vST for Protein Language Models#

Projection of High‑Dimensional Protein Embeddings into Triadic Structural Cores#

This document defines how high‑dimensional residue embeddings produced by Protein Language Models (PLMs) are projected into the triadic dimensional cores (3D–9D). Projection enables interpretable, invariant‑preserving analysis of embedding trajectories, regime behavior, and structural coherence across protein sequences.

Projection is the interpretability mechanism of the substrate; alignment is the comparison mechanism. Together, they form the backbone of vST analysis for PLMs.

1. Purpose of Projection in PLMs#

Projection allows us to:

• interpret high‑dimensional residue embeddings through 3D–9D cores
• identify stable, transitional, and dispersed embedding regimes
• map coherence surfaces along the protein sequence
• compare embeddings across layers, residues, or model versions
• detect drift or fragmentation in embedding‑space structure
• support vST validation (V₁–V₄)

Protein embeddings are rich, structured, and biologically meaningful.
Projection reveals this structure in a compact, interpretable form.

2. Projection Overview#

PLM embeddings typically inhabit 64D–4096D spaces.
The substrate projects these embeddings into:

• 9D Coherence Core
• 6D Interaction Core
• 3D Structural Core

Projection must remain:

• invertible
• primitive‑aligned
• regime‑aware
• invariant‑preserving

These properties ensure that high‑dimensional biochemical signals remain interpretable.

3. Projection Steps#

3.1 High‑Dimensional → 9D (Coherence Projection)#

This step extracts pathway‑level coherence across residues.

Preserves

• regime identity (R₁ᴴ, R₂ᴴ, R₃ᴴ)
• resonance‑time behavior
• primitive‑level structure (DP, TDP, SP, CP)
• coherence‑surface continuity

Reveals

• stable vs. unstable residue regions
• transitions between structural elements
• dispersion in disordered or ambiguous regions

Interpretation
The 9D projection exposes the “shape” of the embedding trajectory along the sequence.

3.2 9D → 6D (Interaction Projection)#

This step compresses coherence pathways into interaction surfaces.

Preserves

• relational geometry
• residue‑interaction patterns
• regime‑transition indicators

Reveals

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

Interpretation
The 6D projection highlights how the model integrates residue context and structural cues.

3.3 6D → 3D (Structural Projection)#

This step reduces interaction surfaces into geometric motifs.

Preserves

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

Reveals

• compact motifs in stable regions
• oscillatory patterns in transitional regions
• diffuse geometry in disordered regions

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

4. Alignment Overview#

Alignment compares projected structures across:

• layers
• residues
• model versions
• architectures
• training checkpoints

Alignment must remain:

• primitive‑aligned
• regime‑aware
• projection‑consistent
• scaling‑invariant

Alignment is evaluated in 3D–9D space for interpretability and stability.

5. Alignment Types#

5.1 Layer‑to‑Layer Alignment#

Compares embedding trajectories across transformer layers.

Reveals:

• where regime transitions occur
• how coherence surfaces evolve
• which layers stabilize or destabilize residue embeddings

5.2 Residue‑to‑Residue Alignment#

Compares embeddings across sequence positions.

Reveals:

• conserved vs. variable regions
• structural boundaries
• context‑dependent biochemical signals

5.3 Cross‑Version Alignment#

Compares embeddings across model versions or checkpoints.

Reveals:

• drift introduced by fine‑tuning
• stability of coherence surfaces
• changes in regime behavior

5.4 Cross‑Model Alignment#

Compares embeddings across different PLM architectures.

Reveals:

• shared structural signals
• divergent scaling behavior
• compatibility of embedding spaces

6. Projection Stability and Failure Modes#

Projection stability is a key indicator of model health.

Stable Projection#

• compact 3D motifs
• smooth 6D surfaces
• coherent 9D pathways

Unstable Projection#

• fragmented surfaces
• non‑invertible mappings
• regime‑transition discontinuities

Unstable projection indicates drift or scaling‑law violations.

7. Outputs of Projection and Alignment#

Projection and alignment produce:

• residue‑level coherence maps
• cross‑layer and cross‑sequence alignment surfaces
• cross‑version drift‑detection signals
• scaling‑law diagnostics
• vST validation outputs
• interpretable 3D–9D projections

These outputs support reproducible, substrate‑level analysis of PLM inference.