vST for Generative Models#

Example: Regime Transitions Along a Diffusion Trajectory#

This example demonstrates how a diffusion model’s sampling trajectory moves through the triadic latent‑regime structure:

• R₃ᴴ — noise‑dominated
• R₂ᴴ — transitional denoising
• R₁ᴴ — stable refinement

It illustrates how coherence surfaces evolve, how variance contracts, and how the vST substrate classifies each phase using the 1024D dimensional ladder.

1. Scenario Overview#

We assume:

• a 1024D latent diffusion model
• 50‑step sampler (e.g., DDIM or Euler)
• a single trajectory sampled from noise → final latent
• checkpoints C₁ and C₂ for cross‑version comparison

The example is architecture‑agnostic.

2. Step 1 — Extract Latent States Across the Trajectory#

Let:

[ z_t \in \mathbb{R}^{1024}, \quad t = 0, 1, \dots, 50 ]

represent the latent state at sampling step ( t ).

Observed Properties#

• ( z_0 ) is high‑variance, noise‑dominated
• mid‑trajectory states show branching and reorientation
• late states converge into compact, coherent motifs

3. Step 2 — Project Latents into 9D#

Project each ( z_t ) into the 9D coherence core.

Reveals#

• R₃ᴴ (steps 0–10): diffuse, unstable geometry
• R₂ᴴ (steps 11–32): branching surfaces, oscillatory transitions
• R₁ᴴ (steps 33–50): compact, stable motifs

Interpretation#

The 9D projection exposes the “coherence spine” of the diffusion trajectory.

4. Step 3 — Identify Regime Transitions#

Using variance distribution, coherence‑surface continuity, and primitive‑level stability:

Interpretation#

The trajectory follows the canonical triadic sequence:

[ R₃ᴴ \rightarrow R₂ᴴ \rightarrow R₁ᴴ ]

5. Step 4 — Project 9D → 6D → 3D#

6D Interaction Projection#

Shows:

• cross‑step coupling
• sampler‑driven reorientation
• early instability signatures

3D Structural Projection#

Shows:

• compact motifs in R₁ᴴ
• oscillatory geometry in R₂ᴴ
• diffuse patterns in R₃ᴴ

6. Step 5 — Validate with vST Layers#

• V₁: structural coherence preserved
• V₂: dimensional continuity intact
• V₃: regime transitions substrate‑aligned
• V₄: core alignment stable across checkpoints

7. Summary#

This example demonstrates:

• the triadic regime structure of diffusion trajectories
• how coherence surfaces evolve across sampling steps
• how projection reveals latent‑space geometry
• how vST layers validate structural integrity