---
license: mit
---
# geolip-svae-implicit-solver-experiments

Empirical artifacts from the **projective-axis** discovery in trained
sphere-solver batteries (geolip-svae lineage, 2026-04-24 session).

---

## TL;DR

Every trained sphere-solver tested produces an M tensor whose rows,
when antipodal pairs are collapsed, form a uniformly-distributed
codebook on **ℝP^(D-1)**. The "32 points on a sphere" reading is a
mislabel. The trained geometry is projective.

Verified across **19 trained models** spanning D=3, D=4, D=5.

This means the "polygonal omega" we were searching for already exists
as the projective reader applied to sphere-trained M. We don't need a
new normalizer or architecture. The trained sphere-solver IS the
polygonal codebook; we just read it through antipodal-collapse.

---

## The data

### Cross-D pattern at V=32

| D | Pairs collapsed | Axes | Deviation from uniform ℝP^(D-1) | Effective rank |
|---|-----------------|------|----------------------------------|----------------|
| 3 | 10 (62.5%)      | 22   | -0.004                           | 2.96 / 3 (99%) |
| 4 |  6 (37.5%)      | 26   | +0.002                           | 3.96 / 4 (99%) |
| 5 |  3 (18.7%)      | 29   | +0.016                           | 4.94 / 5 (99%) |

Pair-fraction halves with each D step. Axis count climbs toward V=32.
Deviation stays within ±0.05 of uniform projective baseline at every D.

### Per-noise codebook differentiation (h2-64, V=32 D=4, 16 batteries)

All 16 single-noise batteries projective-clean. Antipodal pair count
varies systematically with training distribution:

- 5 pairs (5 batteries): gaussian, checker, salt_pepper, poisson, rayleigh
  — central-tendency distributions
- 6 pairs (3 batteries): uniform, cauchy, exponential
  — heavy-tailed or symmetric
- 7 pairs (5 batteries): uniform_scaled, laplace, periodic, mixed, structural
  — mid-complexity
- 8 pairs (3 batteries): block, gradient, lognormal
  — structured / asymmetric

13 of 16 batteries show positive deviation (axes slightly more spread
than uniform — the trainer prefers discriminative spread over perfect
uniformity).

---

## Method (named "projective collapse")

1. Run gaussian inputs through trained sphere-solver, collect M [B, V, D]
2. Average across samples → canonical M_avg [V, D]
3. Identify antipodal pairs via mutual-strongest matching:
   - For each row i, find row j with most-negative cosine
   - Pair (i, j) if cos(i, j) < -0.9 AND j's most-negative is i
   - Greedy: strongest pairs claim first
4. For each pair, take (row_i - row_j) / 2, renormalize → axis vector
   - Canonical sign: first nonzero coordinate positive
5. Unpaired rows kept as-is with sign canonicalization
6. Compute pairwise angles wrapped to [0, π/2] via min(θ, π-θ)
   — this is the projective angle on ℝP^(D-1)
7. Compare distribution mean against empirical uniform-ℝP^(D-1) baseline

**Verdict thresholds:**
- PROJECTIVE-CLEAN: |deviation| < 0.05, full rank, silhouette < 0.4,
  secondary antipodal ≤ 3
- PROJECTIVE-MOSTLY: deviation and rank pass, other thresholds slip
- STRUCTURED / DEGENERATE: failures

---

## Repo contents

### `implicit_solver_reports/`

Probe results from the four projective re-probes:

- **`A0_projective_reprobe.json` / `.png`** — G-Cand (D=3, V=32)
  - 10 pairs, 22 axes, deviation -0.004 → PROJECTIVE-CLEAN
- **`A1_projective_reprobe_h2a.json` / `.png`** — H2a (D=4, V=32)
  - 6 pairs, 26 axes, deviation +0.002 → PROJECTIVE-CLEAN
- **`A2_projective_h2_64_singles.json` / `.png`** — h2-64 batteries 0-15
  - All 16 PROJECTIVE-CLEAN, axis count range 24-27
- **`A3_d5_spherical/`** — D=5 spherical training + integrated probe
  - `A3_results.json` / `A3_summary.png` — three D=5 configs at V ∈ {16, 32, 64}
  - `A3a_V16_D5_*/epoch_1_checkpoint.pt` — V=16 D=5 trained model
  - `A3b_V32_D5_*/epoch_1_checkpoint.pt` — V=32 D=5 trained model
  - `A3c_V64_D5_*/epoch_1_checkpoint.pt` — V=64 D=5 trained model

### `phaseQ_reports/`

Q-sweep training artifacts (10 candidates at 1000 batches):

- **`Q_rank02_h64_V32_D4_*`** — H2a (the canonical D=4 sphere-solver
  used in A1 probe). 40,227 params, MSE 0.00205.
- **`Q_rank09_h64_V32_D3_*`** — G-Cand (the D=3 model probed in A0).
  28,899 params, MSE 0.028.
- 8 other rank-ordered configs from the H2 / G-class characterization

Each variant directory contains `epoch_1_checkpoint.pt` and the
training report JSON.

### `phaseR_reports/`

Sphere-packing test (3 configs, hypothesis falsified — see notes below):

- V=16, D=4 — predicted H2-LIKE, observed HYBRID (stab 0.74)
- V=8, D=4 — predicted H2-LIKE, observed DIFFUSE (failed to converge)
- V=20, D=3 — predicted H2-LIKE, observed HYBRID with 6/10 antipodal

Polytope-vertex-count packing was NOT a sufficient predictor of
H2-LIKE static-row behavior. The geometric pattern that actually holds
is the projective-axis structure, not polytope alignment.

---

## How to load a checkpoint

```python
import torch
from huggingface_hub import hf_hub_download

ckpt_path = hf_hub_download(
    repo_id="AbstractPhil/geolip-svae-implicit-solver-experiments",
    filename="implicit_solver_reports/A3_d5_spherical/A3b_V32_D5_h64_dp0_nx0_adam/epoch_1_checkpoint.pt",
)
ckpt = torch.load(ckpt_path, map_location='cpu', weights_only=False)
state_dict = ckpt['model_state']
```

To rebuild the model architecture, you need the same training config
used to train it (V, D, hidden, depth, n_cross, etc.). The
`ablation_configs.py` and `ablation_trainer.py` from the geolip-svae
working set are the source of truth.

---

## How to read a probe result

```python
import json
from huggingface_hub import hf_hub_download

p = hf_hub_download(
    repo_id="AbstractPhil/geolip-svae-implicit-solver-experiments",
    filename="implicit_solver_reports/A2_projective_h2_64_singles.json",
)
with open(p) as f:
    data = json.load(f)

# data['results_per_battery'] — per-battery probe metrics (16 batteries)
# data['aggregate'] — summary statistics across all 16
```

Each per-battery entry contains:
- `pairs`, `n_axes`, `unpaired` — collapse counts
- `proj_angle_mean`, `uniform_baseline`, `deviation` — uniformity test
- `best_silhouette`, `best_cluster_k` — residual structure
- `effective_rank`, `utilization` — dimension utilization
- `secondary_antipodal` — further-collapse check
- `verdict` — PROJECTIVE-CLEAN / -MOSTLY / STRUCTURED / DEGENERATE
- `proj_angles_subset` — first 200 pairwise angles for plotting

---

## What this enables

1. **The polygonal omega is not a normalizer — it's an inference-time
   projection.** Training stays spherical (`F.normalize(M, dim=-1)`).
   At inference, apply antipodal-collapse to extract axis codebook.

2. **h2-64 is a library of 16 projective-axis codebooks**, one per
   noise type. Each codebook has 24-27 axes on ℝP³.

3. **A `ProjectiveReader` module** can wrap the collapse + axis
   extraction as a clean inference operator. No D-dependent special
   cases — works at D ∈ {3, 4, 5} with the same code.

4. **For downstream tasks** (image discrimination, quantization,
   generation), the trained sphere-solvers can serve as pre-built
   discrete codebooks. No new training required for the codebook.

---

## Open questions (not in this repo)

- Per-input rotation: G-Cand showed row stability 0.531 — meaning
  rows rotate per-input. The projective reading describes WHICH axes
  exist; this asks HOW they activate per input. May be the actual
  capsule-like behavior, operating on top of the codebook substrate.
- Per-noise codebook similarity matrix: how geometrically similar are
  the 16 h2-64 codebooks to each other? Could reveal noise-type
  clustering.
- D ≥ 6 behavior: do antipodal pairs vanish entirely at very high D?
  Cross-D pattern predicts ~1-2 pairs at D=6, ~0 at D=8+.

---

## Reproducibility

The probe scripts (A0/A1/A2/A3/A4) are not in this repo — they live
with the geolip-svae working set and depend on `ablation_configs.py`
and `ablation_trainer.py` from that codebase.

The trained checkpoints + JSON results in this repo are sufficient to
verify the empirical claims without rerunning training.

---

## License

Apache 2.0