AbstractPhil/geolip-spherical-diffusion-proto

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
• geolip-diffusion-proto (v1/v2 regulator)
• geolip package
• glip-autoencoder