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 |
PHYS2 | Physics | Quantum-jump photon statistics and driven-probe heat transport | Quantum optics and open quantum systems | # PHYS2: Quantum-jump photon statistics and driven-probe heat transport
A two-level atom prepared in the ground state $|g\rangle$ immediately after a quantum jump at $\tau = 0$ is resonantly driven with Rabi frequency $\Omega = 12.00\;\mu\text{s}^{-1}$ and spontaneous emission rate $\Gamma = 4.00\;\mu\text{s}^{-1}$ ($... | {"task_id": "PHYS2", "title": "Quantum-jump photon statistics and driven-probe heat transport", "category": "Quantum optics and open quantum systems", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {"type": "object... | tasks/PHYS2 | trajectory/PHYS2 | 1 | 1 | 1,048,576 |
PHYS3 | Physics | Vlasov–Landau plasma damping and driven-probe heat transport | Kinetic plasma physics | # PHYS3: Vlasov–Landau plasma damping and driven-probe heat transport
In a collisionless Maxwellian electron plasma with Debye wavenumber parameter $x = k\lambda_D = 0.26$ ($\lambda_D = v_{\mathrm{th}}/\omega_{pe}$), let $w = \omega_r / \omega_{pe}$ and $\zeta = \frac{w}{\sqrt{2}\,x}$, and let $D_+(\zeta) = e^{-\zeta^... | {"task_id": "PHYS3", "title": "Vlasov–Landau plasma damping and driven-probe heat transport", "category": "Kinetic plasma 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": {"... | tasks/PHYS3 | trajectory/PHYS3 | 1 | 0 | 1,048,576 |
PHYS4 | Physics | Görtler boundary-layer instability and dispersive waveguide pulses | Fluid mechanics and hydrodynamic stability | # PHYS4: Görtler boundary-layer instability and dispersive waveguide pulses
A laminar Blasius boundary layer with freestream speed $U_\infty = 24.0\text{ m/s}$ and kinematic viscosity $\nu = 1.50e-05\text{ m}^2/\text{s}$ develops along a concave wall of radius of curvature $R_c = 1.80\text{ m}$. With Blasius momentum ... | {"task_id": "PHYS4", "title": "Görtler boundary-layer instability and dispersive waveguide pulses", "category": "Fluid mechanics and hydrodynamic stability", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {"type": ... | tasks/PHYS4 | trajectory/PHYS4 | 1 | 0 | 1,048,576 |
PHYS5 | Physics | Relativistic gyroscope precession and dispersive waveguide pulses | General relativity and spin dynamics | # PHYS5: Relativistic gyroscope precession and dispersive waveguide pulses
A test gyroscope orbits a rotating central mass in a circular orbit of inclination $i = 60.0^\circ$ (angle between the orbit normal $\hat{\mathbf{n}}$ and the primary spin axis $\hat{\mathbf{J}}$), with dimensionless relativistic compactness $u... | {"task_id": "PHYS5", "title": "Relativistic gyroscope precession and dispersive waveguide pulses", "category": "General relativity and spin dynamics", "instruction_file": "instruction.md", "input_data": "All parameters and equations needed for the task are embedded in instruction.md.", "action_schema": {"type": "object... | tasks/PHYS5 | trajectory/PHYS5 | 1 | 0 | 1,048,576 |
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