vST for Scientific Simulators#

Projection of High‑Dimensional Simulation States into Triadic Dimensional Cores#

This document defines how high‑dimensional simulation states are projected into the triadic dimensional cores (3D–9D). Projection enables interpretable, invariant‑preserving analysis of state‑space trajectories, dynamical regimes, solver behavior, and cross‑version drift in scientific simulators.

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

1. Purpose of Projection in Scientific Simulators#

Projection allows us to:

• interpret high‑dimensional simulation states through 3D–9D cores
• identify stable, transitional, and dispersed dynamical regimes
• map coherence surfaces across time and space
• compare states across solver iterations, grid resolutions, or model versions
• detect drift or fragmentation in state‑space structure
• support vST validation (V₁–V₄)

Simulation states are structured, physical, and often multi‑field.
Projection reveals this structure in a compact, interpretable form.

2. Projection Overview#

Simulation state‑spaces often inhabit 64D–10⁶D 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 physical signals remain interpretable.

3. Projection Steps#

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

This step extracts pathway‑level coherence across time, space, or solver iterations.

Preserves

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

Reveals

• stable vs. unstable dynamical regions
• transitions between physical phases
• dispersion in chaotic or poorly conditioned regions

Interpretation
The 9D projection exposes the “shape” of the simulation’s dynamical evolution.

3.2 9D → 6D (Interaction Projection)#

This step compresses coherence pathways into interaction surfaces.

Preserves

• relational geometry across fields or particles
• solver‑driven coupling behavior
• regime‑transition indicators

Reveals

• interaction‑driven reorientation
• multi‑field coupling patterns
• boundary behavior between dynamical phases

Interpretation
The 6D projection highlights how the simulator integrates physical interactions.

3.3 6D → 3D (Structural Projection)#

This step reduces interaction surfaces into geometric motifs.

Preserves

• motif‑level geometry
• spatial or particle‑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 simulation state.

4. Alignment Overview#

Alignment compares projected structures across:

• solver iterations
• spatial or particle domains
• grid resolutions
• solver configurations
• model versions
• multi‑field couplings

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 Iteration‑to‑Iteration Alignment#

Compares state trajectories across solver steps.

Reveals:

• where regime transitions occur
• how coherence surfaces evolve
• which solver stages stabilize or destabilize the system

5.2 Spatial/Particle Alignment#

Compares states across spatial regions or particle subsets.

Reveals:

• coherent vs. divergent regions
• phase boundaries
• localized instabilities

5.3 Cross‑Resolution Alignment#

Compares states across grid refinements or timestep reductions.

Reveals:

• scaling‑law continuity
• resolution‑dependent drift
• stability of coherence surfaces

5.4 Cross‑Version Alignment#

Compares states across simulator versions or parameterizations.

Reveals:

• drift introduced by code changes
• solver‑conditioning effects
• changes in regime behavior

6. Projection Stability and Failure Modes#

Projection stability is a key indicator of simulator 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, scaling‑law violations, or numerical instability.

7. Outputs of Projection and Alignment#

Projection and alignment produce:

• temporal or spatial coherence maps
• cross‑iteration and cross‑resolution 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 scientific simulators.