The Spiraling Intelligence Thesis: Intelligence as a Bounded Non-Convergent Trajectory

1. Introduction

Production AI commonly follows a Train–Freeze–Deploy lifecycle: a model is trained, its weights are frozen, and the system is served as a static function $y=f(x;W)$. When conditions change, adaptation typically requires external retraining and deployment of a new checkpoint. This model of intelligence encourages fixed-point thinking: an optimal state $W^{*}$ is sought, then treated as the system.

Continual learning under distribution shift exposes a structural weakness in this paradigm. New gradients can overwrite old structure, producing catastrophic interference [1]. Considerable research mitigates this via parameter protection, replay, or modularization, but many methods retain a fixed-point intuition: a stable representation is discovered and then preserved.

Biological intelligence offers a different hint. Neural representations drift while remaining functional [2,3]. Sleep reconfigures synaptic structure rather than merely resting it [4,5]. The relevant observation is not that biology “solves” continual learning, but that biological cognition is not naturally described as convergence to a single point.

Scope and Split

Two distinct questions often get conflated: (i) whether bounded, non-convergent dynamics can reduce forgetting under shift, and (ii) whether such dynamics reduce compute cost relative to retraining. This paper addresses (i) as a theoretical and proof-of-concept claim. Cost feasibility and benchmarking are explicitly deferred to a companion paper focused on engineering evaluation.

Contributions

C1. 

A falsifiable thesis framing: intelligence as a bounded trajectory rather than a convergent parameter point, with formal definitions.

C2. 

A minimal mechanistic realization: Rotational Hebbian Learning (RHL) plus an Autopoietic Sleep Cycle.

C3. 

A formal theorem showing phase-based memory separation in RHL.

C4. 

A reproducible toy simulation demonstrating bounded non-convergence and recurrent recovery peaks under distribution switching (the qualitative signature).

C5. 

Clarification of the “non-Markovian” claim relative to deployable checkpoints.

2. Related Work

Continual learning. Approaches mitigate forgetting through parameter regularization (e.g., Elastic Weight Consolidation) [1], replay [6], or architectural strategies. Many methods implicitly assume a stable representation should be preserved; the present thesis instead treats controlled drift as a feature.

Complex-valued neural networks. Complex-valued networks represent oscillations and rotations using phase [7]. Here phase is used as a separation coordinate for temporally distinct traces, implemented as rotation in a complex weight space.

Biological drift and sleep. Representational drift [3] and synaptic homeostasis [4,5] motivate an explicit offline reorganization mode rather than purely online fitting.

Multi-timescale memory. Cascade models [8] emphasize that stable memory can require structured transitions across timescales. Consolidation in this paper plays a related role, reducing plasticity for frequently reinforced traces.

3. The Bounded Trajectory Intelligence Framework

Definition 1

(Bounded Trajectory Intelligence). A learning system implements Bounded Trajectory Intelligence if its parameter state $W_t\in {\mathbb{R}}^d$ satisfies:

1.

Non-convergence:  ${lim}_{t\to \infty }{W}_t$ does not exist (or is not a singleton).

2.

Boundedness:  ${sup}_t\parallel W_t\parallel <\infty$.

3.

Recurrent Revisitation:  For any task-relevant manifold $\mathcal{M}$, $\mathrm{dist}(W_t,\mathcal{M})$ exhibits recurrent local minima.

The Spiraling Intelligence Architecture (SIA) is proposed as a sufficient mechanism. SIA is grounded in the Infinite Transformation Principle (ITP), reframed as design axioms for trajectory-based learning.

3.1. ITP Axioms for AI Systems

Axiom I: Irreversibility. Learning updates are path-dependent and not invertible in practice:

$$W_{t+1}=\Phi (W_t,x_t),{\Phi }^{-1}\mathrm{does}\mathrm{not}\mathrm{exist}\mathrm{in}\mathrm{general}.$$

Irreversibility can be induced by consolidation, pruning, or any many-to-one map.

Axiom II: Spiraling Recurrence. Concepts must be revisited repeatedly but never identically:

$$x_{t_2}=x_{t_1}\nRightarrow W_{t_2}=W_{t_1}.$$

Recurrence implies drift; identical revisitation implies overwriting.

Axiom III: Autopoiesis. The system must include an internal reorganization mode in the absence of external supervision:

$${\displaystyle \frac{dW}{dt}}=-{\nabla }_W\mathcal{H}\left(W\right)+\xi \left(t\right).$$

This is a design claim: consolidation requires offline structure-maintenance.

Axiom IV: Memory Separation. Distinct episodes should reduce destructive interference. Separation may be geometric, temporal, or phase-based.

Remark 1

(On Markovianity). Let the deployable state be ${\tilde{W}}_t$, typically just the weight snapshot. The internal state is ${\mathcal{X}}_t=(W_t,C_t,{\widehat{\theta }}_t,\dots )$. The update ${\mathcal{X}}_{t+1}=F({\mathcal{X}}_t,x_t)$ is Markovian in ${\mathcal{X}}_t$. The internal state includes auxiliary variables ${\widehat{\theta }}_t$ which are not part of the deployable weights ${\tilde{W}}_t$. For a system deployed via frozen checkpoints, this distinction is essential.

4. Rotational Hebbian Learning (RHL)

RHL introduces controlled drift using complex-valued weights. Magnitude stores strength; phase stores temporal context.

Definition 2

(Complex Weights). Let synaptic weights be complex:

$$W_{ij}\left(t\right)=r_{ij}\left(t\right)e^{i{\theta }_{ij}\left(t\right)}\in \mathbb{C}.$$

4.1. Phase-Preserving Activation

A common complex nonlinearity preserves phase while shaping magnitude:

$$f\left(z\right)=\sigma \left(\right|z\left|\right)e^{iarg\left(z\right)}.$$

4.2. RHL Update Rule

$$W_{t+1}=e^{i{\Omega }_t}{W}_t+{\eta }_t\left(y_t{x}_t^{†}\right)(1-C_t).$$

(1)

Here ${\Omega }_t$ is a rotation rate (internal clock), ${\eta }_t$ is plasticity, and $C_t\in [0,1]$ is consolidation.

4.3. Consolidation

A minimal consolidation update:

$$C_{t+1}=(1-\kappa )C_t+\kappa \left|y_t{x}_t^{†}\right|,$$

(2)

with $\kappa \in (0,1)$ and $C_0=0$. Complete reproduction details and figure-generation scripts are provided in Appendix B.5

Theorem 1

(Phase-based Memory Separation). Consider the RHL rule (Eq. 1) with constant rotation Ω and no consolidation ($C_t=0$). For two input patterns $x^{\left(1\right)},x^{\left(2\right)}$ presented at times $t_1$ and $t_2=t_1+\Delta t$, the interference of the first memory on the second, measured by the real part of their inner product in weight space, is:

$$\Re \langle e^{i\Omega \Delta t}{x}^{\left(1\right)},x^{\left(2\right)}\rangle =cos(\Omega \Delta t)\Re \langle x^{\left(1\right)},x^{\left(2\right)}\rangle -sin(\Omega \Delta t)\Im \langle x^{\left(1\right)},x^{\left(2\right)}\rangle .$$

Corollary (Real-valued inputs).

If $x^{\left(1\right)},x^{\left(2\right)}\in {\mathbb{R}}^n$, then $\Im \langle x^{\left(1\right)},x^{\left(2\right)}\rangle =0$, and the interference term reduces to

$$\Re \langle e^{i\Omega \Delta t}{x}^{\left(1\right)},x^{\left(2\right)}\rangle =cos(\Omega \Delta t)\langle x^{\left(1\right)},x^{\left(2\right)}\rangle .$$

Thus, orthogonalisation occurs at $\Omega \Delta t=\pi /2$.

Proof. 

At encoding time $t_1$, the weight component aligned with $x^{\left(1\right)}$ is strengthened. After rotation for $\Delta t$, this component is rotated by $e^{i\Omega \Delta t}$. The interference term follows from the standard inner product in complex vector space. The corollary follows by direct substitution.    □

Remark 2.

This theorem provides a concrete, parameterized mechanism for reducing catastrophic interference without explicit buffers or gradient projections. The phase $\theta =\Omega t$ serves as a temporal coordinate separating memory traces.

5. Autopoietic Sleep Cycle

Sleep is modeled as an explicit mode that reorganizes internal structure without new external labels, implementing Axiom III (Autopoiesis).

5.1. A Homeostatic Functional

A simple energy functional combines consistency with magnitude regulation:

$$\mathcal{H}\left(W\right)=-{\displaystyle \frac{1}{2}}\sum _{i,j}\Re \left(w_{ij}{s}_i{\overline{s}}_j\right)+{\displaystyle \frac{\lambda }{2}}\sum _{i,j}\left(\right|w_{ij}{{|}^2-w_0^2)}^2.$$

Sleep dynamics follow:

$${\displaystyle \frac{dW}{dt}}=-\gamma {\nabla }_W\mathcal{H}\left(W\right)+\xi \left(t\right),$$

where $\xi \left(t\right)$ represents low-amplitude noise.

Figure 1. Wake–sleep cycling with internal observables. Sleep events (markers) coincide with reorganization episodes; stress (and/or related signals) tracks sustained mismatch and distinguishes transient error from persistent failure. This figure makes the autopoietic cycle operational rather than metaphorical.

Figure 1. Wake–sleep cycling with internal observables. Sleep events (markers) coincide with reorganization episodes; stress (and/or related signals) tracks sustained mismatch and distinguishes transient error from persistent failure. This figure makes the autopoietic cycle operational rather than metaphorical.

5.2. Minimal Replay

In the toy setting, replay can be implemented as a “dream” direction derived from the current weights or a small buffer. The present paper treats replay as a minimal mechanism to trigger re-encounter without explicit task labels.

6. Falsifiable Signature: Experiment Design

To test the thesis, the study constructs a minimal experiment where the null hypothesis (convergent dynamics) and the thesis hypothesis (bounded non-convergent dynamics) make qualitatively different predictions.

6.1. Setup

Let $w\in {\mathbb{R}}^2$ and two unit patterns $A,B$ where B is a rotation of A by ${60}^{\circ }$. Inputs are noisy normalized samples:

$$x_t={\displaystyle \frac{p+{\xi }_t}{\parallel p+{\xi }_t\parallel }},{\xi }_t\sim \mathcal{N}(0,{\sigma }^2{I}_2),p\in \{A,B\}.$$

Phases:

1.

Phase A: Task A only ($0\le t<t_A$),

2.

Phase B: Task B only ($t_A\le t<t_B$),

3.

Phase C: mixture ($t_B\le t<T_{max}$).

Alignment with Task A is tracked:

$${\mathrm{align}}_A\left(t\right)={\displaystyle \frac{w_t·A}{\parallel w_t\parallel \parallel A\parallel }}.$$

6.2. Agents

Convergent baseline (Markov). A standard projection-like Hebbian update:

$$y_t=w_t·x_t,\Delta w_t={\eta }_M{y}_t(x_t-y_t{w}_t),w_{t+1}={\displaystyle \frac{w_t+\Delta w_t}{\parallel w_t+\Delta w_t\parallel }}.$$

Spiral agent (drift + sleep). During wake:

$$w_t^{\prime }=R\left(\omega \right)w_t,y_t=w_t^{\prime }·x_t,\Delta w_t={\eta }_W{y}_t(x_t-y_t{w}_t^{\prime }),w_{t+1}={\displaystyle \frac{w_t^{\prime }+\Delta w_t}{\parallel w_t^{\prime }+\Delta w_t\parallel }}.$$

During periodic sleep in Phase C, a small self-consistency update is applied with slower rotation and mild pruning.

6.3. Hypotheses

• H0 (Convergent): After the switch, ${\mathrm{align}}_A\left(t\right)$ converges to a constant.
• H1 (Spiral): After the switch, ${\mathrm{align}}_A\left(t\right)$ exhibits stationary oscillations with recurrent peaks exceeding the convergent baseline’s plateau.

These criteria are independent of implementation details and test the thesis at the level of dynamical behavior rather than task performance.

7. Results: The Qualitative Signature

As predicted by the thesis, the spiral agent’s alignment ${\mathrm{align}}_A\left(t\right)$ in Phase C exhibits a stationary oscillatory process, while the Markov agent’s alignment converges to a constant (Figure 2).

Quantitative analysis: For the spiral agent, the time series ${\left\{{\mathrm{align}}_A\left(t\right)\right\}}_{t\in \mathrm{Phase}\mathrm{C}}$ rejects the null hypothesis of stationarity around a constant mean (Augmented Dickey-Fuller test, $p<0.01$). The autocorrelation function shows significant periodicity with period $\approx 2\pi /\omega$. Recurrent peaks exceed the Markov baseline’s plateau for $>15\%$ of Phase C duration across 100 random seeds.

Interpretation: This constitutes the falsifiable signature implied by the thesis: bounded recurrence rather than fixed-point settling. The spiral agent does not “forget” Task A; it temporarily loses alignment but recurrently re-aligns via the combination of rotational drift and sleep-driven reorganization.

Figure 3. Catastrophic interference under an $A\to B$ switch. The Markov learner rapidly collapses away from Task-A alignment after the switch, while the spiral learner preserves a bounded trajectory that later enables recurrent partial recovery in the mixed regime.

Figure 3. Catastrophic interference under an $A\to B$ switch. The Markov learner rapidly collapses away from Task-A alignment after the switch, while the spiral learner preserves a bounded trajectory that later enables recurrent partial recovery in the mixed regime.

8. Falsifiability and Boundary Conditions

The thesis makes testable predictions. It would be falsified if:

1.

In the minimal simulation, the spiral agent’s recurrent peaks are artifacts of stochasticity and do not exceed the baseline’s plateau with statistical significance.

2.

The boundedness property fails, leading to divergence or collapse.

3.

The qualitative signature disappears when scaling to slightly larger but still tractable models (e.g., a 10-neuron network) under the same principles.

The provided code enables community falsification attempts.

Figure 4. Summary over random seeds for the two-task stream. Reported quantities should include at minimum: Phase-C peak alignment to Task A, Phase-C minimum alignment (floor), and fraction of Phase-C steps where the spiral agent exceeds the Markov plateau. This aggregation guards against cherry-picked trajectories.

Figure 4. Summary over random seeds for the two-task stream. Reported quantities should include at minimum: Phase-C peak alignment to Task A, Phase-C minimum alignment (floor), and fraction of Phase-C steps where the spiral agent exceeds the Markov plateau. This aggregation guards against cherry-picked trajectories.

9. Limitations and Future Work

Theoretical scope. This study has not proven that RHL + sleep leads to bounded trajectories in high-dimensional, deep networks. This is a key open theoretical problem.

Empirical scope. The toy simulation demonstrates a qualitative signature, not state-of-the-art performance. Scaling to benchmarks with cost constraints is addressed in the companion paper.

Safety and governance. A trajectory-defined system complicates auditability relative to static checkpoints. Deployment would require logging, controlled sleep windows, and conservative gating of irreversible operations.

Extensions. Several compatible modules (meta-cognitive stress, phase alignment interfaces, structural mutation) are discussed in the companion paper but are not required for the core thesis.

10. Conclusion

The paper proposes a thesis: intelligence is better modeled as a bounded trajectory than as convergence to a fixed point. The study presents a minimal mechanism—Rotational Hebbian Learning with an Autopoietic Sleep Cycle—that instantiates this thesis and prove it separates memories via phase. Claims regarding compute efficiency and deployment feasibility are intentionally excluded from this paper. In a reproducible toy setting, observe the predicted qualitative signature: bounded non-convergence with recurrent recovery peaks under distribution switching. This work reframes continual learning from preserving fixed points to cultivating regulated trajectories.

Appendix A.1. Rotational Hebbian Learning (RHL) as a Drift-Plus-Plasticity Map

The core wake update in the complex-valued formulation is

$$W_{t+1}=e^{i{\Omega }_t}{W}_t+{\eta }_t\left(y_t{x}_t^{†}\right)(1-C_t),$$

(A1)

where $e^{i{\Omega }_t}$ is a unitary rotation (norm-preserving), ${\eta }_t$ is a bounded plasticity scale, and $C_t\in [0,1]$ is a consolidation factor that reduces further plasticity.

In the 2D real toy simulation, the analogous wake map is

$$\begin{array}{cc}\hfill w_t^{\prime }& =R\left(\omega \right)w_t,\hfill \end{array}$$

(A2)

$$\begin{array}{cc}\hfill w_{t+1}& ={\displaystyle \frac{w_t^{\prime }+{\eta }_{\mathrm{W}}{y}_t(x_t-y_t{w}_t^{\prime })}{\parallel w_t^{\prime }+{\eta }_{\mathrm{W}}{y}_t(x_t-y_t{w}_t^{\prime })\parallel }},\hfill \end{array}$$

(A3)

with $R\left(\omega \right)\in SO\left(2\right)$.

Appendix A.2. Phase Separation (Interference Suppression by Rotation)

Lemma A1

(Rotation suppresses real interference at quadrature). Let a trace W encode an episode aligned with $x^{\left(1\right)}$ at time $t_1$, and let a second query $x^{\left(2\right)}$ arrive at time $t_2=t_1+\Delta t$. Under pure rotation between episodes with constant Ω,

$$\Re \langle e^{i\Omega \Delta t}{x}^{\left(1\right)},x^{\left(2\right)}\rangle =\Re \left(e^{i\Omega \Delta t}\langle x^{\left(1\right)},x^{\left(2\right)}\rangle \right).$$

(A4)

If $\Omega \Delta t=\pi /2$ and $\langle x^{\left(1\right)},x^{\left(2\right)}\rangle$ is real, then $\Re \langle e^{i\Omega \Delta t}{x}^{\left(1\right)},x^{\left(2\right)}\rangle =0$.

Proof. 

Immediate from linearity and $\Re \left(e^{i\pi /2}a\right)=\Re \left(ia\right)=0$ for real a.    □

Appendix A.3. Boundedness in the Toy Setting (What Is Actually Guaranteed)

Two different boundedness mechanisms appear in the implementation:

(i) Explicit renormalization (2D toy).

If the update ends with normalization $w_{t+1}\leftarrow w_{t+1}/\parallel w_{t+1}\parallel$, then $\parallel w_t\parallel =1$ for all t by construction.

(ii) Homeostatic contraction (general complex form).

Let the sleep dynamics include a contraction term derived from a radial homeostatic potential

$${\mathcal{H}}_{\mathrm{homeo}}\left(W\right)={\displaystyle \frac{\lambda }{2}}{\left({\parallel W\parallel }^2-w_0^2\right)}^2,$$

(A5)

and a sleep update of the form

$$W\leftarrow W-\gamma {\nabla }_W{\mathcal{H}}_{\mathrm{homeo}}\left(W\right)+\xi ,$$

(A6)

with bounded noise $\parallel \xi \parallel \le {\xi }_{max}$. Then, outside a neighborhood of $\parallel W\parallel =w_0$, the deterministic part points inward and reduces $\parallel W\parallel$.

Theorem A1

(Toy boundedness under explicit normalization or sufficient contraction). If either (a) the algorithm renormalizes $w_t$ at each step, or (b) the sleep operator includes a contraction satisfying $\parallel W_{t+1}\parallel \le \rho \parallel W_t\parallel +b$ with $\rho <1$ applied infinitely often, then ${sup}_t\parallel W_t\parallel <\infty$.

Proof. 

Case (a) is immediate. Case (b) follows from standard affine contraction bounds.    □

Appendix A.4. Fixed Points Are not the Generic Attractor When Drift Is Persistent

In the simplest heuristic form: if consolidation saturates so that the plastic term becomes small, the wake update approaches $W_{t+1}\approx e^{i\Omega }{W}_t$. For $\Omega \ne 0\left(\mathrm{mod}2\pi \right)$, the only fixed point of the pure rotation map is $W=0$. Therefore, non-trivial fixed points are not generically stable under persistent drift; the natural long-run behaviors are bounded cycles or bounded recurrent trajectories shaped by sleep and pruning.

Appendix A.5. Clarifying the “Non-Markovian” Claim

The extended state

$${\mathcal{X}}_t=(W_t,C_t,{\mathcal{S}}_t,{\mathcal{F}}_t,{\widehat{\theta }}_t)$$

(A7)

makes the system Markovian in ${\mathcal{X}}_t$. The non-Markovian claim concerns weights alone under the deployment lens: $W_{t+1}$ depends on history through $(C_t,{\mathcal{S}}_t,{\mathcal{F}}_t,{\widehat{\theta }}_t)$, so a frozen-checkpoint view of W misses the operative state required to reproduce behavior.

Appendix B.1. Task Stream and Phases

Two unit vectors $A,B\in {\mathbb{R}}^2$ are used, with B obtained by a fixed rotation of A (e.g., ${60}^{\circ }$). Inputs are noisy normalized samples:

$$x_t={\displaystyle \frac{p+{\xi }_t}{\parallel p+{\xi }_t\parallel }},p\in \{A,B\},{\xi }_t\sim \mathcal{N}(0,{\sigma }^2{I}_2).$$

(A8)

Phases:

• Phase A: $t\in [0,t_A)$ presents only A
• Phase B: $t\in [t_A,t_B)$ presents only B
• Phase C: $t\in [t_B,T_{max})$ presents a mixture of A and B

Appendix B.2. Metrics

Alignment to Task A:

$${\mathrm{align}}_A\left(t\right)=\langle {\widehat{w}}_t,\widehat{A}\rangle .$$

(A9)

Additional summary metrics:

• Peak recovery: ${max}_{t\in \mathrm{Phase}\mathrm{C}}{\mathrm{align}}_A\left(t\right)$
• Floor (worst-case): ${min}_{t\in \mathrm{Phase}\mathrm{C}}{\mathrm{align}}_A\left(t\right)$
• Fraction above Markov plateau: ${\displaystyle \frac{1}{\left|\mathcal{T}\right|}}{\sum }_{t\in \mathcal{T}}\mathbb{I}\{{\mathrm{align}}_A^{\left(S\right)}\left(t\right)>{\mathrm{align}}_A^{\left(M\right)}\left(t\right)\}$
• Sleep density: number of sleep steps per 1000 updates (or per Phase C window)

Appendix B.3. Hyperparameters to Report (Minimum Set)

• Noise level: $\sigma$
• Phase endpoints: $t_A,t_B,T_{max}$
• Markov agent: ${\eta }_M$
• Spiral agent wake: $\omega ,{\eta }_W$
• Sleep cadence: K (one sleep step every K wake steps)
• Sleep rotation: ${\omega }_{\mathrm{sleep}}$ (e.g., $\omega /2$)
• Pruning threshold or shrink rule used during sleep

Appendix B.4. Pseudocode (Toy 2D Version)

Appendix B.5. Reproducibility and Figure Generation

All simulations reported in this paper are generated from a minimal, self-contained Python codebase released as a public reference implementation. The repository is titled:

ITP Spiraling Intelligence Simulations

and accompanies:

Atalebe, S.

The Spiraling Intelligence Architecture: Toward a Non-Markovian AI based on the Infinite Transformation Principle (ITP).

Implemented agents.

The code implements two learning agents:

• Markov Agent: a conventional Hebbian-style learner with normalization, converging to a fixed compromise representation under distributional switching.
• Spiral Agent: a non-Markovian learner incorporating

  -

  rotational Hebbian updates (phase-based drift),

  -

  autopoietic sleep cycles (replay, pruning, and contraction),

  -

  stress-driven modulation of learning rate and rotation speed.

Simulation scripts.

The following scripts generate all figures used in the paper:

• spiral_vs_markov_demo.py
  A minimal two-dimensional demonstration of weight trajectories constrained to the unit circle. This script visualizes qualitative differences between convergent and spiraling dynamics.
• two_task_sim.py
  A two-task continual learning experiment with Task A and Task B defined by rotated input patterns. This script reproduces catastrophic forgetting in the Markov agent and bounded recurrence in the Spiral agent.
• long_run_sim.py
  A long-horizon simulation illustrating spiraling representational drift, sleep-driven consolidation, and phase-based oscillations in alignment over extended time.

Generated figures.

When executed with default parameters, the scripts produce the following figures (file names shown exactly as generated):

• fig_spiral_vs_markov.png — weight trajectories for Markov and Spiral agents.
• fig_two_task_alignment.png — alignment to Task A across sequential task phases.
• fig_long_run_alignment.png — long-term alignment behavior under wake–sleep cycling.

Figures included in the main text correspond directly to these outputs, either unchanged or with cosmetic formatting adjustments only (e.g., axis labels, legend placement).

Execution environment.

All simulations were run using Python 3 with standard scientific libraries. A minimal environment can be created as follows:

python3 -m venv venv

source venv/bin/activate

pip install numpy matplotlib

No GPU acceleration or specialized hardware is required. All experiments run in seconds on a standard CPU.

Determinism and variability.

Unless otherwise stated, simulations use fixed random seeds for reproducibility. Summary plots aggregate statistics over multiple seeds where indicated. Exact seed values and hyperparameters are specified in the corresponding script headers.

Scope of reproducibility.

The released code is intended to reproduce the qualitative behaviors discussed in this paper—bounded non-convergence, recurrent recovery, and contrast with Markovian learning—rather than to optimize performance on standard benchmarks. The simplicity of the code is deliberate, to make the underlying dynamics transparent and inspectable.

Appendix C.1. Constraint-First Definition

Let $A\left(t\right)=\langle {\widehat{w}}_t,\widehat{A}\rangle$ over a horizon $t\in [t_{\mathrm{sw}},t_{\mathrm{sw}}+H]$. Feasibility:

$$\begin{array}{cc}\hfill & \exists t\in [t_{\mathrm{sw}},t_{\mathrm{sw}}+H]\mathrm{s}.\mathrm{t}.A\left(t\right)\ge A_{\mathrm{target}},\hfill \end{array}$$

(A10)

$$\begin{array}{cc}\hfill & \underset{t\in [t_{\mathrm{sw}},t_{\mathrm{sw}}+H]}{min}A\left(t\right)\ge A_{\mathrm{floor}}.\hfill \end{array}$$

(A11)

Recovery time:

$$T_{\mathrm{rec}}=min\{t-t_{\mathrm{sw}}:A\left(t\right)\ge A_{\mathrm{target}}\}.$$

(A12)

Cost model:

$$C_{\mathrm{total}}=N_{\mathrm{wake}}+c_{\mathrm{sleep}}{N}_{\mathrm{sleep}}+c_{\mathrm{retrain}}{N}_{\mathrm{retrain}}.$$

(A13)

Any compute claim is only meaningful inside the feasible set.