vST for Generative Models#

Substrate Definition#

The substrate is architecture‑agnostic and applies to diffusion models, autoregressive generators, VAEs, flow models, GANs, and hybrid systems.

1. Purpose of the Generative‑Model Substrate#

The generative‑model substrate provides a structured, reproducible framework for:

• interpreting high‑dimensional latent‑space structure
• identifying stable, transitional, and dispersed generative regimes
• mapping coherence surfaces across sampling trajectories
• analyzing scaling behavior across model size and latent dimensionality
• detecting drift across checkpoints, fine‑tuning, or sampler changes
• projecting latent states into 3D–9D triadic cores for interpretability

Generative models produce structured, regime‑rich trajectories.
The substrate ensures these remain interpretable across the full dimensional ladder (3D → 1024D).

2. Substrate Overview#

Generative‑model latent spaces typically inhabit 64D–4096D regions.
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 latent trajectories, sampling phases, and generative 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 Generative Models#

3.1 Dimensional Primitive (DP)#

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

• local coherence within latent neighborhoods
• variance behavior across sampling steps
• projection stability
• regime alignment

DPs appear in diffusion steps, autoregressive hidden states, flow‑model transformations, and VAE latent transitions.

3.2 Triadic Dimensional Primitive (TDP)#

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

• stable (R₁) generative phases
• transitional (R₂) sampling or decoding phases
• dispersed (R₃) noisy or unstable phases

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 latent‑space capacity expands with model size, sampler complexity, or latent dimensionality.

3.4 Coherence Primitive (CP)#

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

• coherent generative phases
• transitional sampling regions
• dispersed or noisy latent states
• regime transitions

CPs are essential for drift detection and vST validation.

4. Triadic Dimensional Cores for Generative Models#

4.1 3D Structural Core#

Captures motif‑level geometry in latent activations:

• compact motifs in stable phases
• oscillatory motifs in transitional phases
• diffuse motifs in noisy or unstable phases

4.2 6D Interaction Core#

Captures relational structure across sampling steps:

• cross‑step coupling
• sampler‑driven reorientation
• early instability signatures

4.3 9D Coherence Core#

Captures pathway‑level coherence across generative trajectories:

• 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)#

Generative‑model latent spaces naturally inhabit high‑dimensional regimes.
The substrate models these using the dimensional ladder:

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

Each step preserves:

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

This ensures stable interpretation across architectures and sampling methods.

6. Generative‑Trajectory Structure#

Generative models produce trajectories that move through:

• compact stable regions (R₁ᴴ)
• branching transitional regions (R₂ᴴ)
• dispersed or noisy 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 latent states 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 generative‑model substrate produces:

• generative‑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 generative models.