## 2 Materials and Methods
### Mathematical Framework  
A generic pattern of the 3 MT power profile showed an initial steep decline in power output that was then followed by a more gradual decrease in intensity before settling to an asymptotic level during the last 30 to 45 s. In general, the behavior of power output versus time can be effectively described by a mono-exponential (MONO) decay function, which is modeled by the following equation:  
\[\mathrm{P}=\mathrm{a}e^{-\mathrm{b}\mathrm{t}}+\mathrm{C}\mathrm{P} \tag{1}\]  
where \(\mathrm{P}\) is the instantaneous power, \(\mathrm{t}\) is time, \(\mathrm{C}\mathrm{P}\) is critical power, \(\mathrm{b}\) is the decaying constant, and a represents the initial power output at the start of the decay process. In addition, a bi-exponential (BI) decay function models power output with two distinct phases: an initial rapid decay, reflecting the depletion of anaerobic energy stores, followed by a slower decay phase associated with sustained aerobic energy production. This dual-phase function captures the characteristic decline and stabilization of power during exhaustive exercise, making it suitable for estimating critical power and anaerobic capacity. \[\mathrm{P}=\mathrm{a}e^{-\mathrm{b}\mathrm{t}}+\mathrm{c}e^{-\mathrm{d} \mathrm{t}}+\mathrm{C}\mathrm{P} \tag{2}\]  
where a and c represent the initial scaling factors associated with the two exponential decay components, while b and d denote the respective decay rates. These parameters are optimized for each participant, as they vary based on individual differences in power output and rate of fatigue. The end power (EP) was calculated as the average power output during the final 30 s of the test, while W\({}^{\prime}\) was estimated by calculating the power-time integral above the EP [9]. To model the relationship between CP and W\({}^{\prime}\), the area under the curve bounded by CP (W\({}^{\prime}\)) was derived using an exponential decay function. For the mono-exponential (MONO) function, this can be mathematically expressed as  
\[\int\mathrm{P}-\mathrm{CP}\,dt=\int\mathrm{a}e^{-\mathrm{bt}}\,dt \tag{3}\]  
By substituting \(\int(P-CP)\,dt=\mathrm{W}^{\prime}\) into Equation (3) and solving algebraically for W\({}^{\prime}\) from time 0 to \(\infty\), the final solution is obtained as  
\[\mathrm{W}^{\prime}_{\mathrm{MONO}}=\frac{\mathrm{a}}{\mathrm{b}} \tag{4}\]  
Similarly, W\({}^{\prime}\) for the bi-exponential (BI) function can be determined as  
\[\mathrm{W}^{\prime}_{\mathrm{BI}}=\frac{\mathrm{a}}{\mathrm{b}}+\frac{ \mathrm{c}}{\mathrm{d}} \tag{5}\]  
where a, b, c, and d are parameter estimates derived specifically for each participant to best fit their data within the decay model. All the analyses were conducted in R software (Version 4.4.1; Vienna, Austria). A custom script was used to fit an exponential decay model to cycling power data, specifically aiming to estimate critical power (CP) and work capacity (W\({}^{\prime}\)) from a 3 min all-out cycling test (3 MT). The script performs the following steps:  
1. Model fitting. A custom function applies the nonlinear least-squares method ('nlsLM' function from the'minipack.lm' library) to fit the exponential decay model to the data. The model includes parameters for exponential decay terms and the critical power value. The function returns the residual sum of squares (RSS) and the fitted model. 2. Grid search for parameter estimation. A grid search is conducted over possible values for the model parameters (a, b, c, d, and CP) to find the best-fitting model. The script generates a parameter grid and applies parallel processing to efficiently search for the optimal model using the'mclapply' function from the 'parallel' library. The defined ranges for each parameter in the grid search are as follows: a = 800 to 1500, by = 200; b = 0.05 to 0.3, by = 0.05; c = 300 to 700, by = 100; d = 0.01 to 0.1, by = 0.02. 3. Model fitting over time intervals. The script iterates over time intervals (60 to 180 s in 15 s increments) and fits the model to subsets of the data within each interval.