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ASTRO1
Astronomy
Coupled Peebles Recombination with $s+p$-Wave Dirac Dark-Matter Annihilation & Silk Damping
Cosmology / recombination and CMB damping
# Coupled Peebles Recombination with $s+p$-Wave Dirac Dark-Matter Annihilation & Silk Damping Consider a Friedmann–Lemaître–Robertson–Walker universe with: - Hubble constant: $H_0 = 67.36\text{ km s}^{-1}\text{ Mpc}^{-1}$ ($h \equiv H_0 / 100$) - Present CMB temperature: $T_0 = 2.7255\text{ K}$ - Physi...
{"task_id": "ASTRO1", "input_type": "self-contained numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "ASTRO1", "numeric_fields": ["residual_ionization_fraction_xe_at_z200", "baryon_matter_temper...
tasks/ASTRO1
trajectory/ASTRO1
1
0
1,048,576
ASTRO2
Astronomy
Kerr Superradiant $2p$ Scalar Cloud with Self-Gravity Shift, Boson Number & Annihilation GW Signal
Strong-field gravity / black-hole superradiance
# Kerr Superradiant $2p$ Scalar Cloud with Self-Gravity Shift, Boson Number & Annihilation GW Signal A stellar-mass Kerr black hole at cosmological redshift $z_{\text{cos}} = 0.065$ with line-of-sight peculiar velocity $v_{\text{pec}} = -850.0\text{ km s}^{-1}$ (approaching the observer relative to the local Hubbl...
{"task_id": "ASTRO2", "input_type": "self-contained numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "ASTRO2", "numeric_fields": ["initial_horizon_angular_velocity_rad_s", "saturated_black_hole_...
tasks/ASTRO2
trajectory/ASTRO2
1
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ASTRO3
Astronomy
MIT Bag Model Strange Quark Star TOV Structure, ISM Dispersion & Bare-Surface Thermal Flux
Compact stars / relativistic stellar structure
# MIT Bag Model Strange Quark Star TOV Structure, ISM Dispersion & Bare-Surface Thermal Flux A bare strange quark star at distance $D = 2.40\text{ kpc}$ ($z = 0$) is governed by the massless-quark MIT Bag Model equation of state $P(\epsilon) = \frac{1}{3}(\epsilon - 4B)$ with bag constant $B = 62.0\text{ MeV fm}^{...
{"task_id": "ASTRO3", "input_type": "self-contained numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "ASTRO3", "numeric_fields": ["quark_star_radius_km", "quark_star_mass_msun", "ism_radio_dispe...
tasks/ASTRO3
trajectory/ASTRO3
1
0
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ASTRO4
Astronomy
Bonnor–Ebert Critical Stability Limit via Emden–Chandrasekhar ODE Turning Point & Submm Flux
Interstellar medium / gravitational stability
# Bonnor–Ebert Critical Stability Limit via Emden–Chandrasekhar ODE Turning Point & Submm Flux A pressure-bounded, self-gravitating isothermal sphere ($P = c_s^2\rho$, $c_s = \sqrt{\frac{k_B T}{\mu m_u}}$) has radial density profile $\rho(\xi) = \rho_c e^{-\psi(\xi)}$ where $\xi \equiv r \sqrt{4\pi G\rho_c} / c_s$ and...
{"task_id": "ASTRO4", "input_type": "self-contained numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "ASTRO4", "numeric_fields": ["critical_dimensionless_radius_xi_crit", "critical_bonnor_ebert_...
tasks/ASTRO4
trajectory/ASTRO4
1
0
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ASTRO5
Astronomy
Eccentric Exoplanet Orbital Insolation: Flux-Averaged vs. Time-Averaged Temperatures & Periastron Flux
Exoplanets / orbital irradiation and radiative transfer
# Eccentric Exoplanet Orbital Insolation: Flux-Averaged vs. Time-Averaged Temperatures & Periastron Flux An exoplanet of radius $R_p = 1.1\,R_J$ with Bond albedo $A_B = 0.22$ and full global heat redistribution (uniform spherical reradiation over $4\pi$ steradians with unit emissivity) orbits a host star of luminosity...
{"task_id": "ASTRO5", "input_type": "self-contained numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "ASTRO5", "numeric_fields": ["flux_averaged_equilibrium_temp_k", "time_averaged_instantaneous...
tasks/ASTRO5
trajectory/ASTRO5
1
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BIO1
Biology
Theorell–Chance enzyme kinetics and coupled carrier-buffered thermodynamics
Enzyme kinetics and biochemical networks
# BIO1: Theorell–Chance enzyme kinetics and coupled carrier-buffered thermodynamics An alcohol dehydrogenase isoform of total concentration $E_{\text{tot}} = 0.04\text{ }\mu\text{M}$ operates via the compulsory-order Theorell–Chance mechanism without a kinetically significant central ternary complex: 1. $E + A \rightl...
{"task_id": "BIO1", "title": "Theorell–Chance enzyme kinetics and coupled carrier-buffered thermodynamics", "category": "Enzyme kinetics and biochemical networks", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {"t...
tasks/BIO1
trajectory/BIO1
1
0
1,048,576
BIO2
Biology
Single-molecule work fluctuations with Donnan partitioning
Molecular biophysics and nonequilibrium thermodynamics
# BIO2: Single-molecule work fluctuations with Donnan partitioning In an optical-tweezers single-molecule RNA unfolding/refolding experiment (with energies measured in units of $k_B T$), the forward unfolding work distribution $P_F(W)$ is Gaussian with mean $\mu_F = 52.3000\,k_B T$ and variance $\sigma_F^2 = 19.6000\,...
{"task_id": "BIO2", "title": "Single-molecule work fluctuations with Donnan partitioning", "category": "Molecular biophysics and nonequilibrium thermodynamics", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {"type...
tasks/BIO2
trajectory/BIO2
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BIO3
Biology
Eyring ion-channel permeation with thermal and electrolyte linkage
Cellular electrophysiology and ion transport
# BIO3: Eyring ion-channel permeation with thermal and electrolyte linkage A monovalent cation channel ($z = +1$) is modeled by an Eyring two-barrier, one-binding-site free-energy profile spanning fractional electrical distance $x \in [0, 1]$ from the extracellular side ($x = 0$) to the intracellular side ($x = 1$), w...
{"task_id": "BIO3", "title": "Eyring ion-channel permeation with thermal and electrolyte linkage", "category": "Cellular electrophysiology and ion transport", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {"type":...
tasks/BIO3
trajectory/BIO3
1
0
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BIO4
Biology
Underdominant-inversion traveling waves with viscoelastic and homeostatic coupling
Population genetics and evolutionary dynamics
# BIO4: Underdominant-inversion traveling waves with viscoelastic and homeostatic coupling A chromosomal inversion that is deleterious in karyotypic heterozygotes (selection reduction $s_u = 0.03$) but advantageous in inversion homozygotes (selection advantage $s_b = 0.09$) spreads across a 1D habitat with dispersal d...
{"task_id": "BIO4", "title": "Underdominant-inversion traveling waves with viscoelastic and homeostatic coupling", "category": "Population genetics and evolutionary dynamics", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_...
tasks/BIO4
trajectory/BIO4
1
0
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BIO5
Biology
Soil nitrification–denitrification with crowding and mass-transfer coupling
Ecological biogeochemistry and nutrient cycling
# BIO5: Soil nitrification–denitrification with crowding and mass-transfer coupling In a hypoxic soil aggregate at dissolved oxygen concentration $[\text{O}_2] = 15.0\text{ }\mu\text{M}$, aerobic nitrification produces nitrate ($\text{NO}_3^-$) at rate $R_{\text{nit}} = V_{\text{nit}}[\text{O}_2]/(K_{O,\text{nit}} + [...
{"task_id": "BIO5", "title": "Soil nitrification–denitrification with crowding and mass-transfer coupling", "category": "Ecological biogeochemistry and nutrient cycling", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schem...
tasks/BIO5
trajectory/BIO5
1
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CHEM1
Chemistry
Electrochemical impedance with coupled conformational equilibrium
Electrochemistry / impedance spectroscopy
# CHEM1 ## Problem Specification Electrochemical impedance spectroscopy (EIS) is recorded at the equilibrium potential ($E = E^{0\prime}$, $T = 298.15\text{ K}$) on a planar electrode of area $A = 0.2000\text{ cm}^2$ with uncompensated solution resistance $R_s = 18.00\,\Omega$ and double-layer capacitance $C_{\text{d...
{"task_id": "CHEM1", "input_type": "self-contained numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "CHEM1", "numeric_fields": ["charge_transfer_resistance_ohm", "rc_characteristic_omega_rad_s",...
tasks/CHEM1
trajectory/CHEM1
2
1
1,048,576
CHEM2
Chemistry
Polyelectrolyte hydrogel swelling and nonlinear mechanics
Polymer / soft-matter thermodynamics
# CHEM2 ## Problem Specification An ionic tetrafunctional polyelectrolyte hydrogel network crosslinked in the dry reference state ($\phi_{2,0} = 1$) has dry polymer density $\rho_2 = 1.150\text{ g cm}^{-3}$ and number-average molar mass between crosslinks $M_c = 8500.0\text{ g mol}^{-1}$ (active elastic chain density...
{"task_id": "CHEM2", "input_type": "self-contained numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "CHEM2", "numeric_fields": ["equilibrium_polymer_volume_fraction", "chi_integrated_swelling_ra...
tasks/CHEM2
trajectory/CHEM2
2
1
1,048,576
CHEM3
Chemistry
Radical-cation Hückel molecular-orbital analysis
Theoretical / quantum chemistry
# CHEM3 ## Problem Specification The planar 13-azaphenalenyl radical cation ($\text{C}_{12}\text{NH}_9^{\bullet +}$, point group $D_{3h}$, $13\text{ }\pi$ electrons) consists of three ortho- and peri-fused six-membered rings sharing a central trivalent nitrogen atom ($\text{N}_{13}$, contributing two $\pi$ electrons ...
{"task_id": "CHEM3", "input_type": "self-contained numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "CHEM3", "numeric_fields": ["total_pi_energy_beta", "delocalization_energy_beta", "somo_huckel...
tasks/CHEM3
trajectory/CHEM3
2
0
1,048,576
CHEM4
Chemistry
Wilkinson-catalyzed hydrogenation microkinetics
Organometallic catalysis / reaction kinetics
# CHEM4 ## Problem Specification Homogeneous hydrogenation of a macrocyclic alkene by Wilkinson's catalyst ($\text{L} = \text{PPh}_3$) at $T = 298.15\text{ K}$ proceeds via four coupled steps, where the substrate exists in rapid conformational pre-equilibrium in solution at total analytical concentration $C_{\text{al...
{"task_id": "CHEM4", "input_type": "self-contained numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "CHEM4", "numeric_fields": ["dihydride_catalyst_fraction_fc3", "resting_state_fraction_fc1", "...
tasks/CHEM4
trajectory/CHEM4
2
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1,048,576
CHEM5
Chemistry
Non-ideal ammonia synthesis using a cubic equation of state
Thermodynamics / reaction equilibrium
# CHEM5 ## Problem Specification A stoichiometric feed of $1\text{ mol N}_2$ and $3\text{ mol H}_2$ reacts to equilibrium at $T = 700.0\text{ K}$: $$\text{N}_2(g) + 3\,\text{H}_2(g) \rightleftharpoons 2\,\text{NH}_3(g)$$ At the reference temperature $T_{\text{ref}} = 298.15\text{ K}$ ($P^\circ = 1\text{ bar}$), the s...
{"task_id": "CHEM5", "input_type": "self-contained numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "CHEM5", "numeric_fields": ["equilibrium_constant_ka_700k", "nh3_pure_fugacity_coeff_300bar", ...
tasks/CHEM5
trajectory/CHEM5
2
0
1,048,576
EARTH1
Earth science
Inverse Dieterich–Ruina Nonlinear Phase-Plane Excursion, Hopf Bifurcation, and Rebound Peak
Earthquake physics / rate-and-state fault dynamics
# EARTH1 — Inverse Dieterich–Ruina Nonlinear Phase-Plane Excursion, Hopf Bifurcation, and Rebound Peak - **Problem ID**: `EARTH1` - **Domain**: Earthquake Physics & Fault Mechanics ## 1. Physical System & Regime A velocity-weakening fault patch under effective normal stress $\sigma_n' = 85.0\text{ MPa}$ obeys the **...
{"task_id": "EARTH1", "input_type": "self-contained advanced Earth science numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "EARTH1", "numeric_fields": ["critical_hopf_stiffness_GPa_m", "margina...
tasks/EARTH1
trajectory/EARTH1
1
1
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EARTH2
Earth science
Inverse Variable-Conductivity Anisotropic Porous Thermal Convection Sturm–Liouville BVP
Geothermal convection / porous-media stability
# EARTH2 — Inverse Variable-Conductivity Anisotropic Porous Thermal Convection Sturm–Liouville BVP - **Problem ID**: `EARTH2` - **Domain**: Geothermal Reservoir Engineering & Hydrothermal Convection ## 1. Physical System & Regime A horizontal fluid-saturated geothermal porous reservoir of thickness $H = 1645.0\text{...
{"task_id": "EARTH2", "input_type": "self-contained advanced Earth science numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "EARTH2", "numeric_fields": ["critical_basal_rayleigh_darcy_number", "...
tasks/EARTH2
trajectory/EARTH2
1
0
1,048,576
EARTH3
Earth science
Antarctic Ice-Shelf Firn-Aquifer Meltwater Percolation & Hydrofracture BVP
Glaciology / Antarctic ice-shelf hydrofracture
# EARTH3 — Antarctic Ice-Shelf Firn-Aquifer Meltwater Percolation & Hydrofracture BVP - **Problem ID**: `EARTH3` - **Domain**: Glaciology & Ice-Shelf Fracture Mechanics ## 1. Physical System & Regime A steady-state 1-D warm-summer ice-shelf perennial firn aquifer downward meltwater finger percolation, latent-heat re...
{"task_id": "EARTH3", "input_type": "self-contained advanced Earth science numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "EARTH3", "numeric_fields": ["J_melt_top", "J_slab_base", "I_slab", "l...
tasks/EARTH3
trajectory/EARTH3
1
0
1,048,576
EARTH4
Earth science
Hydrothermal-to-Abyssal Dissolved Iron Organic-Siderophore Stabilization BVP
Chemical oceanography / marine iron cycling
# EARTH4 — Hydrothermal-to-Abyssal Dissolved Iron Organic-Siderophore Stabilization BVP - **Problem ID**: `EARTH4` - **Domain**: Chemical Oceanography & Trace Metal Cycling ## 1. Physical System & Regime A steady-state 1-D abyssal ocean benthic nepheloid layer dissolved $\text{Fe(III)}$ competition between irreversi...
{"task_id": "EARTH4", "input_type": "self-contained advanced Earth science numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "EARTH4", "numeric_fields": ["J_Fe_benth", "J_FeL_abyss", "I_sid", "la...
tasks/EARTH4
trajectory/EARTH4
1
0
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EARTH5
Earth science
Polar Stratospheric Cloud (PSC) Type Ia NAT Denitrification & ClONO2-HCl Activation BVP
Atmospheric chemistry / polar ozone depletion
# EARTH5 — Polar Stratospheric Cloud (PSC) Type Ia NAT Denitrification & ClONO2-HCl Activation BVP - **Problem ID**: `EARTH5` - **Domain**: Stratospheric Chemistry & Polar Ozone Depletion ## 1. Physical System & Regime A steady-state 1-D polar vortex lower-stratospheric nitric acid trihydrate ($\text{NAT}$, $\text{H...
{"task_id": "EARTH5", "input_type": "self-contained advanced Earth science numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "EARTH5", "numeric_fields": ["J_ClN_top", "J_nat_base", "I_psc", "lamb...
tasks/EARTH5
trajectory/EARTH5
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ENG1
Engineering
Hartmann-channel ohmic dissipation and microwave-transducer heat flow
Magnetohydrodynamics and liquid-metal cooling
# ENG1: Hartmann-channel ohmic dissipation and microwave-transducer heat flow **Engineering Discipline**: Magnetohydrodynamics & Liquid-Metal Cooling ## Problem Statement An incompressible electrically conducting liquid metal flows steadily between two wide parallel electrically insulating plates located at y equal t...
{"task_id": "ENG1", "title": "Hartmann-channel ohmic dissipation and microwave-transducer heat flow", "category": "Magnetohydrodynamics and liquid-metal cooling", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {"ty...
tasks/ENG1
trajectory/ENG1
1
0
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ENG2
Engineering
Strip-foundation stress and causal optical-sensor pulse dispersion
Geotechnical mechanics and optical sensing
# ENG2: Strip-foundation stress and causal optical-sensor pulse dispersion **Engineering Discipline**: Geotechnical Mechanics & Elasticity ## Problem Statement A flexible strip footing of full width 2b applies a uniform vertical compressive pressure q over the interval minus b to plus b on the surface of a homogeneou...
{"task_id": "ENG2", "title": "Strip-foundation stress and causal optical-sensor pulse dispersion", "category": "Geotechnical mechanics and optical sensing", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {"type": "...
tasks/ENG2
trajectory/ENG2
1
1
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ENG3
Engineering
Gaussian source reverse water-filling and coupled quantum ground-state energy
Information theory and signal processing
# ENG3: Gaussian source reverse water-filling and coupled quantum ground-state energy **Engineering Discipline**: Signal Processing, Information Theory & Communications ## Problem Statement A three-dimensional memoryless Gaussian source has independent transform coefficients with standard deviations sigma_1, sigma_2,...
{"task_id": "ENG3", "title": "Gaussian source reverse water-filling and coupled quantum ground-state energy", "category": "Information theory and signal processing", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {...
tasks/ENG3
trajectory/ENG3
1
1
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ENG4
Engineering
Control-loop stability and coupled electrostatic–osmotic free energy
Control systems and electrochemical transduction
# ENG4: Control-loop stability and coupled electrostatic–osmotic free energy **Engineering Discipline**: Closed-Loop Servomechanisms, Phase Margin & Loop Dead-Time ## Problem Statement A closed-loop velocity-tracking servomechanism is governed by a canonical second-order open-loop plant scaled by an outer proportiona...
{"task_id": "ENG4", "title": "Control-loop stability and coupled electrostatic–osmotic free energy", "category": "Control systems and electrochemical transduction", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {"...
tasks/ENG4
trajectory/ENG4
1
0
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ENG5
Engineering
Fixed-bed adsorption breakthrough and Pockels-waveguide pulse propagation
Chemical process engineering and electro-optics
# ENG5: Fixed-bed adsorption breakthrough and Pockels-waveguide pulse propagation **Engineering Discipline**: Chemical & Process Engineering ## 1. Physical System & Modeling Specification An initially clean fixed-bed adsorber of length L_bed, bed voidage eps_b, and adsorbent particle density rho_p is fed at superfici...
{"task_id": "ENG5", "title": "Fixed-bed adsorption breakthrough and Pockels-waveguide pulse propagation", "category": "Chemical process engineering and electro-optics", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema"...
tasks/ENG5
trajectory/ENG5
1
0
1,048,576
MATH1
Mathematics
Frobenius Slopes and Hasse-Witt Invariants of a Genus 2 Curve over $\mathbb{F}_7$
Algebraic geometry / arithmetic curves
# Frobenius Slopes and Hasse-Witt Invariants of a Genus 2 Curve over $\mathbb{F}_7$ Consider the smooth projective genus $g=2$ hyperelliptic curve $C$ defined over $\mathbb{F}_p$ with $p=7$ by the affine equation: $$y^2 = x^5 + 2x^3 + 3x + 1$$ Let $J(C)$ be the Jacobian of $C$, and let $\text{Frob}$ denote the geometr...
{"task_id": "MATH1", "input_type": "self-contained advanced mathematical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "MATH1", "required_fields": ["frobenius_traces", "char_poly_coeffs", "newton_slope...
tasks/MATH1
trajectory/MATH1
3
1
1,048,576
MATH2
Mathematics
Radial Equilibria and Non-Homogeneous Annular Deformations for Polyconvex Hyperelasticity
Partial differential equations / nonlinear elasticity
# Radial Equilibria and Non-Homogeneous Annular Deformations for Polyconvex Hyperelasticity Let $\Omega \subset \mathbb{R}^2$ be a two-dimensional isotropic hyperelastic body undergoing a radial deformation $\mathbf{u}: \Omega \to \mathbb{R}^2$ given in polar coordinates $(r, \theta)$ by $$\mathbf{u}(r, \theta) = \big...
{"task_id": "MATH2", "input_type": "self-contained advanced mathematical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "MATH2", "required_fields": ["total_energy_at_2", "natural_state_lambda", "ivp_out...
tasks/MATH2
trajectory/MATH2
3
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MATH3
Mathematics
Normalized Expected Sum of Singular Values of Gaussian Matrices
Probability / random matrix theory
# Normalized Expected Sum of Singular Values of Gaussian Matrices Let $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$ and let $d \ge 1$ be an integer. Consider a $d \times d$ random matrix $G_{\mathbb{K}}$ with independent and identically distributed entries of mean $0$ and variance $1/d$: - For $\mathbb{K} = \mathbb{R}$,...
{"task_id": "MATH3", "input_type": "self-contained advanced mathematical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "MATH3", "required_fields": ["complex_alpha_d1", "complex_alpha_d2", "complex_alph...
tasks/MATH3
trajectory/MATH3
3
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MATH4
Mathematics
Analytic Invariants of the Normalized Ramanujan Delta L-Function
Analytic number theory / modular forms
# Analytic Invariants of the Normalized Ramanujan Delta L-Function Let $\Delta(z) = q \prod_{n=1}^\infty (1 - q^n)^{24} = \sum_{n=1}^\infty \tau(n) q^n$ ($q = e^{2\pi i z}$) be the weight-$12$ modular discriminant cusp form with Ramanujan tau coefficients $\tau(n)$. Its unnormalized $L$-function is $L(s, \Delta) = \su...
{"task_id": "MATH4", "input_type": "self-contained advanced mathematical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "MATH4", "required_fields": ["normalized_tau_7", "von_mangoldt_lambda_4", "analyti...
tasks/MATH4
trajectory/MATH4
3
1
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MATH5
Mathematics
Determinant, Signature, and Cyclotomic Tristram-Levine Invariants of Pretzel Knots
Topology / knot invariants
# Determinant, Signature, and Cyclotomic Tristram-Levine Invariants of Pretzel Knots Let $P(p_1, p_2, \dots, p_m)$ denote the standard $m$-strand pretzel link with odd integer twist parameters $p_1, \dots, p_m$ and an odd number of strands $m \ge 3$, which is a single-component knot. Orient $P(p_1, \dots, p_m)$ so tha...
{"task_id": "MATH5", "input_type": "self-contained advanced mathematical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "MATH5", "required_fields": ["knot_determinant", "knot_signature", "tristram_levin...
tasks/MATH5
trajectory/MATH5
3
1
1,048,576
MED1
Medicine
Womersley blood flow coupled to a three-element Windkessel
Cardiovascular hemodynamics
# MED1: Womersley blood flow coupled to a three-element Windkessel - **Problem ID:** `MED1` - **Domain:** cardiology_vascular ## Problem Specification Incompressible Newtonian blood (`blood_density_kg_m3`, `blood_viscosity_mpa_s`) flows through a rigid axisymmetric circular arterial segment of radius `vessel_radius_m...
{"task_id": "MED1", "title": "Womersley blood flow coupled to a three-element Windkessel", "category": "Cardiovascular hemodynamics", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {"type": "object", "properties": ...
tasks/MED1
trajectory/MED1
1
0
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MED2
Medicine
Glomerular ultrafiltration with tubuloglomerular feedback and protein transport
Renal physiology
# MED2: Glomerular ultrafiltration with tubuloglomerular feedback and protein transport - **Problem ID:** `MED2` - **Domain:** nephrology_urology ## Problem Specification Glomerular blood is supplied from a renal arterial manifold at pressure `renal_arterial_pressure_pra_mmhg` (mmHg) and enters the capillary tuft at ...
{"task_id": "MED2", "title": "Glomerular ultrafiltration with tubuloglomerular feedback and protein transport", "category": "Renal physiology", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {"type": "object", "pro...
tasks/MED2
trajectory/MED2
1
1
1,048,576
MED3
Medicine
Two-hit retinoblastoma, mosaic selection, and sequencing-biased genetic risk
Cancer genetics and Bayesian risk modeling
# MED3: Two-hit retinoblastoma, mosaic selection, and sequencing-biased genetic risk - **Problem ID:** `MED3` - **Domain:** oncology_cancer_biology ## Problem Specification Across both eyes, the developing retina arises from `retinal_retinoblast_pool_n` diploid retinoblast precursor lineages (split equally between th...
{"task_id": "MED3", "title": "Two-hit retinoblastoma, mosaic selection, and sequencing-biased genetic risk", "category": "Cancer genetics and Bayesian risk modeling", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": ...
tasks/MED3
trajectory/MED3
1
1
1,048,576
MED4
Medicine
SPGR MRI T1 mapping with field scaling, protein binding, and T2* decay
Medical imaging physics
# MED4: SPGR MRI T1 mapping with field scaling, protein binding, and T2* decay - **Problem ID:** `MED4` - **Domain:** Ophthalmology, Audiology & Medical Imaging Physics — MRI Spoiled Gradient-Echo Ernst Equation and DCE-MRI Relaxometry ## Problem Specification A 3D spoiled gradient-echo (SPGR) DCE-MRI acquisition at ...
{"task_id": "MED4", "title": "SPGR MRI T1 mapping with field scaling, protein binding, and T2* decay", "category": "Medical imaging physics", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {"type": "object", "prope...
tasks/MED4
trajectory/MED4
1
0
1,048,576
MED5
Medicine
Non-inferiority trials with constancy drift and crossover correction
Clinical-trial biostatistics
# MED5: Non-inferiority trials with constancy drift and crossover correction - **Problem ID:** `MED5` - **Domain:** Clinical Trial Biostatistics & Epidemiology — Non-Inferiority Trial Fixed-Margin (95-95) and Synthesis Methods ## Problem Specification A non-inferiority clinical trial compares a new test regimen $T$ a...
{"task_id": "MED5", "title": "Non-inferiority trials with constancy drift and crossover correction", "category": "Clinical-trial biostatistics", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {"type": "object", "pr...
tasks/MED5
trajectory/MED5
1
0
1,048,576
PHYS1
Physics
Quantum oscillations and polariton wavepacket dispersion
Condensed-matter physics
# PHYS1: Quantum oscillations and polariton wavepacket dispersion A 3D metal at $T = 2.20\text{ K}$ in a magnetic field $B = 8.00\text{ T}$ has an extremal Fermi-surface cross section $A_{\mathrm{ext}} = 4.20\text{ nm}^{-2}$, cyclotron mass $m_c^* = 0.420\,m_e$, effective $g$-factor $g^* = 2.15$, and Dingle temperatur...
{"task_id": "PHYS1", "title": "Quantum oscillations and polariton wavepacket dispersion", "category": "Condensed-matter physics", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {"type": "object", "properties": {"pr...
tasks/PHYS1
trajectory/PHYS1
1
1
1,048,576
End of preview. Expand in Data Studio

STEM RL Tasks

40 advanced STEM tasks · 8 domains · executable verifiers · 55 recorded model attempts

STEM RL Tasks is a public sample of Ulam's scientific reasoning tasks for reinforcement learning with verifiable rewards (RLVR) and model evaluation. The problems ask models to connect scientific concepts, work through mathematical models, and produce structured answers that can be checked programmatically. Each task includes a problem statement, a machine-readable instance, a reference answer, and an offline Python verifier.

The collection spans astronomy, biology, chemistry, Earth science, engineering, mathematics, medicine, and physics. Problems range from black-hole superradiance and enzyme kinetics to nonlinear fault dynamics, quantum transport, algebraic curves, and cardiovascular hemodynamics. Many combine several calculations, making consistency across assumptions, units, and numerical methods part of the challenge.

Explore Ulam's STEMBench

STEMBench is Ulam's correctness-first evaluation of AI research across biology, chemistry, physics, astronomy, Earth science, and engineering. It measures how far a model advances a well-scoped scientific problem, with credit for supported partial progress, reproducibility, and a clear account of what remains unresolved. Explore the domain comparisons, evaluation methodology, and public research-task atlas to see how today's AI systems approach demanding scientific work. STEMBench uses a research-progress rubric; this repository provides a separate collection of fixed-answer tasks with executable verifiers.

Domains and task coverage

Domain Tasks Examples
Astronomy 5 Cosmological recombination, black-hole superradiance, stellar structure, exoplanet irradiation
Biology 5 Enzyme kinetics, molecular thermodynamics, ion transport, population genetics, nutrient cycling
Chemistry 5 Impedance spectroscopy, hydrogel swelling, molecular orbitals, catalytic kinetics, reaction equilibria
Earth science 5 Fault dynamics, geothermal convection, ice-shelf hydrofracture, ocean iron cycling, ozone chemistry
Engineering 5 Magnetohydrodynamics, geotechnical mechanics, information theory, control systems, adsorption
Mathematics 5 Arithmetic geometry, nonlinear elasticity, random matrices, modular forms, knot invariants
Medicine 5 Blood flow, renal physiology, cancer genetics, MRI physics, clinical-trial statistics
Physics 5 Quantum oscillations, open quantum systems, plasma damping, fluid stability, relativistic spin dynamics

Browse the sample split in the Hugging Face dataset viewer for the task statements and metadata. The complete task catalog, attempt inventory, and technical guide are in docs/tasks.json and docs/README.md.

What's included

tasks/<TASK_ID>/
  instruction.md          Problem statement and submission requirements
  instance.json           Task parameters and answer schema
  task.json               Task metadata and file references
  verify.py               Executable answer verifier
  solution/solution.json  Reference answer
trajectory/<TASK_ID>/attempt_<N>/
  trajectory.json         Available captured model and tool record
  submitted_action.json  Complete submitted answer
  result.json            Recorded outcome and available diagnostics
environment/             Shared grading CLI and one-step STEMEnv interface
data/tasks.jsonl          Flat task index for the dataset viewer
docs/                    Package guide, inventory, and SHA-256 checksums

Some mathematics tasks also include an optional reference solver. Task IDs and the original task and trajectory files are retained from the supplied package. The viewer index provides a consistent browsing layer; the original files remain the source of truth.

Download and grade

Download the repository using the Hugging Face CLI:

hf download ulamai/STEM-RL-Tasks --repo-type dataset --local-dir STEM-RL-Tasks
cd STEM-RL-Tasks

Grading requires Python 3.10 or newer and uses only the standard library. It runs offline, with no additional datasets or private container images.

# Validate every packaged reference answer.
python environment/grade.py --all

# Grade your own JSON answer against a task's verifier.
python environment/grade.py --task CHEM1 --candidate candidate.json

# Regrade saved submissions and compare with their recorded outcomes.
python environment/grade.py --check-trajectories

Follow each task's exact answer schema, required keys, units, types, and tolerances. Schemas appear as submission_schema or action_schema in instance.json. Answers can include numerical fields, arrays, integers, or rational strings. A full-task pass requires the entire answer to satisfy the verifier; any diagnostic partial reward is separate from that outcome.

For programmatic use, environment.STEMEnv exposes reset(seed=...) and step(answer). Each episode accepts one answer, making it convenient to connect a policy to task observations and verifier rewards. See the technical guide for an example.

Recorded attempts

The package includes 55 historical attempts: 19 passes and 36 failures, a pooled attempt pass rate of 34.55%. Chemistry includes two attempts per task, mathematics three, and the other domains one. These deliberately selected examples support inspection of successful and unsuccessful submissions; their pooled rate is not a pass@1 or pass@k estimate or a measure of general model performance.

The source records label the models as gemini-4-argon and, for astronomy, gemini-4-argon-high. These are preserved source labels. The release does not add new model runs. Captured trajectory formats vary, and some records contain excerpts rather than complete conversations. Medical records include truncated tool-code and output snippets. Complete saved submitted answers are included for independent regrading.

Validation reproduced all 40 reference-answer passes and all 55 recorded outcomes, including failures. Regrading checks saved answers against the supplied verifiers; it does not replay historical model or tool execution.

Intended use

Use the tasks to prototype RLVR pipelines, evaluate numerical and scientific reasoning, study tool-assisted problem solving, and inspect failure modes against explicit answer contracts. For a blind evaluation, give the solving model only the problem instructions and instance. Keep reference answers, verifiers, saved trajectories, and results on the evaluator side. STEMEnv is a convenience interface and does not provide sandbox isolation.

These are scientific benchmark problems. Verifier acceptance establishes agreement with the packaged reference values and tolerances. Medicine tasks are for research and evaluation, not clinical decision-making.

About Ulam

Ulam develops reasoning data and evaluations for advanced AI systems. Explore STEMBench for scientific research evaluations, methodology, and domain-level comparisons, or visit our website to discuss custom scientific reasoning tasks and evaluation work.

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