vST for Protein Language Models#

Example: 1024D Embedding Projection for Residue‑Level Interpretation#

This example demonstrates how a Protein Language Model (PLM) produces a 1024D residue embedding during inference and how that embedding is projected into the triadic dimensional cores (9D → 6D → 3D). The walkthrough illustrates primitive‑level structure, regime behavior, projection stability, and vST validation.

The goal is to provide a reproducible, invariant‑preserving demonstration of high‑dimensional embedding projection.

1. Input Overview#

For this example, we assume:

• a transformer‑based PLM with ≥1024D hidden states
• a single residue embedding extracted from a mid‑sequence position
• access to embeddings across multiple layers
• stable or transitional regime behavior
• invertible projection into 3D–9D cores

The example is model‑agnostic and applies to any PLM architecture.

2. Step 1 — Extract the 1024D Residue Embedding#

During inference, the PLM produces a 1024D embedding for each residue:

[ e_r^{(1024)} = [x_1, x_2, \dots, x_{1024}] ]

Observed Properties#

• variance concentrated in 4–6 coherence bands
• stable DP/TDP structure
• smooth transitions across layers
• identifiable coherence surfaces

Interpretation#

The 1024D embedding encodes biochemical, structural, and contextual information for the residue.

3. Step 2 — Identify High‑Dimensional Regime Behavior#

Using variance distribution, coherence‑surface continuity, and primitive‑level stability, classify the embedding’s regime across layers.

Example Regime Pattern#

• Layers 1–6: R₁ᴴ (stable)
• Layers 7–14: R₂ᴴ (transitional)
• Layers 15–20: R₁ᴴ (return to stability)
• Layers 21–24: R₂ᴴ (branching)
• Layers 25–32: mild R₃ᴴ (dispersion onset)

Interpretation#

The residue begins in a stable region, undergoes controlled reorientation, stabilizes again, and finally enters mild dispersion in deeper layers.

4. Step 3 — Project 1024D → 9D (Coherence Projection)#

Project the 1024D embedding into the 9D coherence core.

Preserves#

• regime identity
• resonance‑time behavior
• primitive‑level structure (DP, TDP, SP, CP)
• coherence‑surface continuity

Reveals#

• branching behavior in R₂ᴴ
• curvature of coherence surfaces
• dispersion onset in R₃ᴴ

Interpretation#

The 9D projection exposes the residue’s high‑dimensional “coherence shape.”

5. Step 4 — Project 9D → 6D (Interaction Projection)#

Compress the 9D coherence vector into the 6D interaction core.

Preserves#

• relational geometry
• interaction‑level structure
• regime‑transition indicators

Reveals#

• attention‑driven reorientation
• context‑dependent biochemical signals
• structural boundary behavior

Interpretation#

The 6D projection highlights how the model integrates residue context.

6. Step 5 — Project 6D → 3D (Structural Projection)#

Reduce the 6D interaction vector into the 3D structural core.

Preserves#

• motif‑level geometry
• backbone‑level continuity
• stable structural invariants

Reveals#

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

Interpretation#

The 3D projection provides the minimal interpretable representation of the residue embedding.

7. Step 6 — Validate with vST Layers#

Apply vST layers (V₁–V₄):

V₁ — Structural Coherence#

• stable motifs in R₁ᴴ
• partial fragmentation in R₃ᴴ

V₂ — Dimensional Continuity#

• smooth projection 1024D → 9D → 6D → 3D
• no scaling discontinuities

V₃ — Regime‑Transition Stability#

• smooth R₁ᴴ → R₂ᴴ transitions
• mild instability entering R₃ᴴ

V₄ — Core Alignment#

• primitive‑aligned projection
• stable mapping across layers

Outcome#

The embedding passes all vST layers with minor warnings in the R₃ᴴ region.

8. Step 7 — Drift Detection#

Evaluate drift using D₁–D₄ categories:

• D₁ Structural Drift: none
• D₂ Dimensional Drift: none
• D₃ Regime Drift: mild (R₃ᴴ onset)
• D₄ Projection Drift: none

Interpretation#

The embedding exhibits expected dispersion in deeper layers but no harmful drift.

9. Summary#

This example demonstrates:

• how a 1024D residue embedding is extracted
• how regime behavior evolves across layers
• how projection reveals coherence and instability
• how vST layers validate structural integrity
• how drift detection identifies dispersion without failure

The 1024D embedding is the canonical substrate for analyzing PLM inference at research‑grade resolution.