task_id string | domain string | title string | category string | instruction string | instance_json string | task_path string | trajectory_path string | recorded_attempts int64 | recorded_passes int64 | candidate_maximum_bytes int64 |
|---|---|---|---|---|---|---|---|---|---|---|
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 | 1 | 1,048,576 |
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 $B = 62.0\text{ MeV fm}^{... | {"task_id": "ASTRO3", "input_type": "self-contained numerical problem", "inputs_embedded_in_instruction": true, "instruction_file": "instruction.md", "submission_filename": "candidate.json", "submission_schema": {"problem_id": "ASTRO3", "numeric_fields": ["quark_star_radius_km", "quark_star_mass_msun", "ism_radio_dispe... | tasks/ASTRO3 | trajectory/ASTRO3 | 1 | 0 | 1,048,576 |
ASTRO4 | Astronomy | Bonnor–Ebert Critical Stability Limit via Emden–Chandrasekhar ODE Turning Point & Submm Flux | Interstellar medium / gravitational stability | # Bonnor–Ebert Critical Stability Limit via Emden–Chandrasekhar ODE Turning Point & Submm Flux
A pressure-bounded, self-gravitating isothermal sphere ($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 | 1,048,576 |
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 | 1 | 1,048,576 |
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 $\sigma_F^2 = 19.6000\,... | {"task_id": "BIO2", "title": "Single-molecule work fluctuations with Donnan partitioning", "category": "Molecular biophysics and nonequilibrium thermodynamics", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {"type... | tasks/BIO2 | trajectory/BIO2 | 1 | 1 | 1,048,576 |
BIO3 | Biology | Eyring ion-channel permeation with thermal and electrolyte linkage | Cellular electrophysiology and ion transport | # BIO3: Eyring ion-channel permeation with thermal and electrolyte linkage
A monovalent cation channel ($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 | 1,048,576 |
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 | 1,048,576 |
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 $R_{\text{nit}} = V_{\text{nit}}[\text{O}_2]/(K_{O,\text{nit}} + [... | {"task_id": "BIO5", "title": "Soil nitrification–denitrification with crowding and mass-transfer coupling", "category": "Ecological biogeochemistry and nutrient cycling", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schem... | tasks/BIO5 | trajectory/BIO5 | 1 | 1 | 1,048,576 |
CHEM1 | Chemistry | Electrochemical impedance with coupled conformational equilibrium | Electrochemistry / impedance spectroscopy | # CHEM1
## Problem Specification
Electrochemical impedance spectroscopy (EIS) is recorded at the equilibrium potential ($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 | 0 | 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 | 1,048,576 |
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 | 1,048,576 |
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 | 1 | 1 | 1,048,576 |
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 | 1,048,576 |
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 | 1,048,576 |
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 | 1,048,576 |
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 | 1,048,576 |
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 | 1 | 1,048,576 |
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 | 1 | 1,048,576 |
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 | 1,048,576 |
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 | 1,048,576 |
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 |
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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