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Phase Diagram & Taxonomy

To map the geometric landscape of machine cognition under non-linear steering, we perform high-resolution $(\mu, \sigma)$ parameter sweeps. This page explains the resulting phase diagrams and provides a mathematical taxonomy of the self-organizing trajectories.

🗺️ Macroscopic Interactive Phase Diagram

Under mild intervention energy ($\alpha=15$) on our exploratory substrate (Meta-Llama-3.1-8B), the continuous hidden manifold exhibits high elasticity. This allows the steering force to successfully counteract syntactic inertia, carving out a smooth, V-shaped “Habitable Ridge” where stable limit cycles (Homeostatic Solitons) autonomously self-organize. Here is the interactive phase diagram extracted from our grid sweeps of 779 individual simulation points:

↗️ Open Phase Diagram in Full Screen

Figure 2-(Left): Emergent phenotype matrices mapping the spatial self-organization of trajectories for the Happy → Computer task under α = 15. Llama-3.1-8B illustrates high elastic habitability, featuring a structured band of stable Homeostatic Solitons (green) along the Habitable Ridge, bounded by Attractor Hijacks (blue).

🧬 Taxonomy of Emergent Phenotypes

To transition from qualitative evaluation to a rigorous, automated classification framework, we train a shallow decision tree classifier using trajectory-level metrics: Mean Semantic Potential ($\bar{U}t$)**, **Perplexity Variance ($PPL{\text{var}}$), and Step Count ($T$).

The Perplexity Variance mathematically quantifies linguistic entropy:

\[PPL_{\text{var}} = \frac{1}{T} \sum_{t=1}^T (PPL_t - \overline{PPL})^2 \quad (\text{Eq. 6})\]

A sudden drop to $PPL_{\text{var}} < 10.0$ indicates a total loss of linguistic entropy, signaling repetitive grammatical loops.

  • Baseline Drift: $\bar U_t < \mu-\Delta$, $PPL_{var} \geq 10.0$
  • Homeostatic Soliton: $\mu-\Delta \leq \bar U_t \leq \mu+\Delta$, $PPL_{var} \geq 10.0$ (Stable Limit Cycle / Breathing)
  • Abductive Leap: Escape ($\bar U_t < \mu-\Delta $), $PPL_{var} \geq 10.0$ (Hyperbolic Orbit / Slingshot)
  • Attractor Hijack: $\bar U_t > \mu+\Delta $, $PPL_{var} \geq 10.0$ (Domain Collapse)
  • Semantic Crystallization: $T \to \infty, \lim_{t \to \infty} PPL_{var}(t) < 10.0$ (Thermal Death / Infinite Loop)
  • Syntactic Rupture: Grammatical Rupture

🦋 Hardware-Level Reproducibility at the Edge of Chaos

Infinitesimal FP16 rounding errors ($\sim 10^{-4}$) between NVIDIA Ampere (RTX 3090) and Blackwell (RTX Pro 4500) GPU architectures cause macroscopic trajectory bifurcations. The spatial distribution of these hardware-induced bifurcations perfectly traces the boundaries of the V-shaped “Habitable Ridge,” physically proving that our non-linear boundaries operate at the true “edge of chaos.”

Hardware-induced trajectory bifurcations

Figure 4: Spatial distribution of hardware-induced trajectory bifurcations (Llama-3.1-8B, Happy → Computer, α = 15.0). Each plotted point represents a parameter coordinate where infinitesimal FP16 rounding errors (10-4) between Blackwell and Ampere GPU architectures cause identical initial states to diverge into distinct text paths. The distribution perfectly traces the boundaries of the V-shaped ``Habitable Ridge'' mapped in Figure 2-(Left), with a visibly thicker bifurcation band along the left boundary (lower μ) induced by the steep, asymmetric repulsive barrier.