---
license: apache-2.0
tags:
  - flow-matching
  - diffusion
  - geometric-deep-learning
  - constellation
  - geolip
  - cifar10
  - geometric-lookup
---

# GeoLIP Spherical Diffusion Prototype

**Flow matching diffusion through constellation bottleneck on S^15.**

Four progressive experiments proving that geometric triangulation
on the unit hypersphere is a viable information bottleneck for
diffusion models — and that the binding constant 0.29154 emerges
from velocity matching through geometric lookup.

## Experiments

### v1 — Regulator (baseline)
Constellation as a side-channel regulator on feature maps.
Gate stayed at 6%. Constellation was decorative.
- Loss: 0.1900 | Params: 6.1M | Near 0.29: 0%

### v2 — Skip Bypass (the sneaky test)
268M parameter `Linear(16384, 16384)` skip projection alongside
the constellation bottleneck. The model was given every reason
to bypass the constellation. **It chose the constellation** — gate
at 11.8%, routing 88% through 768 triangulation dimensions.
- Loss: 0.1757 | Params: 287M | Near 0.29: 9%

### v3 — Pure Constellation Bottleneck
Skip projection removed. Everything through S^15. Zero bypass.
Beat the 268M skip version with 8× fewer bottleneck params.
Reconstruction cos_sim ≈ 0 — the bottleneck is a geometric
lookup table, not an autoencoder.
- Loss: 0.1749 | Params: 36.6M | Near 0.29: 30%

### v4 — Geometric Lookup Flow Matching (GLFM)
Three-stage pipeline: Address → Condition → Generate.
Multi-scale addressing (coarse + fine). 46% of anchors
converged within ±0.05 of the binding constant 0.29154.
- Loss: 0.1754 | Params: 35.2M | Near 0.29: 46%

## The 0.29154 Binding Constant

Anchor drift from home position converges toward 0.29154 radians
across all experiments. This constant has now appeared in:

| Domain | Architecture | Training |
|---|---|---|
| MinimalShunts | Binding/separation phase boundary | Contrastive |
| CLIP projections | Geometric transition | Contrastive |
| T5 generation | Alpha convergence | Language modeling |
| CaptionBERT | Phase boundary | Contrastive |
| **Flow matching** | **Max anchor drift** | **Velocity matching** |

The constant marks the boundary where anchors transition from
geometric frame holders to task-specific encoders.

## Key Empirical Results

| Finding | Result |
|---|---|
| CV ≈ 0.20 is geometry of S^15 | Precision-invariant, 1-bit to fp64 |
| Constellation relay preserves 99.4% cos_to_orig at depth 16 | vs 7.4% for attention |
| Model prefers constellation over 268M skip bypass | 88/12 split |
| 768 tri dims match 16384 unconstrained dims for velocity | cos 0.949 |
| Bottleneck doesn't reconstruct — it's a lookup table | cos_sim ≈ 0 to input |
| Anchors self-organize: structural (<0.29) vs semantic (>0.29) | Confirmed across 4 versions |

## Architecture — GLFM (v4)

```
Stage 1 — ADDRESS
  encoder(x_t) → (B, 256, 8, 8)
  coarse: pool → proj → S^15 → triangulate (768d)
  fine: per-pixel → proj → S^15 → triangulate → aggregate (768d)
  address = concat(coarse, fine) = 1536d

Stage 2 — CONDITION
  fuse(address + time_emb + class_emb + noise_emb) → 1024d

Stage 3 — GENERATE
  4× ResBlock(1024d) → proj(16384d) → reshape(256, 8, 8) → decoder
```

## Files

### HuggingFace Integration
- `configuration_flow_match.py` — PretrainedConfig
- `modeling_flow_match.py` — PreTrainedModel (AutoModel compatible)

### Checkpoints (if present)
- `checkpoints/` — best checkpoints from each training run

### Samples (if present)
- `samples/` — v1 regulator samples
- `samples_bn/` — v2/v3 bottleneck samples
- `samples_cd/` — v3 pure constellation samples
- `samples_glfm/` — v4 GLFM samples

### Analysis Outputs (if present)
- `analysis/` — v1 analysis images
- `analysis_bn/` — v2 analysis images
- `analysis_cd/` — v3 analysis images
- `analysis_glfm/` — v4 analysis images

## Part of the GeoLIP Ecosystem

- [geolip-constellation-core](https://huggingface.co/AbstractPhil/geolip-constellation-core)
- [geolip-diffusion-proto](https://huggingface.co/AbstractPhil/geolip-diffusion-proto) (v1/v2 regulator)
- [geolip package](https://pypi.org/project/geolip/)
- [glip-autoencoder](https://github.com/AbstractEyes/glip-autoencoder)