Non-Equilibrium Cognitive Field

A self-modifying dynamical system for proto-cognitive adaptation

This project is in active beta development. Results reported here are preliminary and subject to revision as the research matures. Contributions, critiques, and collaborations are warmly welcome.

Devanik · B.Tech ECE '26, NIT Agartala Samsung Fellowship · Indian Institute of Science

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What Is This?

NECF is an attempt to explore a question that does not yet have a satisfying answer in the literature:

Can a system's learning rules themselves be treated as dynamic state variables — evolving continuously under thermodynamic constraints — while preserving a coherent identity?

Most adaptive systems operate at two levels: a state that changes (neural activations, oscillator phases) and a rule that governs how the state changes (weights, coupling strengths). The rule is typically fixed or updated by a separate, outer-loop optimizer. NECF introduces a third level: the rule governing each node's adaptation is itself a dynamic variable, constrained by an identity curvature functional that prevents both chaotic drift and catatonic collapse.

This is not a claim about intelligence or general reasoning. It is a study of whether Level-3 meta-rule dynamics produces measurably different adaptive behavior from Level-1 (fixed rules) and Level-2 (globally adaptive coupling) baselines — and whether that difference can be characterized rigorously.

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Table of Contents

1. Theoretical Framework
2. The Three Levels of Dynamics
3. Mathematical Specification
4. Identity Curvature Functional
5. Epistemic Contagion — The Core Mechanism
6. External Driving and Non-Equilibrium Structure
7. Seven Falsifiable Predictions
8. Experimental Results (Beta)
9. Relationship to Prior Work
10. What NECF Is Not
11. Codebase Structure
12. Installation and Usage
13. Daily Research Log
14. Known Limitations
15. Roadmap
16. References

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1. Theoretical Framework

1.1 Motivation

The standard picture of an adaptive system is a two-level hierarchy:

$$ \text{Level 1: } \frac{d\phi}{dt} = F(\phi,, \mathcal{L}) $$

$$ \text{Level 2: } \frac{d\mathcal{L}}{dt} = 0 \quad \text{(fixed)} \quad \text{or} \quad G(\mathcal{L}, \nabla_\phi \mathcal{J}) \quad \text{(trained by outer loop)} $$

where $\phi$ is the system state, $\mathcal{L}$ is the learning rule, and $\mathcal{J}$ is a loss function. The key characteristic of Level-2 systems is that the function $G$ itself is fixed — it does not adapt. Meta-learning approaches like MAML (Finn et al., 2017) find a good initialization for $\mathcal{L}$, but the update rule for $\mathcal{L}$ during deployment is still fixed.

NECF proposes a genuine third level:

$$ \text{Level 3: } \frac{d\mathcal{L}_i}{dt} = \underbrace{F_{\text{contagion}}(\mathcal{L}_i,, \varepsilon_i,, W)}_{\text{Boltzmann epistemic contagion}} - \lambda, \underbrace{\nabla_{\mathcal{L}_i} \mathcal{H}[\mathcal{L}]}_{\text{identity gradient}} $$

where $F_{\text{contagion}}$ is an error-driven, thermodynamically-weighted rule propagation mechanism, and $\mathcal{H}[\mathcal{L}]$ is an identity curvature functional that constrains the evolution. Critically, the rule governing $\mathcal{L}$'s evolution is not fixed — it adapts through the Boltzmann-weighted averaging of neighbor rules, making the rule-evolution process itself a genuine spatial field phenomenon.

1.2 Physical Intuition

The system draws on three physical analogies:

Kuramoto model (Kuramoto, 1975): $N$ coupled oscillators with natural frequencies $\omega_i$ synchronize above a critical coupling $K_c$. NECF uses the amplitude-weighted Kuramoto model as its Level-1 substrate, with local coupling strength $\beta_i$ drawn from the rule field rather than a global constant.

Dissipative structures (Prigogine, 1977): Open thermodynamic systems driven far from equilibrium can spontaneously organize into ordered states. NECF is explicitly constructed as an open system — driven by Lorenz chaotic input, periodic signals, and Poisson spikes — to prevent equilibration and maintain perpetual self-organization.

Active inference (Friston, 2017): The Curiosity Engine monitors prediction error and drives the system toward states of minimal free energy. NECF's departure from standard active inference is that the minimization mechanism itself (the $\mathcal{L}$ field) is dynamic.

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2. The Three Levels of Dynamics

The following taxonomy situates NECF in the landscape of adaptive systems:

Level	What evolves	What is fixed	Representative systems
0	Nothing	Everything	Lookup tables, hardcoded controllers
1	State $\phi$	Rule $\mathcal{L}$	Standard Kuramoto, fixed-weight neural networks
2	State $\phi$, coupling $K_{ij}$	Rule for updating $K$	Adaptive Kuramoto (Ha et al., 2016), gradient descent
3	State $\phi$, rule $\mathcal{L}$	Identity curvature $\mathcal{H}$	NECF (this work)
4	State, rule, and $\mathcal{H}$	—	Hypothetical; not yet formalized

The specific claim of this project is that Level-3 dynamics — where $\mathcal{L}$ evolves as a continuous field variable under the constraint of a fixed identity functional — has not been previously formalized as a unified implementable architecture in the coupled oscillator literature. A March 2026 literature search confirmed that:

• Standard Kuramoto (1975): $K$ is a global scalar, fixed
• Adaptive Kuramoto (Ha et al., SIAM 2016): $K_{ij}$ evolves, but the rule governing $K$'s evolution is fixed
• Hopfield-Kuramoto (arXiv 2505.03648, May 2025): joint memory model, no meta-rule evolution
• MAML (Finn et al., 2017): learns an initialization for $\mathcal{L}$, not a continuously evolving rule field with identity constraints
• Friston FEP (2005–2022): minimizes free energy via a fixed variational update rule

The gap NECF occupies is the combination of (a) per-node rule fields, (b) Boltzmann-weighted epistemic contagion between rules, (c) identity curvature constraint on rule evolution, and (d) Lyapunov-gated rollback — unified in a single dynamical system.

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3. Mathematical Specification

3.1 State Representation

Each node $i \in {1, \ldots, N}$ carries a complex state:

$$ \phi_i(t) = A_i(t), e^{i\theta_i(t)} $$

• $A_i \in (0, 1]$: amplitude — interpreted as local confidence or energy
• $\theta_i \in [0, 2\pi)$: phase — causal alignment / temporal coordinate

The Kuramoto order parameter quantifies global synchrony:

$$ r(t), e^{i\psi(t)} = \frac{1}{N} \sum_{j=1}^N A_j, e^{i\theta_j} $$

$r \in [0, 1]$ with $r = 0$ (incoherent) and $r = 1$ (fully synchronized).

3.2 Local Learning Rule Field

Each node carries a rule vector:

$$ \mathcal{L}_i(t) = \bigl(\alpha_i(t),; \beta_i(t),; \gamma_i(t)\bigr) $$

Parameter	Role	Initial value
$\alpha_i$	Error sensitivity — governs amplitude decay rate	0.30
$\beta_i$	Coupling strength — governs phase synchrony pull	0.80
$\gamma_i$	Curiosity weight — governs uncertainty-seeking	0.10

3.3 Level-1 Dynamics: Field Evolution

Phase update (amplitude-weighted generalized Kuramoto with curiosity and external driving):

$$ \frac{d\theta_i}{dt} = \underbrace{\omega_i}_{\text{intrinsic}} + \underbrace{\beta_i \cdot \frac{1}{N} \sum_{j=1}^N W_{ij}, A_j, \sin(\theta_j - \theta_i)}_{\text{synchrony pull}} + \underbrace{\gamma_i \cdot \nabla_{\theta_i} U(\theta_i)}_{\text{curiosity gradient}} + \underbrace{\Delta_{\text{ext}}(t)}_{\text{Lorenz + spikes}} $$

where $W_{ij} \sim \mathcal{U}(0.5, 1.5)$ is the symmetric coupling matrix (zero diagonal) and $\Delta_{\text{ext}}(t)$ captures spatially distributed Lorenz kicks and Poisson phase resets (Section 6).

Curiosity potential — approximated via circular KDE with Silverman bandwidth $h = 1.06,\hat{\sigma}_\theta, N^{-1/5}$:

$$ p(\theta_i) \approx \frac{1}{N} \sum_{j=1}^{N} \frac{1}{h} \exp!\left(-\frac{\sin^2(\theta_i - \theta_j)}{2h^2}\right), \qquad U(\theta_i) = -\log p(\theta_i \mid \text{context}) $$

Nodes in dense, coherent phase regions experience a low $|\nabla U|$; nodes in sparse, chaotic regions experience a high gradient, pulling them toward informationally underexplored territory.

Amplitude update (error-driven with noise and periodic modulation):

$$ \frac{dA_i}{dt} = -\alpha_i, \varepsilon_i(t), A_i + \sigma, \eta_i(t) + \varepsilon_s \sin(2\pi f_s t) $$

where $\varepsilon_i(t) = \sin^2!\bigl((\theta_i - \psi)/2\bigr) \in [0,1]$ is the local prediction error (squared circular distance from the mean-field phase $\psi$), $\eta_i(t) \sim \mathcal{N}(0,1)$ is a Wiener process with amplitude $\sigma = 0.02$ (thermodynamic floor preventing amplitude collapse to zero), and $\varepsilon_s \sin(2\pi f_s t)$ is the periodic amplitude modulation driver ($\varepsilon_s = 0.03$, $f_s = 0.1$).

Mean-field critical coupling (Kuramoto, 1975; Strogatz, 2000):

$$ K_c = \frac{2}{\pi, g(\Omega)} = \frac{2\sigma_\omega\sqrt{2\pi}}{\pi} \approx 0.4787 \quad \text{for } \sigma_\omega = 0.3 $$

At initialization $\beta_i \approx 0.80$, giving an effective coupling $K_{\text{eff}} = \beta_i \cdot K \cdot \bar{W} \approx 0.56 > K_c$, placing the field in the synchronizing regime from the outset.

3.4 Level-3 Dynamics: Meta-Rule Evolution

$$ \frac{d\mathcal{L}_i}{dt} = \underbrace{F_{\text{contagion}}(\mathcal{L}_i,, \varepsilon_i,, W)}_{\text{Boltzmann epistemic contagion}} ;-; \lambda, \underbrace{\nabla_{\mathcal{L}_i} \mathcal{H}[\mathcal{L}]}_{\text{identity gradient}} $$

This is a dynamical system in rule space, not gradient descent on a loss function. The contagion term $F$ propagates rules from low-error nodes to high-error nodes; the identity gradient $\nabla \mathcal{H}$ prevents unlimited drift.

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4. Identity Curvature Functional

The central object distinguishing NECF from prior adaptive systems is $\mathcal{H}[\mathcal{L}]$:

$$ \mathcal{H}[\mathcal{L}] = \underbrace{\frac{1}{N} \sum_{i=1}^N |\mathcal{L}_i(t) - \mathcal{L}_i^{(0)}|^2}_{\text{drift penalty}} + \underbrace{\kappa; \overline{\text{Var}}(\mathcal{L}_i)}_{\text{collapse penalty}} $$

Properties:

• $\mathcal{H}[\mathcal{L}] = 0$ at initialization by construction
• $\mathcal{H} \to \infty$ in both failure modes:
  • Chaotic drift: each $\mathcal{L}_i$ wanders freely → drift term grows
  • Catatonic collapse: all $\mathcal{L}_i \to \bar{\mathcal{L}}$ (identical) → variance penalty activates
• The viable regime corresponds to $\mathcal{H}$ bounded in a system-dependent interval

The gradient used in the meta-dynamics update:

$$ \nabla_{\mathcal{L}_i} \mathcal{H} = \frac{2}{N}\bigl(\mathcal{L}_i - \mathcal{L}_i^{(0)}\bigr) + \frac{2\kappa}{N}\bigl(\mathcal{L}_i - \bar{\mathcal{L}}\bigr) $$

where $\bar{\mathcal{L}} = \frac{1}{N}\sum_i \mathcal{L}_i$ is the field mean. The first term is an elastic restoring force toward each node's structural origin; the second term is a repulsive field centered on the mean — it pushes nodes away from homogenization, preserving spatial rule diversity.

4.1 Rollback Mechanism

At each step, the system checks whether identity curvature has increased too rapidly:

$$ \delta\mathcal{H}(t) = \mathcal{H}[\mathcal{L}(t)] - \mathcal{H}[\mathcal{L}(t - \Delta t)] $$

If $\delta\mathcal{H} > \delta_{\text{thresh}}$ (default $0.30$), the system applies a thermodynamic rollback:

$$ \mathcal{L}(t) \leftarrow \mathcal{L}(t - \Delta t) - \eta_{\text{rb}} \cdot \nabla_\mathcal{L} \mathcal{H}[\mathcal{L}(t - \Delta t)], \quad \eta_{\text{rb}} = 0.05 $$

This is not gradient descent — it is a reversion to the previous state plus a corrective step in the direction of lower identity curvature. The combined effect prevents the same spike from recurring immediately and provides a Lyapunov stability certificate for the rule field over $T \to \infty$.

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5. Epistemic Contagion — The Core Mechanism

The contagion term $F_{\text{contagion}}$ implements thermodynamically-weighted rule propagation. The key design decision — and the site of the primary engineering fix — is the choice of influence weights.

5.1 The Singularity Problem (Rejected Design)

A naive inverse-error weighting:

$$ w_j^{\text{naive}} = \frac{1}{\varepsilon_j + \epsilon} $$

is singular. For $\varepsilon_j = 10^{-5}$, $w_j \approx 10^5$. After normalization this degenerates to a one-hot vector, snapping the entire field to the rule of whichever single node happens to achieve near-zero error on that timestep. This is not field dynamics — it is a discrete logic switch masquerading as a continuous system.

5.2 Boltzmann Softmax Weights (Adopted Design)

$$ w_j(\kappa) = \frac{\exp(-\varepsilon_j / \kappa)}{\displaystyle\sum_{k=1}^N \exp(-\varepsilon_k / \kappa)} $$

implemented with the log-sum-exp trick for numerical stability:

log_w = -eps / kappa
log_w -= log_w.max()          # stability
w = np.exp(log_w)
w /= w.sum() + 1e-15

The weight is globally $C^\infty$; its sensitivity to error is:

$$ \frac{\partial w_j}{\partial \varepsilon_j} = -\frac{1}{\kappa}, w_j(1 - w_j) $$

which is bounded by $[-1/(4\kappa),, 0]$ for all $\varepsilon_j$ — preventing unbounded weight gradients and guaranteeing mathematical stability of the numerical integration step.

Properties of the Boltzmann weights:

$\kappa$	Character	$H_w = -\sum w_j \ln w_j$	Max weight
$\to 0$	Winner-takes-all	$\to 0$	$\to 1$
$0.10$	Default (selective)	$\approx 3.85$	$\approx 0.053$
$0.50$	Diffuse	$\approx 4.15$	$\approx 0.021$
$\to \infty$	Uniform	$\ln N$	$1/N$

At $\kappa = 0.10$, the maximum weight across a 64-node field is approximately $0.016$, compared to $\sim 90{,}000$ under the naive formula — a six-order-of-magnitude reduction that preserves genuine field diffusion.

5.3 Contagion Update

For each node $i$, the Boltzmann-weighted target rule is:

$$ \mathcal{L}_i^{\text{target}} = \frac{\sum_j W_{ij}, w_j, \mathcal{L}_j}{\sum_j W_{ij}, w_j + \epsilon} $$

The contagion contribution to $d\mathcal{L}_i/dt$:

$$ F_{\text{contagion},i} = \boldsymbol{\mu} \odot \bigl(\mathcal{L}_i^{\text{target}} - \mathcal{L}_i\bigr) \cdot \varepsilon_i $$

where $\boldsymbol{\mu} = (\mu_\alpha, \mu_\beta, \mu_\gamma) = (0.05, 0.05, 0.05)$ and $\odot$ is elementwise multiplication. The factor $\varepsilon_i$ acts as receptivity: high-error nodes update more strongly toward the rules of their low-error neighbors; low-error nodes are stubborn — they resist change, acting as stable anchors from which contagion radiates.

Two-group mixing time (analytically derived and empirically verified):

$$ \tau_{\text{mix}}(\kappa) \approx \frac{1}{\mu, \varepsilon_h, w_{\text{low}}(\kappa), \Delta t} $$

At the default $\kappa = 0.10$, $\mu = 0.50$ (accelerated test), $\varepsilon_h = 0.25$: $\tau_{\text{emp}} = 878$ steps, $\tau_{\text{theory}} = 708$ steps — relative error $\approx 19%$, consistent with the non-uniform coupling matrix correction.

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6. External Driving and Non-Equilibrium Structure

NECF is deliberately constructed as an open thermodynamic system. Three driving mechanisms prevent the field from reaching equilibrium:

6.1 Lorenz Chaotic Driver

The Lorenz attractor $(x, y, z)$ with standard parameters $(\sigma=10,, \rho=28,, \beta=8/3)$ injects a spatially-distributed, deterministic chaotic phase perturbation:

$$ \Delta\theta_i^{\text{Lorenz}} = \varepsilon_L \cdot \frac{x(t)}{25} \cdot \sin!\left(\frac{2\pi i}{N}\right), \quad \varepsilon_L = 0.05 $$

The spatial factor $\sin(2\pi i/N)$ is essential: nodes near $i = N/4$ receive a strong positive kick while nodes near $i = 3N/4$ receive an equal negative kick, physically tearing the field in half in a deterministic, aperiodic, structured manner — a far richer perturbation regime than white noise.

6.2 Periodic Signal

A structured sinusoidal driver creates resonance opportunities and modulates node amplitudes globally:

$$ \Delta A_i^{\text{periodic}} = \varepsilon_s \cdot \sin(2\pi f_s t), \quad \varepsilon_s = 0.03,\quad f_s = 0.1 $$

6.3 Poisson Phase Resets

At each step, each node independently spikes with probability $\lambda_s = 0.02$, resetting its phase to $\mathcal{U}(0, 2\pi)$. This is the mechanism responsible for the masked Lyapunov proxy: a spiked node can jump from $\theta = 0.1$ to $\theta = 5.9$ in one step — a phase delta of $5.8$ radians. The raw $|\delta\theta|$ would explode, the rollback would trigger continuously, and all dynamics would freeze. The fix is to compute phase divergence only on nodes that were not spiked at $t$ or $t-1$:

$$ \hat{\lambda}_1 = \frac{1}{\Delta t}, \log!\left(\frac{|\delta\theta_{\sim\text{spike}}|}{\sqrt{N_{\text{valid}}}} + 10^{-10}\right) $$

where $\sim\text{spike}$ denotes exclusion of spiked nodes at both $t$ and $t-1$, and $N_{\text{valid}}$ is the number of unmasked nodes. When $N_{\text{valid}} < 2$ (edge case: near-total spike event), the estimator returns NaN and the step is classified SPIKE_DOMINATED — a transient physical event, not internal chaos. The spike mask is explicitly threaded from environment.step() through field.step() to observer.record() at every timestep.

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7. Seven Falsifiable Predictions

NECF makes seven quantitative predictions, each independently falsifiable:

#	Observable	Prediction	Falsification condition
P1	Order parameter $r(t)$	Rises from $\sim 0.1$ to $> 0.5$ within viable regime	$r < 0.2$ at all times
P2	Identity curvature $\mathcal{H}[\mathcal{L}]$	Remains bounded in $(0.1,, 5.0)$ throughout	$\mathcal{H}$ diverges or collapses to zero
P3	Mean prediction error $\bar{\varepsilon}$	Decreases after $\sim 200$ steps	$\varepsilon$ monotonically increases
P4	Masked Lyapunov proxy $\hat{\lambda}_1$	Stays in $(-0.5,, 0.8)$	$\hat{\lambda}_1 > 1.5$ sustained for $> 50$ steps
P5	Rule diversity $\text{Var}(\mathcal{L}_i)$	Bounded; neither collapses to zero nor diverges	$\text{Var}(\mathcal{L}_i) \to 0$
P6	Curiosity directives	Fires within 30 steps of plateau detection	Never fires, or fires on non-plateau steps
P7	Rollback rate	Decreases over the course of a run	Rate increases monotonically

These predictions are what distinguish NECF from a philosophical proposal. A system that satisfies all seven is behaving as the theory predicts; a system that fails any one provides specific information about which component requires revision.

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8. Experimental Results (Beta)

⚠️ All results below are preliminary. $N$ is small (16–64 nodes), run lengths are short ($T \leq 2000$ steps), and no task-level benchmark has been applied. These are substrate-characterization experiments only.

8.1 Synchronization Onset (T0)

Sweep over $K \in [0.05, 2.50]$, $N = 64$, $\sigma_\omega = 0.3$:

$K$	$\bar{r}$
$0.30$	$0.162$
$K_c^{\text{th}} = 0.479$	$0.109$
$0.60$	$0.405$
$1.00$	$0.820$
$2 K_c^{\text{th}} = 0.958$	$0.928$

Empirical $K_c$ (first $K$ where $r > r_{\text{bg}} + 3\sigma_{\text{noise}}$, with $r_{\text{bg}} = N^{-1/2} \approx 0.125$) is approximately $0.89$, compared to the mean-field prediction $0.479$. The deviation is consistent with finite-size broadening: for $N = 64$ the stochastic background elevates the apparent threshold.

The order parameter scales approximately as $r \sim (K - K_c)^\beta$ with fitted $\hat{\beta}$ near $0.5$, consistent with mean-field universality.

8.2 Boltzmann Temperature Scan (T1)

Over 22 values of $\kappa \in [0.01, 5.00]$, the weight entropy $H_w(\kappa) = -\sum_j w_j \ln w_j$ increases monotonically from near-zero (winner-takes-all) toward $\ln N$ (uniform). The optimal operating point $\kappa^* = \arg\max_\kappa H_w(\kappa) \cdot r(\kappa)$ balances discrimination against field diversity. The default $\kappa = 0.10$ lies in the selective regime with $H_w \approx 3.85$ (out of $\ln 32 \approx 3.47$ maximum for $N=32$).

8.3 Identity Stability Landscape (T2)

Grid sweep over $\lambda \in {0.01, 0.05, 0.10, 0.25, 0.50, 1.00}$ and $\delta_{\text{thresh}} \in {0.10, 0.20, 0.30, 0.50, 0.80}$:

• 67% of the explored $(\lambda, \delta)$ space produces viable dynamics at $T = 400$ steps
• Regime boundaries: VIABLE / CATATONIC ($r < 0.04$) / DRIFTED ($\mathcal{H} > 3$) / ROLLBACK-HEAVY
• Default values $(\lambda = 0.10,, \delta = 0.30)$ fall in the VIABLE cell

8.4 Lyapunov Spectrum (T3)

Full spectrum via continuous QR decomposition (Benettin et al., 1980), $N = 16$, $K = 0.70$:

$$ \lambda_1 = +0.173,\quad \lambda_2 = +0.049,\quad \lambda_3 = -0.060,; \ldots $$

Kaplan–Yorke dimension:

$$ D_{\text{KY}} = j + \frac{\sum_{k=1}^{j} \lambda_k}{|\lambda_{j+1}|} = 4.84 $$

indicating a fractal strange attractor of dimension $\approx 4.8$ embedded in the 16-dimensional phase space. This places the system in the bounded-chaos regime — neither frozen ($\lambda_1 < 0$) nor explosively divergent ($\lambda_1 \gg 1$).

8.5 Epistemic Contagion Rate (T4)

Two-group mixing time as a function of $\kappa$ (pure contagion, $\mu = 0.50$, $T = 2500$):

$\kappa$	$\tau_{\text{emp}}$	$\tau_{\text{theory}}$	Rel. error
$0.02$	$666$	$667$	$0.1%$
$0.05$	$670$	$671$	$0.1%$
$0.10$	$708$	$721$	$1.8%$
$0.30$	$878$	$956$	$8.2%$
$1.00$	$1027$	$1186$	$13.4%$

Power law: $\tau_{\text{mix}} \sim \kappa^{0.121}$, $R^2 = 0.92$. Higher temperature slows mixing; the analytical formula $\tau = 1/(\mu, \varepsilon_h, w_{\text{low}}(\kappa), \Delta t)$ predicts the trend with good accuracy at low $\kappa$.

8.6 Free Energy Topology (T5)

From 40 random initialisations ($N = 32$, $K = 0.80$, $T = 600$ settling steps): 11 distinct attractor basins detected by clustering final mean phases. Lorenz-perturbation escape rate: $k_{\text{esc}} \approx 0.38$, indicating a moderately fragile landscape where roughly 38% of settled trajectories are dislodged by a 50-step Lorenz kick of amplitude 0.10.

8.7 Ablation Study (T6)

$N = 32$, $T = 600$, $K = 0.55$, $n = 25$ trials per condition:

Condition	$\bar{r} \pm \sigma$	$\text{Var}(\mathcal{L})$
Level-1 (frozen $\mathcal{L}$)	$0.0335 \pm 0.037$	$9.5 \times 10^{-5}$
Level-2 (global $\beta$ adapts)	$0.0335 \pm 0.037$	$9.5 \times 10^{-5}$
Level-3 (full NECF)	$0.0399 \pm 0.043$	$7.2 \times 10^{-5}$

Welch's $t$-test (L3 vs L1): $t = 0.436$, $p = 0.665$, Cohen's $d = 0.31$.

The difference in $r$ is not statistically significant at $T = 600$. This is expected: the contagion timescale at $\mu = 0.05$ is $\tau_{\text{mix}} \approx 8{,}000$ steps, so $T = 600$ captures only very early adaptation. The $\text{Var}(\mathcal{L})$ ratio (Level-3 / Level-1 $\approx 0.76\times$) confirms that the meta-dynamics is active and measurably modifying the rule field, even when the effect on $r$ is not yet visible.

Interpretation: The current experiments are substrate characterization, not performance benchmarking. The ablation will be re-run at $T \geq 5{,}000$ once compute budget permits.

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9. Relationship to Prior Work

NECF is built on and departs from several bodies of literature:

9.1 Kuramoto Model

The standard Kuramoto model (Kuramoto, 1975; Strogatz, 2000) uses a fixed global coupling $K$. Adaptive extensions (Ha, Kim & Zhang, SIAM 2016; Berner et al., 2021) allow $K_{ij}(t)$ to evolve, but the rule governing $K_{ij}$'s evolution is fixed. NECF departs by making the local coupling strength $\beta_i$ a component of the dynamically-evolving rule field $\mathcal{L}_i$.

9.2 Free Energy Principle

Friston's Free Energy Principle (2005–2022) provides the theoretical foundation for prediction-error-driven adaptation and the curiosity/active inference components. NECF's departure is that the variational update rule (what FEP holds fixed) is itself a dynamic variable — the $\mathcal{L}$ field plays the role of a variational family whose parameters co-evolve with the belief state.

9.3 Meta-Learning

MAML (Finn, Abbeel & Levine, 2017) and its descendants learn a good initialization $\mathcal{L}_0$ such that few gradient steps suffice for a new task. NECF is not an initialization-finding algorithm. It is a continuous-time dynamical system in which $\mathcal{L}$ evolves in deployment, not through explicit gradient computation on a loss, but through thermodynamically-weighted peer-to-peer rule exchange.

9.4 Dissipative Structures

Prigogine's theory of dissipative structures (1977) demonstrates that open thermodynamic systems far from equilibrium can spontaneously organize into ordered states through energy dissipation. NECF uses this principle architecturally: the Lorenz driving and Poisson spikes ensure the system never equilibrates, creating the conditions for persistent self-organization.

9.5 Causal Entropy / Empowerment

The proto-will mechanism draws on causal entropy maximization (Klyubin, Polani & Nehaniv, 2005; Salge, Glackin & Polani, 2014), where an agent selects actions that maximize the logarithm of future state-space volume accessible under its policy. NECF's attractor selection formula:

$$ a^* = \arg\max_a \bigl[H_{\text{causal}}(a) - \mu_{\text{will}}, D_{\text{identity}}(a)\bigr] $$

where $H_{\text{causal}}(a) = -\sum_{S'} p(S' \mid a)\log p(S' \mid a)$ is the Shannon entropy of future states reachable from attractor $a$, and $D_{\text{identity}}(a) = \frac{1}{N}\sum_i |\mathcal{L}{i,a} - \mathcal{L}{i,\text{current}}|^2$ is the structural cost of transitioning to that attractor. At $\mu_{\text{will}} = 0$: pure exploration (destructive). At $\mu_{\text{will}} \to \infty$: complete paralysis. The default $\mu_{\text{will}} = 1.0$ is a constrained empowerment maximizer — seeking influence while preserving identity.

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10. What NECF Is Not

In the interest of accuracy, several things should be clearly stated:

NECF is not a cognitive architecture. It has no perception layer, no task representation, and no mechanism for mapping external stimuli to field states or extracting decisions from phase patterns.

NECF has not been benchmarked on any task. No classification accuracy, no ARC score, no game score. The experiments reported are purely substrate-characterization: does the field behave as the theory predicts?

NECF does not demonstrate intelligence or reasoning. It demonstrates that a coupled oscillator field with Level-3 meta-rule dynamics behaves differently from Level-1 and Level-2 baselines in specific, measurable ways. Whether that difference is relevant to cognitive function is an open question.

NECF is not a refutation of neural networks. It is an exploration of a different computational primitive — one that may complement, rather than replace, existing approaches.

The honest position is that NECF is a proto-cognitive substrate — a dynamical system with properties that may be useful building blocks for more complete architectures, but which is not itself a reasoning system.

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11. Codebase Structure

necf/
├── app.py               # Streamlit live dashboard — entry point
├── config.py            # All hyperparameters (single source of truth)
├── field.py             # Level-1: oscillator field state + Numba-compiled Kuramoto
├── meta_dynamics.py     # Level-3: Boltzmann epistemic contagion + rule evolution
├── identity.py          # H[L] curvature functional + Lyapunov-gated rollback
├── curiosity.py         # CuriosityEngine: plateau detection + exploration directives
├── environment.py       # External drivers: Lorenz attractor, periodic, Poisson spikes
├── observer.py          # All observables: r(t), masked λ_max, entropy, regime classifier
├── experiment.py        # Run loop with ablation helpers (Level-1, Level-2, Level-3)
└── requirements.txt

.github/
└── workflows/
    └── daily_necf_research.yml   # Automated daily research session (no LLM)

docs/
├── INDEX.md             # Session index — one entry per day
└── sessions/
    └── YYYY-MM-DD_<slug>.md    # Technical research note per session

Key Design Decisions

field.py — Numba JIT kernel: The $O(N^2)$ Kuramoto phase-difference computation is compiled via @numba.njit(parallel=True, fastmath=True), pre-warmed during __init__ to avoid first-step latency in Streamlit. Falls back to NumPy broadcasting if Numba is unavailable.

observer.py — Masked Lyapunov: The Lyapunov proxy excludes nodes spiked at $t$ and $t-1$ (a spike at $t-1$ still corrupts the delta at $t$). Returns NaN with n_valid_lyapunov when coverage is insufficient, rather than a spurious value.

experiment.py — Ablation factories: NECFExperiment.level1_only() and .level2_adaptive_coupling() return pre-configured instances for controlled comparison. The ablation factories override meta.step rather than duplicating logic, ensuring identical dynamics up to the rule-update layer.

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12. Installation and Usage

Requirements

python >= 3.11
numpy >= 1.24
scipy >= 1.11
streamlit >= 1.32
matplotlib >= 3.7
numba >= 0.58      # optional but strongly recommended

Install

git clone https://github.com/Devanik21/NECF.git
cd NECF
pip install -r requirements.txt

Launch Dashboard

streamlit run app.py

Navigate to http://localhost:8501. The dashboard provides:

• Real-time phase polar plot and order parameter trace
• All 7 observable metrics updated live
• Rule field distribution histograms
• Identity curvature $\mathcal{H}[\mathcal{L}]$ phase diagram
• Curiosity Engine directive timeline
• Regime classifier (VIABLE / CHAOTIC / CATATONIC)

Scripted Usage

from config import NECFConfig
from experiment import NECFExperiment

# Full NECF (Level-3)
exp = NECFExperiment(NECFConfig(seed=42))
for snap in exp.stream():
    print(f"t={snap.t:4d}  r={snap.r:.4f}  H={snap.H:.4f}  λ={snap.lyapunov_proxy:+.4f}")

# Ablation baselines
exp_l1 = NECFExperiment.level1_only()          # fixed rules
exp_l2 = NECFExperiment.level2_adaptive_coupling()  # global β only

Running the Ablation Study

import numpy as np
from scipy.stats import ttest_ind
from experiment import NECFExperiment
from config import NECFConfig

N_TRIALS = 25
results = {1: [], 2: [], 3: []}

for seed in range(N_TRIALS):
    cfg = NECFConfig(seed=1000 + seed)
    for level, factory in [(1, NECFExperiment.level1_only),
                            (2, NECFExperiment.level2_adaptive_coupling),
                            (3, NECFExperiment)]:
        exp = factory(cfg)
        history = exp.run_batch()
        results[level].append(history.snapshots[-1].r)

r1, r3 = np.array(results[1]), np.array(results[3])
t, p = ttest_ind(r3, r1)
print(f"L3 vs L1: t={t:.3f}, p={p:.4f}, d={( r3.mean()-r1.mean())/np.sqrt((r3.var()+r1.var())/2):.3f}")

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13. Daily Research Log

A GitHub Actions workflow (.github/workflows/daily_necf_research.yml) runs once per day at 06:00 UTC and performs a real numerical experiment on the NECF substrate, writing the results to docs/sessions/. Seven analysis types rotate by day-of-year modulo 7:

Topic index	Analysis
T0	Synchronization onset and critical coupling — sweep $K$, fit $\beta$
T1	Boltzmann temperature scan — optimal $\kappa$ for rule-field diffusion
T2	Identity curvature stability landscape — phase diagram in $(\lambda, \delta)$ space
T3	Lyapunov spectrum via continuous QR decomposition
T4	Epistemic contagion rate constant — two-group mixing time vs $\kappa$
T5	Free energy topology — attractor basin counting and escape rates
T6	Ablation study — Level-1 vs Level-2 vs Level-3 comparison

Each session generates a markdown file with:

• Full parameter specification and seed
• LaTeX-formatted mathematical derivation of the relevant theory
• Numerical results table
• Comparison to analytical predictions where available
• Interpretation and connection to the broader NECF framework

The workflow uses no LLM and no external API — all content is computed from first principles using NumPy and SciPy. The commit message encodes the key numerical result, e.g.:

necf(NECF-2026-077-T6): ablation level comparison [L3_r=0.0399 L1_r=0.0335 p=0.2852 d=0.3120]

See docs/INDEX.md for the full session archive.

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14. Known Limitations

These are not future work items — they are current, known gaps in the architecture that affect the interpretation of all results:

No task interface. The field has no mechanism for receiving structured input or producing structured output. All experiments are self-contained dynamical characterizations. Whether the field's synchrony and rule diversity are relevant to any task-level performance is currently unknown.

Short run lengths. The contagion timescale at $\mu = 0.05$ is $\tau_{\text{mix}} \approx 8{,}000$ steps. Current experiments run to $T \leq 2{,}000$, capturing only the earliest phase of adaptation. The ablation results in particular should be treated as preliminary until longer runs are available.

Small $N$. Results are for $N \in {16, 32, 64}$. Scaling behavior — whether the viable regime survives as $N \to 256$ or $N \to 1024$ — has not been studied.

Lyapunov proxy accuracy. The masked $\hat{\lambda}_1$ is an approximation of the true maximal Lyapunov exponent. It has not been cross-validated against the full QR-method spectrum at matched parameters.

Analytical gap. The Kaplan–Yorke dimension formula $D_{\text{KY}} = j + \sum_{k=1}^j \lambda_k / |\lambda_{j+1}|$ assumes a smooth, ergodic attractor. Whether the NECF attractor satisfies these conditions under open-system driving is an open question.

No comparison to learned baselines. Level-1 and Level-2 ablations are the only baselines. A comparison to a gradient-trained RNN or a transformer-based adaptive system on the same substrate task would significantly strengthen the ablation.

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15. Roadmap

The following phases are planned but have not yet begun. Timelines are estimates only.

Phase 1 — Longer runs and scaling study (next 4–6 weeks) Run ablation at $T = 10{,}000$ steps to cross the contagion timescale. Sweep $N \in {32, 64, 128, 256}$ to characterize finite-size effects on the viable regime boundary.

Phase 2 — Task interface (6–12 weeks) Design a perception layer that maps discrete input patterns to structured phase perturbations, and a readout mechanism that extracts categorical decisions from the order parameter. Apply to a simple pattern classification task.

Phase 3 — Learned transformation spaces (12–20 weeks) Replace the fixed DSL primitives (from the companion LAteNT project) with embeddings learned from ARC-style input-output pairs. This connects NECF's rule-field dynamics to the program synthesis literature.

Phase 4 — Cross-domain generalization measurement (20–28 weeks) Train on Task Domain A, validate on Domain B, test on Domain C (entirely unseen transformation types). Measure transfer rates as a function of $\tau_{\text{mix}}$ and field size $N$.

Phase 5 — Publication (28–32 weeks) Target: NeurIPS 2027 Workshop on Emergence in Complex Systems, or a journal submission to Physical Review E (dynamical systems track) for the mathematical results.

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16. References

Benettin, G., Galgani, L., Giorgilli, A., & Strelcyn, J.-M. (1980). Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems. Meccanica, 15, 9–30.

Berner, R., Gross, T., Kuehn, C., Kurths, J., & Yanchuk, S. (2023). Adaptive dynamical networks. Physics Reports, 1031, 1–59.

Finn, C., Abbeel, P., & Levine, S. (2017). Model-agnostic meta-learning for fast adaptation of deep networks. ICML 2017.

Friston, K., et al. (2017). Active inference and learning. Neuroscience & Biobehavioral Reviews, 68, 862–879.

Friston, K. (2010). The free-energy principle: a unified brain theory? Nature Reviews Neuroscience, 11(2), 127–138.

Ha, S.-Y., Kim, D., & Zhang, X. (2016). On the complete synchronization of the Kuramoto phase model. SIAM Journal on Applied Dynamical Systems, 15(1).

Kaplan, J. L., & Yorke, J. A. (1979). Chaotic behavior of multidimensional difference equations. Lecture Notes in Mathematics, 730, 204–227.

Klyubin, A. S., Polani, D., & Nehaniv, C. L. (2005). Empowerment: a universal agent-centric measure of control. IEEE Congress on Evolutionary Computation, 128–135.

Kuramoto, Y. (1975). Self-entrainment of a population of coupled non-linear oscillators. International Symposium on Mathematical Problems in Theoretical Physics.

Parr, T., Pezzulo, G., & Friston, K. J. (2022). Active Inference: The Free Energy Principle in Mind, Brain, and Behavior. MIT Press.

Pearl, J. (2009). Causality: Models, Reasoning, and Inference (2nd ed.). Cambridge University Press.

Prigogine, I. (1977). Self-Organization in Non-Equilibrium Systems. Wiley. (Nobel Lecture).

Salge, C., Glackin, C., & Polani, D. (2014). Empowerment — an introduction. Guided Self-Organization: Inception, 67–114. Springer.

Strogatz, S. H. (1994). Nonlinear Dynamics and Chaos. Addison-Wesley.

Strogatz, S. H. (2000). From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D, 143(1–4), 1–20.

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Author: Devanik · B.Tech ECE '26, NIT Agartala Samsung Fellowship · Indian Institute of Science

This is an independent research project in active development. Feedback, questions, and collaboration proposals are always welcome.

License: Apache License Version 2.0, January 2004 · Status: Beta