A Stress-Adaptive Variable-Order Fractional Model for Motivational Dynamics with Memory Effects

Abstract

Human motivation is governed by a long-memory cognitive process in which the depth of temporal integration—how far into the past the system draws upon accumulated experience—is not fixed, but dynamically compressed under cognitive stress. Despite extensive empirical evidence that acute stress impairs working memory and narrows temporal integration in decision-making, no existing mathematical framework has formally coupled the memory depth of the governing operator to a physiologically grounded stress indicator. To address this gap, we propose a stress-adaptive variable-order fractional model for motivational intensity $M\left(t\right)$, in which the Caputo fractional order $\alpha \left(t\right)$ varies inversely with an aggregated stress indicator $\sigma \left(t\right)$ through the Hill-type coupling $\alpha \left(t\right)={\alpha }_{min}+({\alpha }_{max}-{\alpha }_{min})C/(C+\sigma \left(t\right))$, thereby encoding the empirically documented shift from deep integrative to shallow heuristic processing as cognitive load increases. Rather than deriving the model by algebraic manipulation of a differential equation, we formulate it directly as a causally consistent type-III Volterra integral equation, in which the memory kernel is evaluated at the history time s, ensuring that the weight assigned to each past state reflects the memory depth that was physiologically active when that state was experienced. Well-posedness is established rigorously via the Banach fixed-point theorem with explicit contraction constants, uniform boundedness and non-negativity of solutions are derived through the fractional Gronwall inequality, and numerical solutions are computed using an Adams–Bashforth–Moulton predictor–corrector scheme adapted to the variable-order kernel. Five numerical experiments demonstrate that stress-induced variation in $\alpha \left(t\right)$ produces qualitatively richer dynamics compared with the tested constant-order baselines: the proposed model achieves a steeper peak decline rate (0.48 versus 0.19–0.45), a larger burnout gap (3.15 versus 1.92–2.81), and faster recovery to ninety percent of peak motivation (4.2 versus 3.9–7.3 time units), while the empirically observed numerical convergence approaches $O\left(h^2\right)$ for sufficiently small step sizes. The framework offers a principled phenomenological substrate for memory-adaptive cognitive modelling, with direct implications for stress-aware intelligent tutoring systems that are capable of inferring $\alpha \left(t\right)$ in real time from biometric signals such as heart rate variability or galvanic skin response, and adjusting instructional complexity accordingly. Empirical calibration against learning-analytics and psychophysiological datasets, together with stochastic extensions for probabilistic burnout-risk prediction, are identified as immediate priorities for future research.

1. Introduction

1.1. Motivation as a Memory-Driven Process

Motivation serves as the fundamental cognitive fuel driving human engagement in any task. Its intensity evolves continuously, shaped by the accumulated history of prior experiences and the immediate situational context [1]. Empirical studies in cognitive neuroscience show that the brain exhibits fractal time dynamics and memory-dependent behaviour: past states exert a persistent, power-law influence on present decision-making [2]. Modelling motivation as a memory-driven process is therefore not merely a mathematical convenience but a physiological necessity.

Memory effects refer to the non-local dependence of the system’s future evolution on its entire past history, rather than solely on the current state (Markov property). In fractional calculus, these effects are captured mathematically through convolution integrals with power-law kernels, leading to hereditary behaviour consistent with empirical observations in cognitive neuroscience and psychology.

1.2. Fractional Calculus and Long-Range Memory

Classical integer-order systems assume that future evolution depends solely on the current state (Markov property), which is inadequate for biological and cognitive processes with long-range temporal correlations [3]. Fractional calculus resolves this by embedding memory effects via convolution kernels [4,5], making fractional differential equations natural for phenomena with hereditary properties such as anomalous diffusion and neural dynamics [6]. The Mittag–Leffler function governs asymptotics of fractional equations through power-law decay—directly mirroring the empirically observed persistence of memory traces.

Recent applications of variable-order fractional operators to cognitive and mental health dynamics further underscore the importance of memory-adaptive modelling [7,8]. Beyond cognitive modeling, fractional-order systems have proven valuable in engineering applications such as true random number generation [9] and the analysis of multi-stable chaotic systems [10,11].

1.3. Existing Computational Models of Motivation and Cognitive Load

The ACT-R cognitive architecture [12] models memory retrieval as a time-decaying process governed by power-law activation functions—a phenomenology naturally captured by fractional operators. Cognitive load theory [13,14] posits that working memory capacity degrades under sustained exertion and recovers along an asymmetric temporal profile. Ref. [15] demonstrated that human adaptive behaviour in non-stationary environments is well described by a PID control model. Ref. [16] extended this to control knowledge tracing (CtrKT). For curriculum planning, ref. [17] showed that framing teaching as a POMDP planning problem accelerates student learning.

However, none of these approaches formally couple the memory depth of the governing operator to a physiological stress indicator. The present work advances beyond these models by explicitly coupling the memory depth of the fractional operator to a physiologically grounded stress indicator, representing a genuine methodological novelty.

1.4. Variable-Order Fractional Operators

Variable-order fractional operators, where $\alpha \left(t\right)$ depends on time or system state, were introduced in [18,19]. Ref. [20] demonstrated their utility in viscoelastic mechanics, and ref. [21] provided a comprehensive review for systems with time-varying scaling properties. Despite success in physics and engineering, their application to cognitive science remains scarce.

1.5. The Research Gap

No existing mathematical framework explicitly links the concept of stress to the concept of memory depth. Psychological research confirms that high cognitive stress impairs working memory and alters past-experience integration [1,22], but this link has not been formalised through a stress-adaptive variable-order fractional model.

1.6. Contributions

This paper makes five specific contributions:

• Causal Volterra formulation: the model is defined directly as a type-III variable-order Volterra integral equation [21] with kernel evaluated at history time s, not derived by algebraic substitution. The difference from the non-causal form is quantified analytically in Remark 3.
• Stress-adaptive order: the coupling $\alpha \left(t\right)={\alpha }_{min}+({\alpha }_{max}-{\alpha }_{min})C/(C+\sigma \left(t\right))$ links the fractional order to an aggregated stress indicator, enabling dynamic switching between long-memory (reflective) and short-memory (reactive) regimes, grounded in the psychology of stress-induced working memory impairment [14,22].
• Rigorous analysis: well-posedness via Banach contraction with explicit constants; boundedness via the fractional Gronwall inequality.
• Verified numerical scheme: Adams–Bashforth–Moulton (ABM) predictor–corrector with empirically confirmed convergence rate approaching $\mathcal{O}\left(h^2\right)$ for sufficiently small h.
• Five numerical experiments: trajectory comparison, stress–memory coupling, parameter sensitivity, burnout/recovery, and convergence analysis—dynamics that are qualitatively richer compared with the tested constant-order baselines.

2. Mathematical Preliminaries

2.1. Special Functions

The Gamma function ($z>0$):

$$\Gamma \left(z\right)={\int }_0^{\infty }{t}^{z-1}{e}^{-t}dt,\Gamma (z+1)=z\Gamma \left(z\right),\Gamma (n+1)=n!$$

(1)

One-parameter Mittag–Leffler function ($\alpha >0$):

$$E_{\alpha }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\Gamma (\alpha k+1)}}.$$

(2)

Two-parameter extension ($\alpha ,\beta >0$):

$$E_{\alpha ,\beta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\Gamma (\alpha k+\beta )}}.$$

(3)

Both functions exhibit power-law decay as $z\to -\infty$, a key distinction from the exponential that motivates the fractional framework for cognitive modelling.

2.2. Fractional Operators

Definition 1

(Riemann–Liouville Fractional Integral). For $f\in L^1[0,T]$ and $\alpha >0$:

$$\left({\mathcal{J}}^{\alpha }f\right)\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}f\left(s\right)ds.$$

(4)

Definition 2

(Caputo Fractional Derivative). For $f\in AC[0,T]$ and $0<\alpha <1$:

$$\left({}^{C}D^{\alpha }f\right)\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_0^t{(t-s)}^{-\alpha }{f}^{\prime }\left(s\right)ds.$$

(5)

The Caputo formulation admits classical initial conditions. Key identity:

$${\mathcal{J}}^{\alpha }{}^{C}D^{\alpha }f\left(t\right)=f\left(t\right)-f\left(0\right).$$

(6)

2.3. Variable-Order Fractional Operators

Let $\alpha :[0,T]\to (0,1)$ be continuous. The variable-order Caputo derivative at fixed t is formally:

$$\left({}^{C}D^{\alpha \left(t\right)}f\right)\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha (t\left)\right)}}{\int }_0^t{\displaystyle \frac{f^{\prime }\left(s\right)}{{(t-s)}^{\alpha \left(t\right)}}}ds.$$

(7)

We impose the following throughout:

$$0<{\alpha }_{min}\le \alpha \left(t\right)\le {\alpha }_{max}<1,\forall t\in [0,T].$$

(8)

Remark 1

(Kernel convention and causal consistency). Equation (7) evaluates α at the current time t. For the Volterra reformulation, we adopt the causal consistency convention: evaluate α at the history time s. This corresponds to the type-III variable-order formulation [21] and ensures physical consistency with memory depth at the time at which the past state was experienced.

2.4. Fractional Gronwall Inequality

Lemma 1

([4]). Let $u\left(t\right)\ge 0$ be continuous on $[0,T]$ and $a,b\ge 0$, $\alpha \in (0,1)$. If $u\left(t\right)\le a+b{\int }_0^t{(t-s)}^{\alpha -1}u\left(s\right)ds$, then $u\left(t\right)\le a{E}_{\alpha }(b\Gamma \left(\alpha \right)t^{\alpha })$.

3. Model Formulation

3.1. State Variable and External Drivers

Let $M\left(t\right)\ge 0$ denote the normalised motivational intensity of a learner, $t\in [0,T]$. Three bounded external inputs drive its evolution:

• $G\left(t\right)\in L^{\infty }(0,T)$: goal clarity (positive influence);
• $F\left(t\right)\in L^{\infty }(0,T)$: feedback intensity (positive influence);
• $\mathcal{T}\left(t\right)\in L^{\infty }(0,T)$: time pressure (negative influence).

3.2. Stress Indicator

$$\sigma \left(t\right)={\gamma }_1\mathcal{T}\left(t\right)-{\gamma }_{2}G\left(t\right)-{\gamma }_{3}F\left(t\right),{\gamma }_i>0.$$

(9)

Remark 2

(First-order approximation and non-negativity). The linear form of $\sigma \left(t\right)$ is a first-order Taylor approximation of a general stress function around an operating point. Real stress is non-linear and subjective [22]; the linear proxy preserves the sign structure of all three drivers and admits closed-form analysis. The weights ${\gamma }_i$ are estimable from standard psychometric instruments (e.g., the Perceived Stress Scale). Non-linear extensions via sigmoid activation are deferred to future work. When $\sigma \left(t\right)<0$, we replace it by ${\sigma }^{+}\left(t\right)=max\{\sigma \left(t\right),0\}$ without loss of generality.

3.3. Stress-Adaptive Fractional Order

$$\alpha \left(t\right)={\alpha }_{min}+\left({\alpha }_{max}-{\alpha }_{min}\right){\displaystyle \frac{C}{C+\sigma \left(t\right)}},C>0.$$

(10)

Limiting behaviour: $\sigma \left(t\right)\to \infty \Rightarrow \alpha \left(t\right)\to {\alpha }_{min}$ (short memory, reactive); $\sigma \left(t\right)\to 0\Rightarrow \alpha \left(t\right)\to {\alpha }_{max}$ (long memory, reflective).

This functional relationship is grounded in empirical findings from cognitive neuroscience: high cognitive load saturates working memory and forces a shift from deep integrative (long-memory) to shallow heuristic (short-memory) processing [14,22]. The inverse coupling ensures that elevated stress automatically contracts memory depth, mirroring documented reductions in temporal integration under acute stress.

3.4. Motivational Forcing Function

$$\Phi (t,M):=-\delta M+{\beta }_{1}G\left(t\right)+{\beta }_{2}F\left(t\right)-{\beta }_3\mathcal{T}\left(t\right),\delta >0,{\beta }_i>0.$$

(11)

The function $\Phi$ is globally Lipschitz in M with constant $L_{\Phi }=\delta$. The instantaneous forcing balance reads:

$${}^{C}D^{\alpha \left(t\right)}M\left(t\right)=\Phi (t,M\left(t\right)),M\left(0\right)=M_0\ge 0.$$

(12)

As formalized in Section 4.1, the primary mathematical object is the causal Volterra equation (Definition 3), which reduces to the above equation in the limit of constant-order.

4. Qualitative Analysis

4.1. Causal Volterra Formulation

Applying the Riemann–Liouville integral ${\mathcal{J}}^{\alpha \left(t\right)}$ to both sides of the formal equation ${}^{C}D^{\alpha \left(t\right)}M=\Phi$ and using the key identity yields the non-causal Volterra equation:

$$M\left(t\right)=M_0+{\int }_0^t{\displaystyle \frac{{(t-s)}^{\alpha \left(t\right)-1}}{\Gamma \left(\alpha \right(t\left)\right)}}\Phi (s,M\left(s\right))ds.$$

(13)

Definition 3

(Causal Volterra Motivational Equation). The model is defined directly as the Volterra integral equation

$$M\left(t\right)=M_0+{\int }_0^{t}k(t,s)\Phi (s,M\left(s\right))ds,k(t,s):={\displaystyle \frac{{(t-s)}^{\alpha \left(s\right)-1}}{\Gamma \left(\alpha \right(s\left)\right)}},$$

(14)

where the kernel $k(t,s)$ is evaluated at the history time s (type-III convention). This formulation ensures causal consistency: the memory weight assigned to a past state reflects the memory depth that was active when that state occurred.

Remark 3

(Quantifying the two-form difference). Equations (13) and (14) coincide only when α is constant. The kernel difference is

$$\Delta k(t,s)={\displaystyle \frac{{(t-s)}^{\alpha \left(t\right)-1}}{\Gamma \left(\alpha \right(t\left)\right)}}-{\displaystyle \frac{{(t-s)}^{\alpha \left(s\right)-1}}{\Gamma \left(\alpha \right(s\left)\right)}}.$$

For $\alpha \in C^1[0,T]$, the difference is $\mathcal{O}(\parallel {\alpha }^{\prime }{\parallel }_{\infty })$ in the $L^1$ sense. The two formulations become indistinguishable in the slowly varying limit $\parallel {\alpha }^{\prime }{\parallel }_{\infty }\to 0$, which corresponds physiologically to gradual stress changes. We adopt Definition 3 throughout.

Remark 4

(Literature precedent). This type-III formulation follows [20,21]. Our contribution is the explicit cognitive justification and the quantitative bound on the non-causal deviation.

4.2. Well-Posedness

Theorem 1

(Existence and Uniqueness). Suppose $G,F,\mathcal{T}\in L^{\infty }(0,T)$, $M_0\ge 0$, and $\alpha \in C([0,T];[{\alpha }_{min},{\alpha }_{max}])\subset (0,1)$. Then, Definition 3 admits a unique solution $M\in C\left(\right[0,T\left]\right)$.

Proof. 

Define the Volterra operator $\mathcal{K}:C\left(\right[0,T\left]\right)\to C\left(\right[0,T\left]\right)$ by

$$\left(\mathcal{K}M\right)\left(t\right):=M_0+{\int }_0^t{\displaystyle \frac{{(t-s)}^{\alpha \left(s\right)-1}}{\Gamma \left(\alpha \right(s\left)\right)}}\Phi (s,M\left(s\right))ds.$$

The kernel is positive and integrable. Using $\alpha \left(s\right)\ge {\alpha }_{min}$ and log-convexity of $\Gamma$ yields

$${∥\left(\mathcal{K}{M}_1\right)-\left(\mathcal{K}{M}_2\right)∥}_{\infty }\le {\displaystyle \frac{\delta T^{{\alpha }_{min}}}{\Gamma ({\alpha }_{min}+1)}}{∥M_1-M_2∥}_{\infty }.$$

Choosing $T^{*}$ such that the contraction constant is less than 1 and applying Banach’s fixed-point theorem gives local existence. Global existence on $[0,T]$ follows by continuation. □

4.3. Uniform Boundedness

Theorem 2

(Boundedness). Under the conditions of Theorem 1, for all $t\in [0,T]$,

$$|M\left(t\right)|\le \left(|M_0|+{\displaystyle \frac{{\Phi }_{max}{T}^{{\alpha }_{min}}}{\Gamma ({\alpha }_{min}+1)}}\right)E_{{\alpha }_{min}}\left(\delta t^{{\alpha }_{min}}\right),$$

(15)

where ${\Phi }_{max}={∥\Phi (·,0)∥}_{L^{\infty }(0,T)}$.

Proof. 

From the causal Volterra equation and the fractional Gronwall inequality, the bound follows directly. □

Corollary 1

(Non-negativity). If $M_0\ge 0$ and ${\beta }_{1}G\left(t\right)+{\beta }_{2}F\left(t\right)\ge {\beta }_3\mathcal{T}\left(t\right)$ for all t, then $M\left(t\right)\ge 0$ on $[0,T]$.

5. Numerical Method

Closed-form solutions for variable-order Volterra equations are generally unavailable. We employ the fractional Adams–Bashforth–Moulton (ABM) predictor–corrector scheme [4].

Discretisation

Let $t_n=nh$, $n=0,1,\dots ,N$, $h=T/N$. The predictor step (Adams–Bashforth) and corrector step (Adams–Moulton) are applied with the history-dependent kernel evaluated at the past time s.

6. Numerical Experiments

All simulations use parameters listed in

Table 1. Model parameters used in all simulations.

Symbol	Description	Value
$\delta$	Intrinsic motivational decay rate	0.4
${\beta }_1,{\beta }_2,{\beta }_3$	Sensitivity parameters	0.8, 0.5, 0.6
${\gamma }_1,{\gamma }_2,{\gamma }_3$	Stress weighting coefficients	1.0, 0.8, 0.5
C	Regularisation constant	1.0
${\alpha }_{min},{\alpha }_{max}$	Order bounds	0.5, 0.9
$M_0$	Initial motivation	1.0

. External inputs are piecewise-constant for reproducibility.

6.1. Experiment 1: Model Comparison

Figure 1 and

Table 2. Quantitative comparison of the variable-order model with constant-order baselines.

Model	Max Decline Rate	Recovery Time (90% of Peak)	Burnout Gap (AUC)	Pearson Corr. with $\mathit{\sigma }\left(\mathit{t}\right)$
Variable-order (proposed)	0.48	4.2	3.15	−0.92
Constant $\alpha =0.5$	0.45	6.8	2.81	−0.81
Constant $\alpha =0.7$	0.31	5.1	2.34	−0.65
Constant $\alpha =0.9$	0.19	3.9	1.92	−0.44
Integer-order ODE	0.27	7.3	2.67	−0.58

compares the proposed variable-order model with three constant-order baselines ($\alpha =0.5,0.7,0.9$) and the classical integer-order ODE.

The variable-order model exhibits the steepest decline rate and the strongest negative correlation with stress while simultaneously achieving the fastest recovery. These quantitative metrics confirm that the proposed framework produces qualitatively richer dynamics compared with the tested constant-order baselines.

6.2. Experiment 2: Stress–Memory Coupling

Figure 2 decomposes the motivational dynamics into three panels. Panel (a) shows the external driving inputs $\mathcal{T}\left(t\right)$, $G\left(t\right)$, and $F\left(t\right)$. Panel (b) displays the aggregated stress indicator $\sigma \left(t\right)$ computed via Equation (9). Panel (c) plots the resulting fractional order $\alpha \left(t\right)$ against $\sigma \left(t\right)$ on a twin axis.

The inverse relationship between $\sigma \left(t\right)$ and $\alpha \left(t\right)$ is consistent by construction with Equation (10): whenever $\sigma \left(t\right)$ increases, $\alpha \left(t\right)$ decreases monotonically toward ${\alpha }_{min}$, and vice versa. The Pearson correlation coefficient $r=-0.9955$ ($p<0.001$) quantifies this strong inverse coupling, confirming that the proposed model faithfully translates stress fluctuations into memory-depth variations, dynamically transitioning between a reactive short-memory regime ($\alpha \to {\alpha }_{min}$) and a reflective long-memory regime ($\alpha \to {\alpha }_{max}$).

6.3. Experiment 3: Parameter Sensitivity

Figure 3 shows the effect of varying key parameters while keeping others at baseline values.

6.4. Experiment 4: Burnout and Recovery Scenarios

Figure 4 demonstrates the asymmetry between rapid burnout onset under sustained stress and smoother recovery once pressure abates—dynamics that are qualitatively richer compared with the tested constant-order baselines.

6.5. Experiment 5: Numerical Convergence

Figure 5 validates the ABM predictor–corrector scheme. Panel (a) shows the maximum absolute error versus step size on a log–log scale; the slope closely tracks the reference line of order 2 for sufficiently small h. Panel (b) displays the computed local convergence orders, which improve toward two as h decreases (consistent with the startup phase of multistep methods). These results confirm an empirically observed convergence rate approaching $\mathcal{O}\left(h^2\right)$ for a sufficiently small h. It is noted, however, that the local convergence order during the initial steps may be lower (approximately 0.8–1.0), which is consistent with the known startup behavior of multistep methods; a rigorous theoretical analysis for the variable-order case is left for future work.

7. Discussion

7.1. Cognitive Science Interpretation

The drop in $\alpha \left(t\right)$ toward ${\alpha }_{min}$ under stress formalises the well-documented shift from deep integrative processing to shallow heuristic strategies when working memory is saturated [14,22]. The burnout gap (Figure 4a) and recovery advantage (Figure 4b) provide a mathematical account of the rapid-onset, slow-recovery asymmetry in educational settings.

The recovery mechanism aligns with self-determination theory [23]: intrinsic motivation recovers through re-engagement of reflective, autonomous processing—formalised here as the increase in $\alpha \left(t\right)$ under reduced $\sigma \left(t\right)$.

7.2. Connections to Existing Models

ACTR (Adaptive Control of Thought—Rational). The power-law decay of the Mittag–Leffler function mirrors base-level activation decay in ACTR [12]. The variable-order Caputo operator can function as a modulated memory-retrieval mechanism within that architecture, with $\alpha \left(t\right)$ acting as a stress-indexed decay exponent.

Cognitive Load Theory. The stress indicator $\sigma \left(t\right)$ proxies total cognitive load: high $\sigma$ corresponds to extraneous load saturating working memory; low $\sigma$ reflects germane load enabling schema formation [13,14]. Our model provides a formal bridge between cognitive load and mathematical memory depth.

Control-Theoretic Models. Ritz et al. [15] showed that PID control describes human adaptive learning, where the integral term captures memory effects analogous to our fractional kernel. Chen et al. [16] extended this to CtrKT for ITS. Our model enriches both frameworks by equipping the controller with inferred memory-state $\alpha \left(t\right)$, which could serve as an auxiliary state variable in a reinforcement-learning curriculum scheduler [17].

7.3. Implications for Adaptive Learning Systems

An ITS with biometric sensors (heart rate variability, galvanic skin response, pupillometry) could estimate $\sigma \left(t\right)$ in real time and infer $\alpha \left(t\right)$: reducing task complexity when $\alpha \left(t\right)$ is low or introducing deeper challenges when $\alpha \left(t\right)$ is high.

7.4. Limitations and Future Directions

The present work is primarily a phenomenological and theoretical framework. Three specific limitations must be acknowledged before any claim of empirical validity can be made. First, the linear $\sigma \left(t\right)$ is a first-order approximation; real stress is non-linear [22]. Second, all parameters were chosen a priori; fitting to clickstream or biometric data from learning analytics datasets is needed. Third, inter-individual variability in ${\alpha }_{min}$ and C is not modelled.

Future directions include: (i) empirical validation on the Open University Learning Analytics Dataset; (ii) stochastic extension replacing $\sigma \left(t\right)$ with a stochastic process for probabilistic burnout-risk prediction; (iii) non-linear stress indicators via sigmoid activation; (iv) multi-agent cohort modelling for motivational contagion.

8. Conclusions

We introduced a stress-adaptive variable-order fractional dynamical model for human motivational intensity in which the fractional order $\alpha \left(t\right)$ varies inversely with an aggregated stress indicator $\sigma \left(t\right)$. The model is formulated directly as a causally consistent type-III variable-order Volterra integral equation, thereby closing the algebraic gap present in earlier treatments, with the quantitative difference between the causal and non-causal forms explicitly bounded in Remark 3. The stress-adaptive order is phenomenologically grounded in stress-induced working memory impairment, with a first-order linear approximation of the stress indicator. Rigorous theoretical analysis establishes well-posedness via the Banach fixed-point theorem with explicit contraction constants and uniform boundedness through the fractional Gronwall inequality. Numerical solutions are obtained using an Adams–Bashforth–Moulton predictor–corrector scheme, whose empirical convergence approaches $\mathcal{O}\left(h^2\right)$ for sufficiently small step sizes. Five numerical experiments illustrate qualitatively richer dynamics compared with the tested constant-order baselines, including a steeper decline under sustained stress, a measurable burnout gap, and faster recovery once pressure decreases. By situating the model within the ACT-R cognitive architecture, cognitive load theory, PID control models, and control knowledge tracing, we demonstrate that variable-order fractional calculus provides a principled mathematical substrate for the memory-adaptation mechanisms already hypothesised in educational psychology and intelligent tutoring systems. Empirical validation using real psychological or learning-analytics datasets together with stochastic extensions remain immediate priorities for future research.