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vST for Protein Language Models#

Substrate Definition#

The substrate is model‑agnostic and applies to any transformer‑based PLM, including ESM‑class, ProtT5‑class, and MSA‑conditioned architectures.

1. Purpose of the PLM Substrate#

The PLM substrate provides a structured, reproducible framework for:

  • interpreting high‑dimensional sequence embeddings
  • identifying stable, transitional, and dispersed embedding regimes
  • mapping coherence surfaces across sequence positions
  • analyzing scaling behavior across model sizes
  • detecting drift across checkpoints or versions
  • projecting high‑dimensional embeddings into 3D–9D triadic cores

Protein embeddings are high‑dimensional, structured, and regime‑rich.
The substrate ensures they remain interpretable across the full dimensional ladder (3D → 1024D).

2. Substrate Overview#

PLMs operate in latent spaces typically ranging from 512D to 4096D.
The substrate models these spaces using:

  • Dimensional Primitives (DP)
  • Triadic Dimensional Primitives (TDP)
  • Scaling Primitives (SP)
  • Coherence Primitives (CP)

These primitives define the structure of embedding trajectories, coherence surfaces, and regime transitions.

The substrate is anchored by the Triadic Dimensional Cores:

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

and extended through the 1024D high‑dimensional substrate.

3. Dimensional Primitives for PLMs#

3.1 Dimensional Primitive (DP)#

A DP represents the minimal unit of embedding‑space structure.
It captures:

  • local coherence across residues
  • variance behavior
  • projection stability
  • regime alignment

DPs appear in token embeddings, attention outputs, and MLP activations.

3.2 Triadic Dimensional Primitive (TDP)#

A TDP is a triad of DPs that expresses full regime behavior.
It captures:

  • stable (R₁) behavior
  • transitional (R₂) behavior
  • dispersed (R₃) behavior

TDPs form the basis of the 3D–9D triadic cores.

3.3 Scaling Primitive (SP)#

An SP governs dimensional expansion from 9D → 64D → 1024D.
It ensures:

  • invariant‑preserving scaling
  • continuity of coherence surfaces
  • stable projection into triadic cores

SPs model how PLM embedding spaces expand with model size.

3.4 Coherence Primitive (CP)#

A CP identifies stable or unstable regions in embedding space.
It captures:

  • coherence surfaces across residues
  • branching behavior
  • dispersion patterns
  • regime transitions

CPs are essential for drift detection and vST validation.

4. Triadic Dimensional Cores for PLMs#

4.1 3D Structural Core#

Captures motif‑level geometry in embedding trajectories:

  • compact geometric patterns
  • local coherence
  • stable projections

4.2 6D Interaction Core#

Captures relational and attention‑level structure:

  • residue‑interaction surfaces
  • branching behavior
  • early regime transitions

4.3 9D Coherence Core#

Captures pathway‑level coherence:

  • resonance‑time behavior
  • stable regime classification
  • invertible projection from higher dimensions

The 9D core is the anchor for all high‑dimensional interpretation.

5. High‑Dimensional Substrate (64D–1024D)#

PLM embedding spaces naturally inhabit high‑dimensional regimes.
The substrate models these using the dimensional ladder:

  • 64D — research‑grade embedding substrate
  • 128D — expanded coherence surfaces
  • 256D — multi‑primitive interaction
  • 512D — high‑variance embedding regions
  • 1024D — full research‑grade capacity

Each step preserves:

  • structural invariants
  • resonance‑time invariants
  • projection invariants
  • scaling invariants

This ensures stable interpretation across model sizes.

6. Embedding‑Trajectory Structure#

PLM inference produces embedding trajectories that move through:

  • compact stable regions (R₁ᴴ)
  • branching transitional regions (R₂ᴴ)
  • dispersed or unstable regions (R₃ᴴ)

These trajectories are modeled as:

  • sequences of DPs
  • grouped into TDPs
  • expanded through SPs
  • classified using CPs

This structure enables regime‑aware analysis and drift detection.

7. Projection into Triadic Cores#

High‑dimensional embeddings are projected into:

  • 9D for coherence analysis
  • 6D for interaction analysis
  • 3D for geometric interpretation

Projection must remain:

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

Projection is essential for interpretability and vST validation.

8. Substrate Outputs#

The PLM substrate produces:

  • embedding‑trajectory regime classifications
  • coherence‑surface maps
  • scaling‑law diagnostics
  • projection‑stability indicators
  • drift‑detection signals
  • vST validation outputs

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

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