vST for Generative Models#
Projection of Latent States and Alignment Across Sampling Trajectories, Checkpoints, and Samplers#
This document defines how high‑dimensional latent states from generative models are projected into the triadic dimensional cores (3D–9D), and how latent‑space alignment is performed across sampling steps, checkpoints, architectures, and sampler configurations.
Projection provides interpretability.
Alignment provides comparability.
Together, they form the backbone of vST analysis for generative systems.
1. Purpose of Projection in Generative Models#
Projection enables us to:
- interpret high‑dimensional latent states through 3D–9D cores
- identify stable, transitional, and dispersed generative regimes
- map coherence surfaces across sampling trajectories
- compare latent states across checkpoints, samplers, or architectures
- detect drift or fragmentation in latent‑space structure
- support vST validation (V₁–V₄)
Generative latents are structured, sampler‑conditioned, and often multi‑modal.
Projection reveals this structure in a compact, interpretable form.
2. Projection Overview#
Generative‑model latent spaces often inhabit 64D–4096D regions.
The substrate projects these states 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 generative signals remain interpretable.
3. Projection Steps#
3.1 High‑Dimensional → 9D (Coherence Projection)#
This step extracts pathway‑level coherence across sampling trajectories.
Preserves
- regime identity (R₁ᴴ, R₂ᴴ, R₃ᴴ)
- resonance‑time behavior
- primitive‑level structure (DP, TDP, SP, CP)
- coherence‑surface continuity
Reveals
- stable refinement phases
- branching mid‑trajectory transitions
- noise‑dominated or unstable regions
3.2 9D → 6D (Interaction Projection)#
This step compresses coherence pathways into interaction surfaces.
Preserves
- relational geometry across sampling steps
- sampler‑driven reorientation
- regime‑transition indicators
Reveals
- cross‑step coupling
- sampler‑dependent behavior
- early instability signatures
3.3 6D → 3D (Structural Projection)#
This step reduces interaction surfaces into geometric motifs.
Preserves
- motif‑level geometry
- temporal continuity
- stable structural invariants
Reveals
- compact motifs in R₁ᴴ
- oscillatory geometry in R₂ᴴ
- diffuse patterns in R₃ᴴ
4. Latent‑Space Alignment Overview#
Alignment compares projected structures across:
- sampling steps
- noise levels
- checkpoints
- samplers
- architectures
- training runs
- fine‑tuning variants
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 Step‑to‑Step Alignment#
Reveals:
- regime transitions
- coherence‑surface evolution
- sampler‑driven reorientation
Used for:
- diffusion trajectories
- autoregressive decoding
- flow‑model transformations
5.2 Cross‑Checkpoint Alignment#
Reveals:
- training‑driven drift
- latent‑space maturation
- collapse or recovery of coherence surfaces
Used for:
- fine‑tuning
- long‑run training
- checkpoint comparison
5.3 Cross‑Sampler Alignment#
Reveals:
- sampler‑induced divergence
- noise‑schedule sensitivity
- stability of refinement phases
Used for:
- DDPM vs. DDIM
- Euler vs. Heun
- ancestral vs. deterministic samplers
5.4 Cross‑Architecture Alignment#
Reveals:
- structural compatibility
- scaling‑law continuity
- architecture‑driven drift
Used for:
- diffusion → autoregressive hybrids
- VAE → diffusion pipelines
- flow‑model integration
6. Projection Stability and Failure Modes#
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, scaling‑law violations, or sampler instability.
7. Alignment Failure Modes#
Alignment failures include:
- cross‑checkpoint divergence
- sampler‑induced fragmentation
- architecture‑dependent incompatibility
- loss of primitive‑aligned projection
- inconsistent 3D–9D mapping
These failures signal structural drift or instability.
8. Outputs of Projection and Alignment#
Projection and alignment produce:
- temporal coherence maps
- cross‑checkpoint alignment surfaces
- cross‑sampler drift‑detection signals
- scaling‑law diagnostics
- vST validation outputs
- interpretable 3D–9D projections
These outputs support reproducible, substrate‑level analysis of generative models.