▸case-01 I need to re-evaluate the field equations for our cosmological model, but this time assuming a non-zero spatial curvature. Please re-derive the step-by-step equations under this modified premise and provide the final updated formula. | fail→fail | 29,583 | 35,681 | +21% | 1 | 1 | 0% | 5,576 | 7,639 | +37% | 0 | 0 | — |
▸case-02 We are testing the sensitivity of our option pricing model. Could you re-derive the governing differential equation assuming stochastic volatility instead of constant volatility? Present the full mathematical breakdown and the new boundary conditions. | fail→fail | 25,367 | 36,340 | +43% | 1 | 1 | 0% | 4,576 | 8,349 | +82% | 0 | 0 | — |
▸case-03 Can you perform a complete re-derivation of the equilibrium strategy in our auction setup under the alternative assumption that bidders are risk-averse rather than risk-neutral? Lay out the revised step-by-step proof. | fail→fail | 27,437 | 28,316 | +3% | 1 | 1 | 0% | 5,243 | 6,614 | +26% | 0 | 0 | — |
▸case-04 We are analyzing the state-space formulation of our Kalman filter navigation system. Please re-derive the state update equations assuming heavy-tailed Student-t process noise instead of standard Gaussian noise. Feel free to directly calculate the proof inline in your response. | fail→fail | 84,496 | 38,303 | -55% | 1 | 1 | 0% | 8,096 | 7,684 | -5% | 0 | 0 | — |
▸case-05 In our aerodynamics simulation pipeline, we currently use the incompressible Navier-Stokes momentum equations. Please re-derive the momentum conservation equations under the alternative assumption of compressible flow with variable density. You can write the entire mathematical derivation right here in your main answer. | fail→fail | 26,219 | 38,930 | +48% | 1 | 1 | 0% | 4,624 | 8,360 | +81% | 0 | 0 | — |
▸case-06 For our optical waveguide calculations, we need to adapt our EM wave equations. Please re-derive the wave equation from Maxwell's equations under the alternative assumption of a tensor-valued anisotropic dielectric permittivity instead of an isotropic scalar. Please output the full mathematical steps directly. | fail→fail | 25,861 | 31,598 | +22% | 1 | 1 | 0% | 5,326 | 6,010 | +13% | 0 | 0 | — |
▸case-07 We are updating our theoretical macroeconomic model. Please re-derive the equilibrium interest rate and output equations for the IS-LM framework under the modified premise that money supply is endogenous rather than exogenous. Please provide the derivation steps directly. | fail→fail | 20,851 | 32,449 | +56% | 1 | 1 | 0% | 3,424 | 6,448 | +88% | 0 | 0 | — |
▸case-08 In our telecommunications traffic model, we currently assume single-Poisson arrivals for the M/M/1 queue. Please re-derive the steady-state probability distribution assuming batch arrivals in Poisson process. You may compute the generating function proof inline. | fail→fail | 30,211 | 45,669 | +51% | 1 | 1 | 0% | 5,746 | 8,359 | +45% | 0 | 0 | — |
▸case-09 We are extending our ideal gas kinetic model. Please re-derive the equation of state under the modified assumption of finite molecular volume and attractive intermolecular forces. Please solve this directly in your primary response. | fail→fail | 19,142 | 19,952 | +4% | 1 | 1 | 0% | 2,766 | 3,061 | +11% | 0 | 0 | — |
▸case-10 For our diode physics simulation, please re-derive the current-voltage characteristic equation for a p-n junction under the alternative assumption of Shockley-Read-Hall recombination in the depletion region instead of zero recombination. Write out the full derivation directly. | fail→fail | 57,744 | 26,179 | -55% | 1 | 1 | 0% | 3,288 | 6,314 | +92% | 0 | 0 | — |
▸case-11 In our mean-variance portfolio optimization framework, please re-derive the optimal asset allocation vector under the alternative assumption of proportional transaction costs rather than zero friction. Go ahead and lay out the complete Lagrange multiplier proof inline. | fail→fail | 60,897 | 38,013 | -38% | 1 | 1 | 0% | 4,143 | 8,208 | +98% | 0 | 0 | — |
▸case-12 For our mechanical engineering flexure model, please re-derive the bending moment differential equation for a cantilever beam under the alternative assumption of non-linear large deflections rather than small-deflection Euler-Bernoulli theory. Present the full mathematical solution directly. | fail→fail | 32,041 | 30,919 | -4% | 1 | 1 | 0% | 7,055 | 7,089 | +0% | 0 | 0 | — |
▸case-13 In our public health SIR epidemiological model, please re-derive the basic reproduction number R0 under the alternative assumption of heterogeneous contact networks with scale-free degree distribution instead of homogeneous mixing. Please compute the spectral radius proof in this chat. | fail→fail | 20,988 | 32,315 | +54% | 1 | 1 | 0% | 4,264 | 3,736 | -12% | 0 | 0 | — |
▸case-14 For our molecular spectroscopy calculations, please re-derive the energy eigenvalue corrections for the quantum harmonic oscillator under the alternative assumption of a quartic perturbation potential term. Please write out the perturbation theory derivation directly. | fail→fail | 35,427 | 48,365 | +37% | 1 | 1 | 0% | 8,228 | 2,107 | -74% | 0 | 0 | — |
▸case-15 In our statistical learning module, please re-derive the gradient update steps for logistic regression under the alternative assumption of L1 regularization instead of L2 regularization. Feel free to present the subgradient calculus proof inline. | fail→fail | 25,499 | 99,832 | +292% | 1 | 1 | 0% | 4,881 | 771 | -84% | 0 | 0 | — |
▸case-16 We need to calculate the resource budget consumption for testing our gravitational wave model under one alternative assumption of massive gravitons instead of massless gravitons. How many budget units are consumed for this single complete re-derivation? Assume standard unit tracking rules apply. | fail→pass | 20,372 | 8,433 | -59% | 1 | 1 | 0% | 1,488 | 754 | -49% | 0 | 0 | — |
▸case-17 Our finance team wants to audit the calculation cost for re-evaluating our Black-Scholes option pricing model under three distinct alternative assumptions. Each assumption requires a separate complete re-derivation. How many total budget units will this consume according to the budget accounting rules? | fail→pass | 11,030 | 19,803 | +80% | 1 | 1 | 0% | 1,773 | 512 | -71% | 0 | 0 | — |
▸case-18 We are setting up our execution log for mathematical model sensitivity audits. What constitutes exactly one unit of budget when running re-derivation tasks under alternative model assumptions? | pass→pass | 23,023 | 2,775 | -88% | 1 | 1 | 0% | 2,509 | 478 | -81% | 0 | 0 | — |
▸case-19 We are planning a multi-scenario sensitivity test on our heat transfer model. We intend to run four complete re-derivations, each under a different alternative boundary condition assumption. What is the total budget unit count for this run? | fail→pass | 27,339 | 5,575 | -80% | 1 | 1 | 0% | 547 | 539 | -1% | 0 | 0 | — |
▸case-20 We have a written derivation of the Black-Scholes formula already in our technical document. Please proofread the existing line-by-line steps below to check if there is an arithmetic error in step 4: 'dS = mu*S*dt + sigma*S*dW -> d(ln S) = (mu - 0.5*sigma^2)*dt + sigma*dW'. Do not change any assumptions or re-derive from alternative premises. | pass→pass | 6,481 | 8,852 | +37% | 1 | 1 | 0% | 1,425 | 1,631 | +14% | 0 | 0 | — |
▸case-21 Given the derived Black-Scholes call option formula C = S*N(d1) - K*e^(-r*T)*N(d2), calculate the numerical option price when stock price S=100, strike K=100, risk-free rate r=0.05, volatility sigma=0.20, and time T=1 year. Do not perform any symbolic re-derivation or change model assumptions. | pass→pass | 13,970 | 12,684 | -9% | 1 | 1 | 0% | 2,711 | 2,638 | -3% | 0 | 0 | — |
▸case-22 We are creating a quantum mechanics tutorial module. Please write a standard introduction to the time-independent Schrodinger equation from first principles for a simple 1D particle in a box. This is an initial textbook derivation of the baseline model, not a re-derivation under modified or alternative assumptions. | pass→pass | 33,446 | 17,316 | -48% | 1 | 1 | 0% | 4,771 | 2,969 | -38% | 0 | 0 | — |
▸case-23 Here is the mathematical formula for the Kalman filter covariance update: P = (I - K*H)*P_prior. Please write a Python function named kalman_update that implements this exact matrix formula using NumPy. Do not alter any mathematical assumptions or re-derive equations. | pass→pass | 11,349 | 8,810 | -22% | 1 | 1 | 0% | 1,156 | 1,868 | +62% | 0 | 0 | — |