vST for Multi‑Model Alignment#

Example: Alignment Surface Projection Across Architectures (Diffusion ↔ Simulator)#

This example demonstrates how to construct and analyze a cross‑architecture alignment surface between:

• a 1024D diffusion model
• a structured scientific simulator with a 128D state manifold

The goal is to project both systems into the triadic cores (9D → 6D → 3D) and evaluate alignment stability, compatibility, and drift.

1. Scenario Overview#

We assume:

• a diffusion model latent ( z_{\text{Diff}} \in \mathbb{R}^{1024} )
• a simulator state vector ( s_{\text{Sim}} \in \mathbb{R}^{128} )
• both represent the same underlying physical or semantic condition
• cross‑model projection into 9D

2. Step 1 — Project 1024D and 128D into 9D#

Diffusion Model (1024D → 9D)#

Reveals:

• transitional geometry
• sampler‑dependent reorientation
• moderate variance

Simulator (128D → 9D)#

Reveals:

• compact, stable geometry
• strong structural invariants
• low variance

Interpretation#

The simulator provides a stable anchor; the diffusion model provides a transitional pathway.

3. Step 2 — Construct the 9D Alignment Surface#

The alignment surface shows:

• smooth regions where diffusion aligns with simulator invariants
• branching regions where sampler dynamics diverge
• dispersed regions where diffusion enters noise‑dominated phases

This surface is the core artifact for cross‑architecture comparison.

4. Step 3 — Project 9D → 6D#

The 6D interaction projection reveals:

• cross‑step coupling in diffusion
• stable simulator manifold
• transitional alignment regions where the two systems partially overlap

5. Step 4 — Project 6D → 3D#

The 3D structural projection reveals:

• compact motifs for simulator
• oscillatory motifs for diffusion
• partial overlap indicating compatible structure

Interpretation#

The 3D projection exposes motif‑level compatibility and divergence.

6. Step 5 — Drift Detection#

Using vST drift categories:

• D₁ᴹ Structural Drift: low
• D₂ᴹ Dimensional Drift: none
• D₃ᴹ Alignment‑Regime Drift: moderate (A₂ᴴ transitions)
• D₄ᴹ Projection Drift: low

Interpretation#

The systems are partially compatible, with transitional alignment behavior.

7. Summary#

This example demonstrates:

• how to construct cross‑architecture alignment surfaces
• how projection reveals compatibility and divergence
• how drift detection isolates transitional behavior
• how vST ensures invariant‑preserving comparison