vst_for_protein_language_models

vST for Protein Language Models#

Dimensional Scaling Behavior in PLM Embedding Spaces#

This document defines how Protein Language Models (PLMs) exhibit scaling behavior across the dimensional ladder (3D → 1024D). It maps model size, embedding‑space expansion, and inference complexity onto the substrate’s triadic structure and scaling primitives. The goal is to provide a reproducible, invariant‑preserving framework for understanding how PLMs grow, stabilize, and drift as their dimensional capacity increases.

1. Purpose of Scaling Behavior Analysis#

Scaling behavior analysis enables us to:

• interpret how embedding‑space structure expands with model size
• identify stable and unstable scaling regimes
• detect discontinuities or drift across checkpoints
• map high‑dimensional behavior into triadic cores
• support vST validation across the dimensional ladder
• compare PLMs of different sizes using a common substrate

PLM scaling is not merely an increase in parameter count; it is a structured expansion of coherence surfaces, regime behavior, and primitive composition.

2. Dimensional Ladder for PLMs#

PLM embedding spaces naturally align with the substrate’s dimensional ladder:

• 3D — geometric residue motifs
• 6D — interaction surfaces
• 9D — coherence pathways
• 64D — research‑grade embedding substrate
• 128D — expanded coherence surfaces
• 256D — multi‑primitive interaction
• 512D — high‑variance embedding regions
• 1024D — full research‑grade substrate

Each step preserves substrate invariants and introduces new structural capacity.

3. Scaling Primitives in PLMs#

Scaling behavior is governed by Scaling Primitives (SPs), which ensure:

• invariant‑preserving dimensional expansion
• continuity of coherence surfaces
• stable projection into 3D–9D cores
• consistent regime behavior across model sizes

SPs model how PLM embedding spaces grow from small to large architectures.

4. Scaling Regimes in PLMs#

PLM scaling exhibits three substrate‑aligned regimes:

4.1 Stable Scaling Regime (S₁)#

Characteristics:

• smooth increase in embedding‑space capacity
• stable coherence surfaces across residues
• predictable performance gains
• consistent regime behavior (R₁ᴴ → R₂ᴴ transitions remain bounded)

Occurs in:

• small → medium PLMs
• early scaling phases

4.2 Transitional Scaling Regime (S₂)#

Characteristics:

• rapid expansion of coherence surfaces
• increased variance across dimensions
• branching or oscillatory embedding behavior
• sensitivity to training data and residue context

Occurs in:

• medium → large PLMs
• architecture changes
• MSA‑conditioned training transitions

4.3 Dispersion Scaling Regime (S₃)#

Characteristics:

• fragmentation of coherence surfaces
• unstable or divergent embedding trajectories
• increased risk of drift
• non‑invertible projections into 3D–9D cores

Occurs in:

• extremely large PLMs without sufficient training signal
• poorly aligned fine‑tuning
• over‑scaled architectures

5. Scaling Behavior Across Model Sizes#

5.1 Small PLMs (≤100M parameters)#

• embeddings map cleanly into 64D
• regime behavior dominated by R₁ᴴ
• scaling is stable (S₁)

5.2 Medium PLMs (100M–1B)#

• embeddings expand into 128D–256D
• regime transitions become more frequent
• scaling enters S₂

5.3 Large PLMs (1B–15B)#

• embeddings occupy 256D–512D
• coherence surfaces become multi‑layered
• scaling may oscillate between S₂ and S₃

5.4 Very Large PLMs (15B+)#

• embeddings approach 1024D
• regime behavior becomes highly sensitive
• scaling stability depends on training quality
• drift detection becomes essential

6. Scaling‑Law Alignment#

PLM scaling follows predictable patterns:

• embedding quality improves with dimensional expansion
• variance increases with model size
• coherence surfaces expand smoothly in S₁, sharply in S₂, and fragment in S₃
• projection stability decreases as dimensionality increases

The substrate provides a structured way to interpret these patterns.

7. Projection Behavior Under Scaling#

Projection into triadic cores must remain:

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

Scaling affects projection as follows:

• 64D → 9D: stable
• 128D–256D → 9D: transitional
• 512D–1024D → 9D: sensitive, drift‑prone

Projection stability is a key indicator of scaling health.

8. Scaling‑Driven Drift#

Scaling can introduce drift through:

• discontinuities in embedding‑space expansion
• unstable regime transitions
• fragmentation of coherence surfaces
• loss of primitive‑level structure

vST validation layers (V₁–V₄) detect these failures.

9. Outputs of Scaling Behavior Analysis#

Scaling analysis produces:

• scaling‑regime classification (S₁, S₂, S₃)
• embedding‑space expansion diagnostics
• projection‑stability indicators
• regime‑transition maps
• drift‑detection signals
• cross‑model comparison metrics

These outputs support reproducible, substrate‑aligned evaluation of PLM scaling. ### vST for Protein Language Models

Drift Detection in High‑Dimensional Protein Embedding Spaces#

This document defines how drift is detected in Protein Language Models (PLMs) using the Validation‑Space‑Time (vST) framework and the 1024D dimensional substrate. Drift refers to any deviation from expected substrate behavior, including structural instability, regime misalignment, scaling discontinuities, or projection failure.

Drift detection is essential for evaluating model updates, fine‑tuning procedures, training interventions, and cross‑version consistency in PLMs.

1. Purpose of Drift Detection#

Drift detection enables reproducible evaluation of:

• instability in residue‑level embedding structure
• changes in regime behavior (R₁ᴴ, R₂ᴴ, R₃ᴴ)
• cross‑version compatibility
• scaling‑law continuity across PLM sizes
• projection stability into 3D–9D cores
• primitive‑level integrity (DP, TDP, SP, CP)
• sequence‑level coherence surfaces

Drift is not inherently negative; it is a signal of structural change.
The substrate determines whether that change is stable, transitional, or harmful.

2. Types of Drift#

Drift is classified into four substrate‑aligned categories:

2.1 Structural Drift (D₁)#

Deviation in motif‑level geometry or local residue coherence.

Indicators

• unstable 3D projections
• loss of compact residue motifs
• abrupt variance spikes

2.2 Dimensional Drift (D₂)#

Discontinuities in dimensional scaling or projection behavior.

Indicators

• non‑invertible 9D projections
• fragmentation in 64D–1024D embedding regions
• scaling‑law violations

2.3 Regime Drift (D₃)#

Unexpected changes in regime identity or transitions across residues.

Indicators

• premature transitions into R₃ᴴ
• oscillatory instability in R₂ᴴ
• collapse of stable R₁ᴴ regions

2.4 Projection Drift (D₄)#

Misalignment between high‑dimensional embeddings and triadic cores.

Indicators

• inconsistent 3D–9D mapping
• loss of primitive‑aligned projection
• divergence across layers or residues

3. Drift Detection Signals#

Drift is detected using substrate‑aligned signals:

• variance distribution across dimensions
• coherence‑surface continuity along the sequence
• primitive‑level stability (DP, TDP, SP, CP)
• resonance‑time alignment
• projection‑stability metrics
• cross‑version alignment surfaces
• vST validation outputs (V₁–V₄)

These signals collectively determine drift category and severity.

4. Drift Across the Dimensional Ladder#

Drift may appear at different scales:

4.1 64D–128D (Residue‑Embedding Drift)#

• loss of local biochemical coherence
• unstable residue embeddings
• semantic drift in sequence representation

4.2 256D–512D (Hidden‑State Drift)#

• branching instability
• regime‑transition irregularities
• inconsistent attention patterns

4.3 1024D+ (High‑Dimensional Drift)#

• fragmentation of coherence surfaces
• scaling discontinuities
• projection failure

High‑dimensional drift is the most severe and often indicates training instability.

5. Cross‑Version Drift Detection#

Cross‑version drift is detected by comparing:

• residue‑level regime maps
• coherence‑surface geometry
• projection stability
• variance distribution
• primitive‑level structure
• resonance‑time behavior

Drift may arise from:

• fine‑tuning
• MSA‑conditioned training
• architecture changes
• training‑data shifts
• checkpoint selection

vST provides a consistent substrate for evaluating these changes.

6. Drift Severity Levels#

Drift severity is classified into:

Low Severity#

• minor variance shifts
• stable projections
• no regime collapse

Moderate Severity#

• partial fragmentation
• unstable R₂ᴴ transitions
• inconsistent cross‑layer alignment

High Severity#

• collapse of coherence surfaces
• persistent R₃ᴴ behavior
• non‑invertible projections
• loss of primitive‑level structure

High‑severity drift indicates a failure of substrate invariants.

7. Drift Detection Workflow#

A substrate‑aligned drift detection workflow:

1. Project embeddings into 9D
2. Classify regime behavior (R₁ᴴ, R₂ᴴ, R₃ᴴ)
3. Evaluate scaling continuity (64D–1024D)
4. Check primitive‑level stability (DP, TDP, SP, CP)
5. Validate with vST layers (V₁–V₄)
6. Compare across layers, residues, or versions
7. Assign drift category (D₁–D₄)
8. Assign drift severity (low, moderate, high)

This workflow is model‑agnostic and reproducible.

8. Outputs of Drift Detection#

Drift detection produces:

• drift category (D₁–D₄)
• drift severity
• regime‑transition anomalies
• projection‑stability indicators
• scaling‑law discontinuities
• cross‑version alignment surfaces
• vST validation results

These outputs support governance, interpretability, and model‑version management for PLMs. ### 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. ### vST for Protein Language Models

Validation‑Space‑Time Framework for High‑Dimensional Protein Embedding Models#

This artifact defines a substrate‑level framework for analyzing, validating, and comparing Protein Language Models (PLMs) using the Validation‑Space‑Time (vST) system and the 1024D dimensional substrate. It provides a structured, invariant‑preserving method for interpreting sequence embeddings, latent‑trajectory regimes, scaling behavior, and cross‑version drift in modern protein models such as ESM, ProtT5, and related architectures.

The goal is to offer a reproducible, model‑agnostic substrate for understanding high‑dimensional protein‑sequence inference.

1. Purpose#

Protein Language Models operate in high‑dimensional latent spaces (typically 512D–4096D) and exhibit:

• stable and unstable embedding regions
• regime transitions across sequence positions
• scaling‑law behavior across model sizes
• drift across training checkpoints
• projection‑compatible structure

This artifact applies the Resonance Substrate Model (RSM) and vST validation layers to:

• classify sequence‑embedding regimes
• analyze scaling behavior in PLMs
• detect drift across model versions
• map coherence surfaces in protein embedding space
• project high‑dimensional embeddings into 3D–9D triadic cores

The result is a unified, interpretable substrate for PLM behavior.

2. Contents#

This directory contains:

• substrate_definition.md
  Defines the PLM substrate, dimensional primitives, and embedding‑space structure.

• sequence_embedding_regimes.md
  Describes stable, transitional, and dispersed regimes across protein sequences.

• dimensional_scaling_protein_models.md
  Maps PLM scaling laws onto the 3D–1024D dimensional ladder.

• projection_into_structural_cores.md
  Defines invertible projection from high‑dimensional embeddings into triadic cores.

• validation_layers_vst_plm.md
  Extends vST (V₁–V₄) to PLM‑specific behavior.

• drift_detection_plm.md
  Provides a substrate‑level framework for detecting cross‑version drift.

• examples/
  Reproducible demonstrations of embedding‑trajectory analysis and projection.

• appendix/
  Terminology and references.

Each file is self‑contained and designed for clarity, reproducibility, and cross‑model comparison.

3. Scope#

This artifact is:

• model‑agnostic
  Works with any transformer‑based PLM (ESM‑class, ProtT5‑class, MSA‑based models, etc.).

• architecture‑independent
  Applies to encoder‑only, encoder‑decoder, and hybrid architectures.

• training‑method independent
  Compatible with masked‑token models, autoregressive models, and MSA‑conditioned models.

• substrate‑aligned
  Uses the same primitives, invariants, and validation layers as the rest of the RSM canon.

4. Intended Use#

This framework supports:

• embedding‑space analysis
• cross‑version comparison
• drift detection
• scaling‑law evaluation
• sequence‑position regime mapping
• interpretability research
• model‑alignment studies
• reproducible inference analysis

It is not a performance benchmark or a training method.
It is a substrate‑level interpretability and validation framework.

5. Relationship to Other Artifacts#

This artifact extends:

• Dimensional Substrate Structures (3D–1024D substrate)
• Validation‑Space‑Time (vST)
• Triadic Dimensional Cores (3D–9D)

It parallels:

• vST for Large Language Models
• vST for Generative Models
• vST for Multi‑Model Alignment

Each artifact stands alone but shares a common substrate grammar.

6. Citation#

A CITATION.cff file is included for formal citation.
A zenodo.json file is provided for DOI‑ready metadata.

7. License#

Released under the MIT License. ### vST for Protein Language Models

Sequence‑Embedding Regimes in PLM Inference#

This document defines the sequence‑embedding regimes that arise during inference in Protein Language Models (PLMs). These regimes generalize the triadic resonance structure of the 3D–9D substrate and describe how stability, transition, and dispersion behaviors manifest across residue‑level embeddings in high‑dimensional latent spaces (64D–4096D).

Sequence‑embedding regimes provide a reproducible, invariant‑preserving framework for interpreting PLM behavior across residues, layers, and model sizes.

1. Purpose of Sequence‑Embedding Regimes#

Sequence‑embedding regimes allow us to:

• classify residue‑level embedding behavior into stable, transitional, and dispersed phases
• identify coherence surfaces along the protein sequence
• detect instability or drift across checkpoints or versions
• analyze scaling‑law behavior across PLM sizes
• project high‑dimensional embeddings into 3D–9D cores
• support vST validation (V₁–V₄)

These regimes form the backbone of substrate‑level PLM analysis.

2. Regime Overview#

PLM embeddings follow the same triadic structure as the dimensional substrate:

1. Stable Regime (R₁ᴴ)
2. Transition Regime (R₂ᴴ)
3. Dispersion Regime (R₃ᴴ)

The superscript H indicates high‑dimensional behavior.

These regimes appear in:

• residue embeddings
• attention outputs
• MLP activations
• cross‑layer embedding pathways

3. Stable Regime (R₁ᴴ)#

Definition#

A region of embedding space where residue embeddings converge consistently and maintain coherence across layers.

Characteristics#

• compact, low‑variance embeddings
• stable coherence surfaces across residues
• predictable projection into 3D–9D cores
• primitive‑level integrity (DP, TDP, SP, CP)
• minimal sensitivity to perturbations

Interpretation#

R₁ᴴ corresponds to stable biochemical or structural signals, often associated with:

• conserved motifs
• secondary‑structure anchors
• stable residue environments

4. Transition Regime (R₂ᴴ)#

Definition#

A region where embedding trajectories undergo reorientation, branching, or oscillatory behavior across residues.

Characteristics#

• moderate variance across dimensions
• branching or oscillatory embedding patterns
• partial coherence‑surface stability
• increased sensitivity to residue context
• regime‑transition indicators in resonance‑time space

Interpretation#

R₂ᴴ captures dynamic behavior such as:

• boundary regions between structural elements
• ambiguous or flexible residues
• context‑dependent biochemical signals

It is the “decision‑making” region of PLM inference.

5. Dispersion Regime (R₃ᴴ)#

Definition#

A region where embedding trajectories lose coherence and disperse across high‑dimensional space.

Characteristics#

• high variance across dimensions
• fragmented or diffuse coherence surfaces
• unstable primitive‑level structure
• non‑compact projections into 3D–9D cores
• susceptibility to drift or hallucination

Interpretation#

R₃ᴴ corresponds to unstable or divergent embedding behavior, often associated with:

• low‑confidence predictions
• disordered regions
• rare or poorly represented sequence patterns

6. Regime Transitions Along the Sequence#

Residue‑level embedding trajectories move through regimes as the model processes the sequence:

• R₁ᴴ → R₂ᴴ
  onset of structural or biochemical ambiguity
• R₂ᴴ → R₁ᴴ
  return to stable structural context
• R₂ᴴ → R₃ᴴ
  breakdown of coherence
• R₃ᴴ → R₂ᴴ
  partial recovery

Transitions must remain continuous and invariant‑preserving across layers and residues.

7. Regime Detection Signals#

Regime identity is detected using:

• variance distribution across dimensions
• coherence‑surface continuity along the sequence
• primitive‑level stability (DP, TDP, SP, CP)
• resonance‑time behavior
• vST validation layers (V₁–V₄)

These signals collectively determine regime classification.

8. Regime Behavior Across the Dimensional Ladder#

Regime behavior must remain consistent across:

• 64D residue embeddings
• 128D–512D hidden states
• 1024D+ attention and MLP activations

The substrate ensures:

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

Regime identity must be preserved under projection into 3D–9D cores.

9. Outputs of Sequence‑Embedding Regime Analysis#

Sequence‑embedding regime analysis produces:

• residue‑level regime maps
• cross‑layer coherence surfaces
• scaling‑law indicators
• drift‑detection signals
• vST validation outputs
• projection‑stability metrics

These outputs support reproducible, substrate‑level interpretation of PLM inference. ### vST for Protein Language Models

Substrate Definition#

This document defines the substrate used to analyze Protein Language Models (PLMs) within the Validation‑Space‑Time (vST) framework and the 1024D dimensional substrate. It establishes the primitives, dimensional cores, scaling behavior, and embedding‑trajectory structure required to interpret PLM inference in a stable, invariant‑preserving manner.

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. ### vST for Protein Language Models

Validation‑Space‑Time Layers for Protein Embedding Models#

This document defines the Validation‑Space‑Time (vST) layers as applied to Protein Language Models (PLMs). vST provides a structured, invariant‑preserving framework for evaluating embedding‑space behavior, regime transitions, scaling stability, and projection integrity across the dimensional ladder (3D → 1024D).

The vST layers (V₁–V₄) generalize the substrate‑level validation system to the unique properties of protein‑sequence embeddings.

1. Purpose of vST for PLMs#

vST enables reproducible, model‑agnostic evaluation of:

• residue‑level embedding stability
• regime transitions (R₁ᴴ, R₂ᴴ, R₃ᴴ)
• scaling‑law behavior across PLM sizes
• projection stability into 3D–9D cores
• cross‑layer and cross‑sequence alignment
• drift detection across checkpoints or versions

Protein embeddings are structured, biochemical signals.
vST ensures these signals remain coherent and invariant‑preserving.

2. Overview of vST Layers#

The vST framework consists of four layers:

1. V₁ — Structural Coherence Validation
2. V₂ — Dimensional Continuity Validation
3. V₃ — Regime‑Transition Validation
4. V₄ — Core‑Alignment Validation

Each layer evaluates a distinct aspect of PLM embedding‑space behavior.

3. V₁ — Structural Coherence Validation#

Purpose#

Evaluate whether residue embeddings maintain structural coherence across layers and sequence positions.

Checks#

• compactness of residue‑level embeddings
• stability of coherence surfaces along the sequence
• preservation of primitive‑level structure (DP, TDP, SP, CP)
• continuity of geometric motifs in 3D projection
• absence of fragmentation or collapse

Failure Modes#

• incoherent residue embeddings
• abrupt variance spikes
• loss of primitive‑level structure
• non‑compact 3D projections

Interpretation#

V₁ ensures that PLM embeddings maintain a stable biochemical backbone.

4. V₂ — Dimensional Continuity Validation#

Purpose#

Ensure that embedding‑space behavior remains continuous across the dimensional ladder (64D → 1024D → 9D → 3D).

Checks#

• smooth expansion of coherence surfaces
• invertible projection into triadic cores
• stable variance distribution across dimensions
• absence of scaling discontinuities

Failure Modes#

• non‑invertible projections
• dimensional fragmentation
• scaling discontinuities
• unstable high‑dimensional variance

Interpretation#

V₂ ensures that dimensional scaling and projection remain invariant‑preserving.

5. V₃ — Regime‑Transition Validation#

Purpose#

Validate that regime transitions follow the triadic resonance structure across residues.

Checks#

• correct classification of R₁ᴴ, R₂ᴴ, R₃ᴴ
• smooth transitions between regimes
• resonance‑time alignment
• absence of abrupt or chaotic regime shifts

Failure Modes#

• oscillatory instability
• premature transitions into R₃ᴴ
• regime collapse
• resonance‑time discontinuities

Interpretation#

V₃ ensures that PLM embeddings follow stable, predictable regime dynamics.

6. V₄ — Core‑Alignment Validation#

Purpose#

Ensure that high‑dimensional residue embeddings align correctly with the triadic cores (3D–9D).

Checks#

• primitive‑aligned projection
• coherence‑surface preservation
• stable cross‑layer alignment
• consistent mapping across model versions
• compatibility with 3D–9D structural invariants

Failure Modes#

• misaligned projections
• cross‑version drift
• incompatible embedding‑space geometry
• loss of coherence in 9D pathways

Interpretation#

V₄ ensures that PLM behavior remains interpretable and comparable across models.

7. vST Outputs for PLMs#

vST produces:

• structural‑coherence diagnostics
• dimensional‑continuity indicators
• regime‑transition maps
• core‑alignment metrics
• drift‑detection signals
• cross‑version comparison surfaces

These outputs support reproducible, substrate‑aligned evaluation of PLM inference.

8. Summary#

The vST layers provide a complete validation framework for PLMs:

• V₁ ensures structural coherence
• V₂ ensures dimensional continuity
• V₃ ensures regime‑transition stability
• V₄ ensures core alignment

Together, they form a rigorous, invariant‑preserving system for analyzing high‑dimensional protein‑sequence embeddings.

If you want to keep the momentum, I can move directly into drift_detection_plm.md so the core of this artifact is fully complete. ### vST for Protein Language Models

References#

This appendix lists references relevant to protein language models, high‑dimensional embedding analysis, scaling laws, structural biology, and validation frameworks. Citations are grouped by category for clarity and presented in a substrate‑agnostic, model‑independent format consistent with the RSM and vST canon.

1. Protein Language Models and Sequence Embeddings#

• Rives, A., Meier, J., Sercu, T., et al.
  Biological Structure and Function Emerge from Scaling Unsupervised Learning to 250 Million Protein Sequences.
  PNAS 118, e2016239118 (2021).

• Elnaggar, A., Heinzinger, M., Dallago, C., et al.
  ProtTrans: Towards Cracking the Language of Life’s Code Through Self‑Supervised Deep Learning and High Performance Computing.
  IEEE TPAMI (2021).

• Rao, R., Liu, J., Verkuil, R., et al.
  MSA Transformer.
  ICML (2021).

• Madani, A., McCann, B., Naik, N., et al.
  ProGen: Language Modeling for Protein Generation.
  arXiv:2004.03497 (2020).

2. Structural Biology and Protein Representation#

• Jumper, J., Evans, R., Pritzel, A., et al.
  Highly Accurate Protein Structure Prediction with AlphaFold.
  Nature 596, 583–589 (2021).

• Baek, M., DiMaio, F., Anishchenko, I., et al.
  Accurate Prediction of Protein Structures and Interactions Using a Three‑Track Neural Network.
  Science 373, 871–876 (2021).

• AlQuraishi, M.
  End‑to‑End Differentiable Learning of Protein Structure.
  Cell Systems 8, 292–301 (2019).

3. High‑Dimensional Modeling and Representation Learning#

• Bengio, Y., Courville, A., & Vincent, P.
  Representation Learning: A Review and New Perspectives.
  IEEE TPAMI 35, 1798–1828 (2013).

• Coifman, R. R., & Lafon, S.
  Diffusion Maps.
  Applied and Computational Harmonic Analysis 21, 5–30 (2006).

• Tenenbaum, J. B., de Silva, V., & Langford, J. C.
  A Global Geometric Framework for Nonlinear Dimensionality Reduction.
  Science 290, 2319–2323 (2000).

4. Scaling Laws and Model Dynamics#

• Kaplan, J., McCandlish, S., Henighan, T., et al.
  Scaling Laws for Neural Language Models.
  arXiv:2001.08361 (2020).

• Hoffmann, J., Borgeaud, S., Mensch, A., et al.
  Training Compute‑Optimal Large Language Models.
  arXiv:2203.15556 (2022).

• Bahri, Y., Kadmon, J., Pennington, J., et al.
  Statistical Mechanics of Deep Learning.
  Annual Review of Condensed Matter Physics 11, 501–528 (2020).

5. Regime Behavior, Stability, and Dynamics#

• Strogatz, S.
  Nonlinear Dynamics and Chaos.
  Westview Press (2014).

• Ott, E.
  Chaos in Dynamical Systems.
  Cambridge University Press (2002).

• Guckenheimer, J., & Holmes, P.
  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields.
  Springer (1983).

6. Validation, Drift Detection, and ML Systems#

• Breck, E., Cai, S., Nielsen, E., et al.
  The ML Test Score: A Rubric for ML Production Readiness and Technical Debt Reduction.
  Google Research (2017).

• Sculley, D., Holt, G., Golovin, D., et al.
  Hidden Technical Debt in Machine Learning Systems.
  NIPS (2015).

• Amershi, S., Begel, A., Bird, C., et al.
  Software Engineering for Machine Learning: A Case Study.
  ICSE‑SEIP (2019).

7. Substrate‑Level and Triadic‑Frameworks Canon#

• Loswin, N.
  Resonance Substrate Model (RSM): Structural Foundations for High‑Dimensional Inference.
  TriadicFrameworks (2025).

• Loswin, N.
  Triadic Dimensional Cores: A 3D–9D Substrate for Structural and Inference‑Level Alignment.
  TriadicFrameworks (2025).

• Loswin, N.
  Validation‑Space‑Time (vST): A Substrate‑Level Framework for Reproducibility and Drift Detection.
  TriadicFrameworks (2025).

• Loswin, N.
  Dimensional Substrate Structures: Scaling Laws and High‑Dimensional Regimes.
  TriadicFrameworks (2026).

• Loswin, N.
  vST for Protein Language Models.
  TriadicFrameworks (2026). ### vST for Protein Language Models

Terminology#

This appendix defines the terminology used throughout the vST for Protein Language Models artifact. Terms are presented in a substrate‑agnostic, model‑independent manner and apply to any transformer‑based PLM operating across the full dimensional ladder (3D → 1024D). Definitions emphasize primitive‑level structure, regime behavior, scaling continuity, and invariant preservation.

1. Substrate Terms#

PLM Substrate#

A structured, invariant‑preserving framework for representing and interpreting protein‑sequence embeddings across 64D–4096D.

Dimensional Ladder#

The ordered sequence of dimensional regimes used for projection and scaling analysis:
3D → 6D → 9D → 64D → 128D → 256D → 512D → 1024D.

Coherence Surface#

A stable region in embedding space where residue‑level trajectories converge and maintain structural continuity.

2. Primitive Terms#

Dimensional Primitive (DP)#

The minimal unit of embedding‑space structure, capturing local coherence and variance behavior across residues.

Triadic Dimensional Primitive (TDP)#

A triad of DPs forming the smallest unit capable of expressing full regime behavior (R₁, R₂, R₃).

Scaling Primitive (SP)#

A rule‑based expansion unit that preserves invariants during dimensional scaling.

Coherence Primitive (CP)#

A minimal unit identifying stable, transitional, or dispersed regions in high‑dimensional embedding space.

3. Core Terms#

Triadic Dimensional Core (TDC)#

The 3D–9D substrate composed of one or more TDPs, used for interpretable projection of residue embeddings.

3D Structural Core#

Captures motif‑level geometry and compact residue‑level structure.

6D Interaction Core#

Captures relational and attention‑driven structure across residues.

9D Coherence Core#

Captures pathway‑level coherence and resonance‑time behavior across the sequence.

4. Regime Terms#

High‑Dimensional Regimes (R₁ᴴ, R₂ᴴ, R₃ᴴ)#

The triadic regime structure expressed in 64D–1024D embedding space.

Stable Regime (R₁ / R₁ᴴ)#

Compact, coherent, low‑variance embedding behavior.

Transition Regime (R₂ / R₂ᴴ)#

Branching, oscillatory, or reorientation behavior across residues.

Dispersion Regime (R₃ / R₃ᴴ)#

Diffuse, fragmented, or unstable embedding behavior.

5. Scaling Terms#

Scaling Behavior#

The structured expansion of embedding‑space capacity as PLM size increases.

Scaling Regimes (S₁, S₂, S₃)#

Triadic scaling behavior describing stable, transitional, and dispersion‑prone scaling phases.

Dimensional Continuity#

The requirement that embedding‑space expansion remains smooth and invariant‑preserving.

6. Projection Terms#

Invertible Projection#

A projection from high‑dimensional embedding space into 3D–9D that preserves primitive‑level structure and regime identity.

Regime‑Aware Projection#

A projection that maintains correct mapping of R₁, R₂, and R₃ behaviors.

Primitive‑Aligned Projection#

A projection that preserves DP, TDP, SP, and CP structure.

7. Alignment Terms#

Layer‑to‑Layer Alignment#

Comparison of residue‑level embedding trajectories across transformer layers.

Residue‑to‑Residue Alignment#

Comparison of embeddings across positions in a protein sequence.

Cross‑Version Alignment#

Comparison of embedding‑space structure across model versions or checkpoints.

Cross‑Model Alignment#

Comparison of embedding‑space geometry across different PLM architectures.

8. Validation Terms#

vST (Validation‑Space‑Time)#

A substrate‑level validation framework evaluating structural coherence, dimensional continuity, regime behavior, and core alignment.

Validation Layers (V₁–V₄)#

Four structured evaluation layers ensuring invariant‑preserving behavior across the dimensional ladder.

9. Drift Terms#

Drift#

A deviation from expected substrate behavior, indicating instability or invariant failure.

Drift Categories (D₁–D₄)#

Classification of drift into structural, dimensional, regime, or projection drift.

Drift Severity#

A measure of drift magnitude (low, moderate, high). ### 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. ### vST for Protein Language Models

Example: Sequence‑Level Regime Transitions in PLM Embeddings#

This example demonstrates how a Protein Language Model (PLM) expresses regime transitions (R₁ᴴ → R₂ᴴ → R₃ᴴ) along a protein sequence. It shows how residue‑level embeddings evolve across layers, how coherence surfaces form and break, and how the vST framework classifies transitions using the 1024D substrate.

The goal is to provide a reproducible, invariant‑preserving demonstration of regime behavior in PLM inference.

1. Input Overview#

For this example, we assume:

• a transformer‑based PLM with ≥1024D hidden states
• a single protein sequence of length L
• access to residue embeddings across all layers
• stable projection into 3D–9D cores

No architecture‑specific mechanisms are required; the example is substrate‑agnostic.

2. Step 1 — Extract Residue Embedding Trajectories#

For each residue position ( r \in [1, L] ), extract the 1024D embeddings across layers:

[ e_r^{(1)},\ e_r^{(2)},\ \dots,\ e_r^{(N)} ]

Observed Properties#

• early layers: compact, low‑variance embeddings
• mid layers: branching and oscillatory behavior
• late layers: partial dispersion in flexible regions

Interpretation#

Residue embeddings trace a high‑dimensional pathway that reflects biochemical context and structural constraints.

3. Step 2 — Identify Regime Behavior Across the Sequence#

Using variance distribution, coherence‑surface continuity, and primitive‑level stability, classify each residue’s regime.

Example Regime Map (Residue Index → Regime)#

Interpretation#

The sequence alternates between stable structural regions and transitional or disordered regions, reflecting typical protein architecture.

4. Step 3 — Project Embeddings into 9D (Coherence Core)#

Project each residue’s 1024D embedding into the 9D coherence core.

What is preserved#

• regime identity
• resonance‑time behavior
• primitive‑level structure
• coherence‑surface continuity

What becomes visible#

• stable surfaces in R₁ᴴ
• branching in R₂ᴴ
• fragmentation in R₃ᴴ

Interpretation#

The 9D projection reveals the “shape” of the embedding landscape along the sequence.

5. Step 4 — Project 9D → 6D → 3D#

6D Interaction Projection#

Reveals:

• residue‑interaction surfaces
• context‑dependent reorientation
• structural boundaries

3D Structural Projection#

Reveals:

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

Interpretation#

The 3D projection provides the minimal interpretable representation of the sequence‑level embedding trajectory.

6. Step 5 — 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 sequence passes all vST layers with warnings localized to the R₃ᴴ region.

7. Step 6 — Drift Detection#

Evaluate drift using D₁–D₄ categories:

• D₁ Structural Drift: low (localized to disordered region)
• D₂ Dimensional Drift: none
• D₃ Regime Drift: moderate (R₃ᴴ onset)
• D₄ Projection Drift: none

Interpretation#

The model exhibits expected dispersion in flexible or disordered regions but no harmful drift.

8. Summary#

This example demonstrates:

• how residue embeddings trace high‑dimensional trajectories
• how regime behavior evolves along a protein sequence
• how projection reveals coherence and instability
• how vST layers validate structural integrity
• how drift detection identifies localized dispersion

Sequence‑level regime transitions are a core interpretability signal in PLM inference.