geolip-svae-implicit-solver-experiments

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

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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.

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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).

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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

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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.

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How to load a checkpoint

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.

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How to read a probe result

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

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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.

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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+.

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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.

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License

Apache 2.0