\!\left(\frac{x}{\phi(q)}\frac{V(z)}{\log z}\right)-O(z^{u})\leq S(x;q,a;z)\leq\frac{x}{\phi(q)}V(z)F(u)+O\!\left(\frac{x}{\phi(q)}\frac{V(z)}{\log z}\right)+O(z^{u}).$

Appendix

Table 8: A data sample from Chimera (Subject: Math).

Example from Chimera
Subject: Computer Science
Topic: Algorithms and randomized algorithms
Problem:
Consider the following idealized version of the Karger–Stein randomized contraction algorithm applied to an arbitrary simple undirected graph on $n\geq 2$ vertices that has a unique minimum cut. Let $t(n)=\frac{n}{\sqrt{2}}$ . Define the per-run cost $T(n)$ and per-run success probability $p(n)$ by the recurrences 1. $T(2)=0\hskip 9.24994pt\text{and}\hskip 9.24994ptT(n)=2T(t(n))+cn^{2}\hskip 9.24994pt\text{for all real }n\geq 2,$ where $c>0$ is a fixed constant. Assume independence between the two recursive calls and across all runs. 2. $p(2)=1\hskip 9.24994pt\text{and}\hskip 9.24994ptp(n)=1-\bigl(1-s(n)p(t(n))\bigr)^{2}\hskip 9.24994pt\text{for all real }n\geq 2,$ where $s(n)=\frac{t(n)\bigl(t(n)-1\bigr)}{n(n-1)}$ is the exact probability that the unique minimum cut survives the contraction from $n$ to $t(n)$ vertices. Let $r(n,\delta)$ be the minimum number of independent repetitions needed so that the overall failure probability is at most $\delta\in(0,1/2)$ , i.e., $r(n,\delta)=\min\left\{r\in\mathbb{N}:(1-p(n))^{r}\leq\delta\right\}.$ Define the minimal expected total work $W(n,\delta)=r(n,\delta)\,T(n).$ Determine the following, with exact leading constants: (a) An exact closed form for $T(n)$ . (b) The limit $L=\lim_{n\to\infty}p(n)\ln n.$ (c) The asymptotic behavior of $r(n,\delta)$ as $n\to\infty$ with $\delta$ fixed, including the exact leading constant: $r(n,\delta)\sim K\ln n\ln(1/\delta),$ and determine $K$ . (d) The asymptotic behavior of $W(n,\delta)$ as $n\to\infty$ with $\delta$ fixed, including the exact leading constant $C(c)$ such that $W(n,\delta)\sim C(c)\,n^{2}(\ln n)^{2}\ln(1/\delta).$

Answer:

Appendix

Table 9: A data sample from Chimera (Subject: Computer Science).

Example from Chimera
Subject: Chemistry
Topic: Green solvents and sustainable synthesis
Problem:
Consider a unimolecular, homogeneous reaction $\mathrm{A}\rightarrow\mathrm{P}$ at $298\ \mathrm{K}$ performed in a binary, ideal liquid mixture of two green solvents: $\mathrm{S}_{1}=\text{gamma-valerolactone (GVL)},\hskip 18.49988pt\mathrm{S}_{2}=\text{2-methyltetrahydrofuran (2-MeTHF)}.$ Let $x\in[0,1]$ denote the mole fraction of $\mathrm{S}_{1}$ (GVL) in the solvent mixture. Assume the following models and parameters hold exactly: Kinetics. The first-order rate constant depends on solvent polarity and viscosity via $k(x)=k_{0}\cdot\exp\!\big[a(ET(x)-ET_{\mathrm{ref}})\big]\cdot[\eta(x)]^{-\beta},$ with $k_{0}=1.0\times 10^{-4}\ \mathrm{s^{-1}},\hskip 9.24994pta=12,\hskip 9.24994ptET_{\mathrm{ref}}=0.30,\hskip 9.24994pt\beta=0.5.$ The Kamlet–Taft polarity parameter mixes linearly: $ET(x)=xET_{1}+(1-x)ET_{2},$ with $ET_{1}=0.60,\hskip 18.49988ptET_{2}=0.20.$ The viscosity follows log-linear mixing: $\ln\eta(x)=x\ln\eta_{1}+(1-x)\ln\eta_{2},$ with $\eta_{1}=1.8\ \mathrm{mPa\cdot s},\hskip 18.49988pt\eta_{2}=0.6\ \mathrm{mPa\cdot s}.$ All logarithms are natural logs, and viscosities in $\mathrm{mPa\cdot s}$ are used consistently in the model. Solubility and solvent requirement. The solubility of A in the mixed solvent is given by a Hildebrand-type correlation: $S(x)=S_{0}\cdot\exp\!\big[-K(\delta(x)-\delta_{A})^{2}\big],$ with $S_{0}=5.0\ \mathrm{mol\cdot L^{-1}},\hskip 9.24994ptK=0.10\ \mathrm{MPa^{-1}},\hskip 9.24994pt\delta_{A}=22\ \mathrm{MPa^{0.5}}.$ The mixture solubility parameter obeys $\delta(x)^{2}=x\delta_{1}^{2}+(1-x)\delta_{2}^{2},$ with $\delta_{1}=25\ \mathrm{MPa^{0.5}},\hskip 18.49988pt\delta_{2}=17\ \mathrm{MPa^{0.5}}.$ The minimum solvent volume needed to dissolve $1$ mol of A is $V_{\min}(x)=\frac{1}{S(x)}\ \mathrm{L}.$ The mixture density is linear in $x$ : $\rho(x)=x\rho_{1}+(1-x)\rho_{2},$ with $\rho_{1}=1.20\ \mathrm{kg\cdot L^{-1}},\hskip 18.49988pt\rho_{2}=0.86\ \mathrm{kg\cdot L^{-1}}.$
Environmental intensity. The cradle-to-gate GWP of the mixed solvent, per unit solvent mass, is linear in $x$ : $g_{\mathrm{mix}}(x)=xg_{1}+(1-x)g_{2},$ with $g_{1}=1.2\ \mathrm{kg\ CO_{2}e\ per\ kg},\hskip 18.49988ptg_{2}=3.0\ \mathrm{kg\ CO_{2}e\ per\ kg}.$ Process objective and constraint. Per mole of product P (assuming 1:1 stoichiometry $\mathrm{A}\rightarrow\mathrm{P}$ and quantitative isolation after reaction), the GWP attributable to solvent usage is $G(x)=m_{\mathrm{solv}}(x)\cdot g_{\mathrm{mix}}(x),$ where $m_{\mathrm{solv}}(x)=\rho(x)\,V_{\min}(x)=\frac{\rho(x)}{S(x)}\hskip 9.24994pt(\mathrm{kg\cdot mol^{-1}}).$ The process must achieve conversion $X^{\ast}=0.99$ within $T_{\max}=3600\ \mathrm{s}.$ For first-order kinetics, this requires $k(x)\geq k_{\mathrm{req}},\hskip 18.49988ptk_{\mathrm{req}}=-\frac{\ln(1-X^{\ast})}{T_{\max}}.$ Task. Determine the unique $x^{\ast}\in[0,1]$ that minimizes $G(x)$ subject to the kinetic constraint $k(x)\geq k_{\mathrm{req}}$ . Report: (1) A proof of uniqueness of the optimizer in the feasible set. (2) An explicit expression for $x^{\ast}$ in terms of the model parameters. (3) The numerical values of $x^{\ast}$ , $S(x^{\ast})$ , $\rho(x^{\ast})$ , $m_{\mathrm{solv}}(x^{\ast})$ , $g_{\mathrm{mix}}(x^{\ast})$ , and the minimal GWP per mole $G(x^{\ast})$ in $\mathrm{kg\ CO_{2}e\cdot mol^{-1}}$ .
Answer:

Appendix

Table 10: A data sample from Chimera (Subject: Chemistry).

Example from Chimera
Subject: Physics
Topic: Rydberg atoms and quantum simulation
Problem:
Consider an infinite one-dimensional chain of identical neutral atoms separated by a fixed lattice spacing $a=5\ \mu\mathrm{m}.$ Each atom has a ground state $\|g\rangle$ and two Rydberg states $\|r_{1}\rangle$ and $\|r_{2}\rangle$ with van der Waals interactions $V_{1}(r)=\frac{C_{6,1}}{r^{6}},\hskip 18.49988ptV_{2}(r)=\frac{C_{6,2}}{r^{6}},$ between atoms both in $\|r_{1}\rangle$ or both in $\|r_{2}\rangle$ , respectively. Two far-off-resonant dressing lasers ( $i=1,2$ ) couple $\|g\rangle$ to $\|r_{i}\rangle$ with Rabi frequencies $\Omega_{i}$ and detunings $\Delta_{i}$ , and the single-atom rotating-frame Hamiltonian term is $\Delta_{i}\|r_{i}\rangle\langle r_{i}\|+\frac{\Omega_{i}}{2}\left(\|g\rangle\langle r_{i}\|+\mathrm{h.c.}\right).$ Assume weak dressing, i.e., $\left\|\frac{\Omega_{i}}{\Delta_{i}}\right\|\ll 1,$ and neglect many-body (three- or more-body) terms beyond pairwise additivity in the effective ground-manifold Hamiltonian. Given numerical parameters: $\frac{\Omega_{1}}{2\pi}=\frac{\Omega_{2}}{2\pi}=3.0\ \mathrm{MHz},$ $\frac{\Delta_{1}}{2\pi}=+40.0\ \mathrm{MHz},$ $\frac{\Delta_{2}}{2\pi}=-\frac{40.0}{\sqrt{2}}\ \mathrm{MHz}=-28.2842712474619\ \mathrm{MHz},$ $\frac{C_{6,1}}{h}=+1250\ \mathrm{GHz\ \mu m^{6}},$ $\frac{C_{6,2}}{h}=-312.5\ \mathrm{GHz\ \mu m^{6}}.$ (a) For a single laser $i$ and two atoms separated by $r$ , derive to fourth order in $\Omega_{i}/\Delta_{i}$ the effective two-body interaction $U_{i}(r)$ that shifts the $\|gg\rangle$ energy in the ground manifold, and show that $U_{i}(r)=\frac{\Omega_{i}^{4}}{8\Delta_{i}^{3}}\,\frac{V_{i}(r)}{2\Delta_{i}+V_{i}(r)},$ where $V_{i}(r)=\frac{C_{6,i}}{r^{6}}.$ State all approximations used.
(b) Show that as $r\to\infty$ , $U_{i}(r)\approx\frac{\Omega_{i}^{4}C_{6,i}}{16\Delta_{i}^{4}r^{6}},$ and, using the numbers above, prove that the total far-tail $1/r^{6}$ coefficient $U_{1}(r)+U_{2}(r)$ vanishes exactly. (c) Using the exact expression in (a) and the parameters above, compute in Hz units the total effective interaction $U(r)=U_{1}(r)+U_{2}(r)$ for $r=a\hskip 9.24994pt\text{and}\hskip 9.24994ptr=2a.$ Provide exact closed-form expressions (in radicals and rational numbers) and numerical values to at least $10^{-3}\ \mathrm{Hz}$ accuracy. (d) Report the ratio $\frac{J_{2}}{J_{1}},$ where $J_{1}=U(a),\hskip 18.49988ptJ_{2}=U(2a),$ as a decimal number with at least four significant digits. All quantities must be computed in frequency units (Hz). Assume pairwise additivity is valid at the stated parameters and that higher-order corrections $O\!\left(\frac{\Omega^{6}}{\Delta^{5}}\right)$ are negligible.
Answer:

Appendix

Table 11: A data sample from Chimera (Subject: Physics).

Example from Chimera
Subject: Literature
Topic: Archive studies and critical bibliography
Problem:
Consider an authority-control deduplication problem in archive studies and critical bibliography. Let $R$ be a finite set of bibliographic records, each record $r\in R$ having fields author (a free-text personal-name string in either “Family, Given …” form), title (a free-text string), and year (an integer). Define the following normalization and similarity functions: 1) Unicode and token normalization For any string $x$ , let $\mathrm{normalize}(x)$ be the result of: (a) Unicode NFKD decomposition and removal of all combining diacritical marks. (b) Lowercasing. (c) Replacing all non-alphanumeric characters (including punctuation and hyphens) by a single ASCII space. (d) Collapsing multiple spaces to one space, then trimming leading/trailing spaces. 2) Author key $\mathrm{AKey}(r)$ Given $r.\mathrm{author}$ in personal-name form “Family, Given …” (assume every record here has exactly one comma, indicating this form), let: (a) Let $L=\mathrm{normalize}(\text{left-of-comma}(r.\mathrm{author}))$ , then remove all spaces from $L$ to get $L^{\ast}$ . (b) Let $G=\mathrm{normalize}(\text{right-of-comma}(r.\mathrm{author}))$ , and let $g$ be the first ASCII letter in $G$ (if none exists, define $g$ as the empty string; in the dataset below every record has one). (c) Define $\mathrm{AKey}(r)=L^{\ast}\,\\|\,g,$ where $\\|$ denotes string concatenation. Define $A(i,j)=\begin{cases}1&\text{if }\mathrm{AKey}(i)=\mathrm{AKey}(j),\\ 0&\text{otherwise}.\end{cases}$ 3) Title tokenization and Jaccard Let stopword set $S=\{a,an,the,of,and,for,to,into,in,on,with,by,from\}.$ Tokenize a title $t$ as follows: apply $\mathrm{normalize}(t)$ , split on spaces to tokens, discard any token in $S$ , and discard any empty token. Let $T(r)$ be the resulting set of unique tokens for record $r$ . Define $J(i,j)=\frac{\|T(i)\cap T(j)\|}{\|T(i)\cup T(j)\|}.$ 4) Year similarity Define $Y(i,j)=\begin{cases}1&\text{if }i.\mathrm{year}=j.\mathrm{year},\\ 0&\text{otherwise}.\end{cases}$ 5) Weighted similarity Let weights be $w_{1}=\frac{13}{25},\hskip 18.49988ptw_{2}=\frac{19}{50},\hskip 18.49988ptw_{3}=\frac{1}{10}.$ For records $i,j$ define $s(i,j)=w_{1}A(i,j)+w_{2}J(i,j)+w_{3}Y(i,j).$
6) Mergeability at threshold $\tau$ Fix $\tau_{0}=\frac{41}{50}.$ Define an undirected graph $G_{\tau}$ on vertex set $R$ with an edge $i$ — $j$ iff $s(i,j)\geq\tau.$ Let $P_{\tau}$ be the partition of $R$ into the connected components of $G_{\tau}$ . Dataset $R$ (9 records) • A1: author = “Smith, John”; title = “Public archive origins and governance”; year = 1999. • A2: author = “Smith, J.”; title = “Public archive origins and governance: an enquiry”; year = 1999. • A3: author = “Smith, John”; title = “Governance of the public archive: its origins”; year = 1999. • B1: author = “García Márquez, Ana-Luisa”; title = “Metadata for colonial archives: a critical bibliography”; year = 2015. • B2: author = “Garcia Marquez, Ana Luisa”; title = “A critical bibliography of colonial archives metadata”; year = 2015. • B3: author = “García-Márquez, Ana L.”; title = “Colonial archives metadata: toward a critical bibliography”; year = 2015. • C1: author = “Lefèvre, Paul”; title = “Archival silences and the politics of records retention”; year = 2008. • C2: author = “Lefevre, P.”; title = “Archival silence and politics of records retention”; year = 2008. • C3: author = “Lefèvre, Paul”; title = “Politics of records retention and archival silences”; year = 2008. Tasks (a) Prove that for any finite $R$ and any $\tau$ , the partition $P_{\tau}$ is unique and independent of the order in which pairwise merges are performed, i.e., clustering by repeatedly merging any pair with $s\geq\tau$ until no merges remain yields the same partition as the connected components of $G_{\tau}$ . (b) For the dataset above, compute $\mathrm{AKey}$ for each record, then compute $J(i,j)$ and $s(i,j)$ for all pairs with $A(i,j)=1$ , determine all edges in $G_{\tau_{0}}$ , and output the partition $P_{\tau_{0}}$ explicitly as a set of disjoint record-id blocks. (c) Let $\tau^{\ast}$ be the supremum of all $\tau\geq\tau_{0}$ such that $P_{\tau^{\prime}}=P_{\tau_{0}}\hskip 9.24994pt\text{for every }\tau^{\prime}\in[\tau_{0},\tau].$ Prove that for any weighted undirected graph, for a fixed component $H$ of $G_{\tau_{0}}$ , the largest $\tau$ at which $H$ remains connected equals the bottleneck value of a maximum spanning tree (MaxST) of $H$ , i.e., the maximum over all spanning trees of the minimum edge weight. Then compute $\tau^{\ast}$ for the dataset.
Answer:

Appendix

Table 12: A data sample from Chimera (Subject: Literature).

Example from Chimera
Subject: History
Topic: Business history and financial history
Problem:
Consider two nineteenth-century bimetallic countries $i\in\{A,B\}$ with legal gold/silver mint ratios $M_{i}>0$ , defined as the number of units of domestic silver coin that constitute the legal-tender face value of one unit of domestic gold coin. Let the unit of account be the domestic silver coin in each country, so one unit of silver coin has face value $1$ . For each country $i$ and metal $m\in\{g,s\}$ (gold or silver), coining (minting) one unit of bullion into coin incurs an ad valorem cost $\kappa_{m}^{i}\in[0,1)$ and a deterministic delay $T_{\mathrm{coin}}(m,i)\geq 0$ before coin is delivered; melting one unit of coin into bullion incurs an ad valorem loss $\mu_{m}^{i}\in[0,1)$ and a deterministic delay $T_{\mathrm{melt}}(m,i)\geq 0$ before bullion is delivered. Shipping one unit of bullion of metal $m$ from country $i$ to country $j$ incurs an ad valorem loss $\tau_{m}^{i\to j}\in[0,1)$ , with $\tau_{m}^{i\to i}=0$ . The risk-free interest rate is $r\geq 0$ (continuous compounding). The world bullion relative price $R>0$ is defined as the number of units of silver bullion obtainable per unit of gold bullion via the bullion market, and is assumed constant over the time scales involved (or locked in via forward contracts), with zero bid–ask spread. Arbitrageurs can freely access bullion markets and mints subject to the costs and delays above, and can repatriate coin value between $A$ and $B$ at par in silver units via bills of exchange (so there is no return shipping cost of coin; only bullion shipping is relevant). Define, for each ordered pair $(i,j)\in\{A,B\}\times\{A,B\}$ , the total processing delays: $T_{\mathrm{GS}}(i,j)=T_{\mathrm{melt}}(g,i)+T_{\mathrm{coin}}(s,j),\hskip 18.49988ptT_{\mathrm{SG}}(i,j)=T_{\mathrm{melt}}(s,i)+T_{\mathrm{coin}}(g,j),$ and the effective one-leg shipping retention factor $s_{i\to j}=\max\{1-\tau_{g}^{i\to j},\,1-\tau_{s}^{i\to j}\}\in(0,1],$ reflecting that an arbitrageur can choose to ship bullion either before or after exchanging metals, selecting the route with the higher retention. An arbitrageur can execute gold $\to$ silver loops (starting with $1$ unit of gold coin in $i$ , ending in silver coin units) or silver $\to$ gold loops (starting with $1$ unit of silver coin in $i$ , ending in silver coin units), choosing any origin $i$ and destination mint $j$ , incurring the corresponding costs, delays, and shipping. Working in present-value terms at $t=0$ , derive the necessary and sufficient no-arbitrage interval $[LB_{\mathrm{global}},\,UB_{\mathrm{global}}]$ for the world bullion ratio $R$ such that no riskless profit can be earned by any such loop. Express $LB_{\mathrm{global}}$ and $UB_{\mathrm{global}}$ in closed form as functions of the parameters. Then, specialize to the single-country case (ignore shipping and set $i=j$ and drop country indices) to simplify the band and provide $\ln(UB/LB)$ in the symmetric-parameters case where $\kappa_{g}=\kappa_{s}=\kappa$ , $\mu_{g}=\mu_{s}=\mu$ , and $T_{\mathrm{melt}}(g)=T_{\mathrm{melt}}(s)=T_{m}$ , $T_{\mathrm{coin}}(g)=T_{\mathrm{coin}}(s)=T_{c}$ .
Answer:
The global no-arbitrage interval for the world bullion price ratio $R$ is $LB_{\mathrm{global}}=\max_{i,j\in\{A,B\}}e^{-r\,[T_{\mathrm{melt}}(s,i)+T_{\mathrm{coin}}(g,j)]}\,(1-\mu_{s}^{i})\,s_{i\to j}\,(1-\kappa_{g}^{j})\,M_{j},$ $UB_{\mathrm{global}}=\min_{i,j\in\{A,B\}}\frac{M_{i}\,e^{r\,[T_{\mathrm{melt}}(g,i)+T_{\mathrm{coin}}(s,j)]}}{(1-\mu_{g}^{i})\,s_{i\to j}\,(1-\kappa_{s}^{j})},$ where $s_{i\to j}=\max\{1-\tau_{g}^{i\to j},\,1-\tau_{s}^{i\to j}\}.$ Special case (single country, no shipping). $LB=M\,(1-\mu_{s})\,(1-\kappa_{g})\,e^{-r\,[T_{\mathrm{melt}}(s)+T_{\mathrm{coin}}(g)]},\hskip 18.49988ptUB=\frac{M\,e^{r\,[T_{\mathrm{melt}}(g)+T_{\mathrm{coin}}(s)]}}{(1-\mu_{g})\,(1-\kappa_{s})}.$

Appendix

Table 13: A data sample from Chimera (Subject: History).

Example from Chimera
Subject: Biology
Topic: Structural biology (X-ray, NMR, cryo-EM)
Problem:
In single-particle cryo-EM, let $S(s)$ denote the true complex Fourier coefficient of the 3D signal at spatial frequency shell $s$ . Two independently reconstructed half-maps $F_{1}(s)$ and $F_{2}(s)$ are modeled as $F_{1}(s)=S(s)+N_{1}(s),\hskip 18.49988ptF_{2}(s)=S(s)+N_{2}(s),$ where $N_{1}(s),N_{2}(s)$ are zero-mean, circular complex Gaussian noise terms, independent of $S(s)$ . Assume stationarity across each shell: $\Phi_{S}(s)=\mathbb{E}\!\left[\|S(s)\|^{2}\right],\hskip 18.49988pt\Phi_{N}(s)=\mathbb{E}\!\left[\|N_{1}(s)\|^{2}\right]=\mathbb{E}\!\left[\|N_{2}(s)\|^{2}\right],$ and a known real-valued inter-half noise correlation coefficient $\rho(s)=\frac{\mathbb{E}\!\left[N_{1}(s)N_{2}(s)^{\ast}\right]}{\Phi_{N}(s)}\hskip 18.49988pt\text{with }-1<\rho(s)<1.$ Define the half-map Fourier shell correlation $c(s)=\frac{\mathbb{E}\!\left[F_{1}(s)F_{2}(s)^{\ast}\right]}{\sqrt{\mathbb{E}\!\left[\|F_{1}(s)\|^{2}\right]\mathbb{E}\!\left[\|F_{2}(s)\|^{2}\right]}}.$ Let $F_{\mathrm{avg}}(s)=\frac{F_{1}(s)+F_{2}(s)}{2}$ be the simple average (the “full map” in the two-half convention). Consider the linear MMSE estimator $\hat{S}(s)=H(s)\,F_{\mathrm{avg}}(s).$ Derive a closed-form expression for the optimal filter $H(s)$ that minimizes $\mathbb{E}\!\left[\left\|S(s)-H(s)F_{\mathrm{avg}}(s)\right\|^{2}\right]$ over complex scalar $H(s)$ , expressed solely in terms of $c(s)$ and $\rho(s)$ . Your result must hold for each shell independently. State all assumptions and show all steps. The final expression for $H(s)$ must be a closed-form function of $c(s)$ and $\rho(s)$ only and must reduce correctly to the uncorrelated-noise special case.
Answer:
$H(s)=\frac{2\,\bigl[c(s)-\rho(s)\bigr]}{\bigl(1-\rho(s)\bigr)\bigl(1+c(s)\bigr)},$ and for $\rho(s)=0$ , $H(s)=\frac{2\,c(s)}{1+c(s)}.$

Appendix

Table 14: A data sample from Chimera (Subject: Biology).

Example from Chimera
Subject: Linguistics
Topic: Syntax (generative, dependency, construction grammar)
Problem:
Let $n\geq 1$ and let $[n]=\{1,2,\dots,n\}$ be a fixed linear order of word positions. A dependency tree $D$ on $[n]$ is a directed tree $(V=[n],E)$ with a designated root $r\in[n]$ , where each node other than $r$ has exactly one incoming arc, and arcs are oriented from heads to dependents. For $i\in[n]$ , let $\mathrm{sub}_{D}(i)$ denote the set of nodes reachable from $i$ in $D$ (including $i$ ). Define the span of $i$ , $\mathrm{sp}_{D}(i)$ , as the set $\{j\in[n]:j\in\mathrm{sub}_{D}(i)\}.$ The dependency tree $D$ is projective if, for every arc $(h,d)\in E$ , every index $k$ strictly between $h$ and $d$ (i.e., $\min(h,d)<k<\max(h,d)$ ) belongs to $\mathrm{sp}_{D}(h)$ . An ordered binary head-annotated phrase-structure tree (PS tree) $P$ on $[n]$ is a full binary ordered rooted tree whose leaves, read left-to-right, are exactly $1,2,\dots,n$ in that order, and where each node $N$ (including leaves) has a head index $\mathrm{hd}(N)\in[n]$ such that: • For a leaf labeled $i$ , $\mathrm{hd}(\text{leaf }i)=i$ . • For an internal node $N$ with left child $L$ and right child $R$ , $\mathrm{hd}(N)\in\{\mathrm{hd}(L),\mathrm{hd}(R)\}$ . Define the head child $H(N)$ as the unique child of $N$ whose head equals $\mathrm{hd}(N)$ ; the other child is the nonhead child. Define the canonical class $\mathrm{HCB}(n)$ of head-annotated PS trees to be those $P$ that satisfy, for each head index $i\in[n]$ , the following canonical spine condition. Let $S_{i}$ be the maximal path from leaf $i$ upward through consecutive nodes $M$ with $\mathrm{hd}(M)=i$ . Along $S_{i}$ from bottom to top: • Every internal node on $S_{i}$ with a nonhead child whose head $d$ satisfies $d<i$ (a left-dependent) appears before any internal node on $S_{i}$ with a nonhead child whose head $d$ satisfies $d>i$ (a right-dependent). • Among left-dependent attachments along $S_{i}$ , the heads $d$ appear in strictly increasing order of $d$ . • Among right-dependent attachments along $S_{i}$ , the heads $d$ appear in strictly increasing order of $d$ . Additionally, if at a node on $S_{i}$ the nonhead child head is $d<i$ , then that nonhead child is the left child; if $d>i$ , it is the right child. Define the mapping $F$ from the set $\mathrm{ProjDep}(n)$ of projective dependency trees on $[n]$ to $\mathrm{HCB}(n)$ recursively as follows. For $D\in\mathrm{ProjDep}(n)$ and $i\in[n]$ , let $L_{i}=\{d\in[n]:d<i\text{ and }(i,d)\in E\},\hskip 18.49988ptR_{i}=\{d\in[n]:d>i\text{ and }(i,d)\in E\},$ sorted increasingly as $l_{1}<\dots<l_{k}$ and $r_{1}<\dots<r_{m}$ . Define $T(i)$ by recursion on the size of $\mathrm{sub}_{D}(i)$ : • If $\mathrm{sub}_{D}(i)=\{i\}$ , then $T(i)$ is the single leaf $i$ with $\mathrm{hd}=i$ . • Otherwise, after constructing $T(l)$ and $T(r)$ for all $l\in L_{i}$ and $r\in R_{i}$ , build a sequence of binary nodes with head $i$ by first attaching left dependents in increasing order on the left and then right dependents in increasing order on the right: $S_{0}:=\text{leaf }i\text{ with }\mathrm{hd}=i.$ For $t=1,\dots,k$ , set $S_{t}$ to be a node with left child $T(l_{t})$ , right child $S_{t-1}$ , $\mathrm{hd}=i$ . For $s=1,\dots,m$ , set $S_{k+s}$ to be a node with left child $S_{k+s-1}$ , right child $T(r_{s})$ , $\mathrm{hd}=i$ . Define $T(i):=S_{k+m}$ . Finally, if $r$ is the root of $D$ , define $F(D):=T(r)$ . Define the mapping $G$ from $\mathrm{HCB}(n)$ to $\mathrm{ProjDep}(n)$ as follows. Given $P\in\mathrm{HCB}(n)$ , let $r:=\mathrm{hd}(\mathrm{root}(P))$ . Define $E_{P}$ by adding, for each internal node $N$ with children $L$ and $R$ : • If $\mathrm{hd}(N)=\mathrm{hd}(L)$ and $\mathrm{hd}(R)=d\neq\mathrm{hd}(N)$ , add the arc $(\mathrm{hd}(N),d)$ . • If $\mathrm{hd}(N)=\mathrm{hd}(R)$ and $\mathrm{hd}(L)=d\neq\mathrm{hd}(N)$ , add the arc $(\mathrm{hd}(N),d)$ .
Define $G(P):=(V=[n],E_{P},\text{root }r)$ . Prove that $F$ is a bijection between $\mathrm{ProjDep}(n)$ and $\mathrm{HCB}(n)$ , and that $G$ is its inverse, i.e., $G\circ F=\mathrm{Id}_{\mathrm{ProjDep}(n)}\hskip 18.49988pt\text{and}\hskip 18.49988ptF\circ G=\mathrm{Id}_{\mathrm{HCB}(n)}.$
Answer:

Appendix

Table 15: A data sample from Chimera (Subject: Linguistics).

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