inZOR-ND: Empirical Law Candidate for Machine-Agnostic Disruption Proximity

Shared temporal structure: collapse test

After per-machine z-score normalization, the temporal trajectories of η(t) collapse onto a common curve across MAST, C-Mod and HL-2A, indicating a shared temporal structure of disruption proximity.

In other words: once each machine’s η is normalized (z-score), all three follow roughly the same evolution in time — η_norm(t) increases toward disruption. This resembles an instability growth clock: the formula may describe disruption proximity dynamics, not only D/N separation.

Fig — Cross-machine collapse test. Left: z-score normalization per machine. Right: min-max [0,1] per machine. After z-score, the three machines follow a common temporal curve; the shared evolution supports a single machine-agnostic temporal structure of the indicator.

1. How We Got Here — Discovery Path with inZOR-ND

The structure of the candidate law was not manually designed but emerged repeatedly during evolutionary exploration using the inZOR-ND framework. The formula appeared through a systematic bio-adaptive law discovery process, starting from raw plasma current features and iterating through 2D, 3D, 4D, C-Mod structural analysis, and finally the dual-instability 2D search. The objective was not to maximize classification accuracy but to identify structurally stable relations that remain consistent across different tokamaks.

Fig 0 — Discovery path: min(sep) staircase from 1-term < 2-term < 3-term. The 3-term law achieves the largest cross-machine robustness (min(sep) = 1.071), discovered by inZOR-ND without manual formula design.

Evolutionary exploration with inZOR-ND

The discovery process relied on the inZOR-ND evolutionary exploration framework, which searches the space of candidate formula structures.

Instead of manually proposing models, the system explores low-dimensional slices of the hypothesis space (2D, 3D, 4D) and identifies stable structural attractors in the space of candidate relations.

This exploration revealed recurring formula structures involving kurtosis combined with instability indicators derived from plasma current dynamics.

Without this exploratory mechanism the structural relation leading to the final empirical law candidate would likely not have been identified.

Dimensional exploration results

These results indicate the presence of a stable structural core centered on the kurtosis term.

2. Feature Definitions (Derived from Ip Only)

All features use a late window D (450 samples ending at t_disrupt − 50 ms) and a reference window N (full shot).

The negative sign on kurtosis means η rises when kurtosis drops — which happens because MHD instabilities (tearing modes, disruption precursors) flatten the Ip distribution before disruption. See Section 6 for the physical interpretation.

3. Validation — Formula Comparison

To confirm the 3-term formula adds genuine value over simpler variants, we compare all 6 formulas (3 single-term, 2 two-term, the 3-term law) on the same data, convention and machines.

Fig 1 — Baseline / Ablation Test. Left: min(sep) increases monotonically: 1-term < 2-term < 3-term. Right: Global ROC-AUC. The 3-term law candidate achieves the highest cross-machine robustness.

The three-term formulation provides the strongest worst-case separation while remaining structurally simple.

4. Leave-One-Machine-Out (LOMO) Validation

Formula coefficients are fixed (no retraining). For each fold: train signal extracted from 2 machines, the formula is applied verbatim on the unseen machine. sep > 0 and ROC > 0.5 required on each test machine.

Fig 2 — LOMO Validation: sep and ROC-AUC for each test machine (left-out machine). All 3 folds PASS. The formula generalises without retraining across MAST, C-Mod, HL-2A.

Overall: ✓ PASS — cross-machine validity confirmed. C-Mod is the hardest machine (highly imbalanced: 414 D / 13 N), yet sep = 1.071 > 0 and ROC = 0.757 > 0.5.

5. Temporal Trajectory η(t) — Continuous Measure of Disruption Proximity

The temporal behaviour of η suggests that the indicator may function not only as a classifier but also as a continuous measure of disruption proximity. Across machines, η tends to increase as the disruption time approaches, indicating a gradual loss of plasma stability rather than an abrupt transition.

η is computed at sliding windows −100, −80, −60, −40, −20 ms before disruption. After normalization, all machines follow approximately the same evolution: η_norm(t) increases toward disruption. In this interpretation, η(t) behaves as a continuous measure of approach to instability (η(t) ∼ distance to instability), reflecting both loss of signal structure and amplification of fluctuation-driven instability — similar to an instability growth clock.

Fig 3 — Temporal trajectory of η(t) for C-Mod, HL-2A and MAST. Mean η (solid line) ± std (shaded). η increases monotonically in all three machines as plasma approaches disruption, confirming the clock-like behaviour of the indicator.

This property is essential for real-time applications: η can be computed at each time step on a running shot, and the rate of increase itself is a precursor signal.

6. Per-Machine Structural Analysis

Pearson |correlation| between each feature and η on each machine reveals the dominant instability regime and confirms the universality of the kurtosis nucleus.

Fig 4 — Per-machine structural analysis: |Pearson correlation| between kurtosis, rate_ratio, cv_ratio and η. kurtosis is a common nucleus across all machines. rate_ratio dominates on MAST; cv_ratio dominates on C-Mod and HL-2A.

Structural dominance per machine

The results indicate two complementary instability channels, which motivates the combined formulation of the final empirical law candidate.

6b. Main Empirical Law Candidate

The resulting candidate relation is:

Interpretation:

Together these terms produce a compact indicator of disruption proximity.

Interpretation: statistical regime transition

The candidate law does not merely separate disruptive from non-disruptive shots. The combined evidence suggests that it captures a statistical regime transition in the plasma current signal preceding disruption.

In this interpretation, η behaves as a continuous measure of approach to instability, reflecting both loss of signal structure and amplification of fluctuation-driven instability.

• Disruption proximity is associated with a loss of structural sharpness in the current signal (kurtosis decrease).
• This is accompanied by growth of instability, through dynamic acceleration (rate_ratio) and/or variability amplification (cv_ratio).
• The shared normalized temporal evolution (collapse test) suggests a common transition pattern across machines.

Figure: Summary of the disruption proximity law candidate and its validation across machines.

7. Physical Interpretation — Why kurtosis Decreases

The use of excess kurtosis as the primary disruption nucleus has a physical basis. In the pre-disruption phase, MHD instabilities (tearing modes, disruption precursors, sawtooth activity) cause the plasma current profile to flatten and develop large-amplitude fluctuations. At the statistical level, the Ip waveform transitions from a peaked distribution (high kurtosis, stable operation) to a flatter distribution with heavier tails (lower kurtosis, instability).

The two instability components capture the dynamics of this process:

• rate_ratio: Late-phase surge in |dIp/dt|, corresponding to rapid current profile changes driven by MHD activity.
• cv_ratio: Amplification of relative current fluctuations, corresponding to increased turbulence or disruption precursor oscillations.

8. Robustness & Sensitivity

Fig 5 — Left: Bootstrap per-machine sep (mean ± std, 300 replicas with replacement). Centre: Global ROC-AUC bootstrap CI₉₅. Right: min(sep) distribution under ±20% coefficient perturbation (sensitivity, 300 replicas). All min > 0; all ROC CI₉₅ > 0.5.

A. Bootstrap Robustness (shot-level resampling, 300 replicas)

PASS Bootstrap CI₂.₅% for min(sep) = 0.3730 > 0 and for Global ROC = 0.9789 > 0.5. The result is robust to shot-level resampling.

B. Sensitivity (coefficients ±20%, 300 replicas)

PASS min(sep) stays in [0.858, 1.254] (always > 0) under ±20% coefficient perturbation. The formula is insensitive to moderate coefficient variation.

9. Role of inZOR-ND in This Discovery

The entire formula discovery was powered by inZOR-ND, the bio-adaptive genomic discovery engine. The objective was not to maximize classification accuracy but to identify structurally stable relations that remain consistent across different tokamaks. Without inZOR-ND, none of this analysis would have been possible:

• 2D, 3D, 4D law-space exploration was driven by the genomic population evolving under cross-machine fitness.
• The dual-instability structure (rate_ratio vs cv_ratio, machine-specific dominance) was revealed by the law-space search, not by manual feature engineering.
• The frozen-elite selection mechanism provided stable formula candidates for structural analysis and validation.
• The feature window convention (D = 450 samples, law-space normalization) emerged from the genomic run configuration.

inZOR-ND is the same engine used across all published tests (PFΔ power systems, TESS prioritization, refraction emergence, social dynamics). Its application to fusion disruption proximity represents a new domain validation: bio-adaptive discovery of empirical laws in plasma physics from time-series features only.

10. Complete Formula Hierarchy

11. Data, Code & Reproducibility

Convention: D = 450 samples ending at t_disrupt − 50 ms; N = full shot. Same convention across all tests.

12. Conclusion

Rather than designing a predictive model, the objective of this study was to explore whether simple structural relations emerge consistently across machines when analysing plasma current dynamics.

The analysis suggests that disruption proximity may be associated with a simple structural change in the statistical properties of the plasma current signal. Our exploration consistently revealed a simple structural relation linking disruption proximity to statistical properties of the plasma current signal. Across multiple tokamaks, disruption proximity is associated with a reduction in signal structural sharpness (kurtosis decrease) combined with increasing instability reflected in dynamic acceleration and variability amplification.

This yields a compact empirical relation:

The structure of the candidate law was not manually designed but emerged repeatedly during evolutionary exploration; the objective was not to maximize classification accuracy but to identify structurally stable relations that remain consistent across different tokamaks. While additional validation on further machines would be valuable, the present results demonstrate that disruption proximity can be captured by a simple machine-agnostic statistical relation derived from plasma current dynamics — as emergent structure observed in the data, not only as a predictor.