results.qmd

---
title: "Results"
subtitle: "Signal Landscape Parallel Exploration Findings"
---

Can a graph neural network recover the connectivity matrix of a neural assembly from its activity alone? This page collects the quantitative results of a closed-loop exploration in which an LLM proposes simulation configurations and training hyper-parameters, a GNN is trained on the resulting synthetic data, and the recovered connectivity is compared to ground truth. The exploration uses UCB tree search with 4 parallel slots per batch to efficiently map the simulation--training landscape.

::: {.callout-tip}
## Current status

**360 iterations completed** across 30 regimes. The number of frames is the dominant lever for dense supercritical networks: 300 and 600 neurons both achieve 100% convergence at 30k frames; **1000 neurons achieves 100% convergence (12/12) at 100k frames with connectivity=1.000** (block 30 breakthrough). Low gain (g=3) is rescued by 30k frames; g=2 is partially rescued (42% at 100 neurons/30k, 92% at 200 neurons/30k). Three structural limits confirmed: subcritical spectral radius (sparse 50%, connectivity~0.44), partial connectivity ceiling (fill=80%/90%, connectivity $\approx$ filling_factor), and fixed-point collapse (g=1, activity rank drops to 1 at 30k). The gain--activity rank relationship depends on network size: at 100 neurons, g=1→5, g=2→17, g=3→26, g=7→35; but at 200 neurons, g=2→35--38 (n compensates for gain reduction). ~99 principles established.
:::

## Outcomes

1. **The number of frames is the dominant lever**: Tripling the frame count from 10k to 30k transforms 300 neurons from 25% to 100% convergence, 600 neurons from 0% to 100%, and g=3 / 200 neurons from 0% to 100%. With sufficient frames, all training parameters become non-critical --- except when gain is too low (g≤2).

2. **Six difficulty axes identified**: Subcritical spectral radius (sparse), parameter count scaling (large n), data abundance (frame count), low gain (g≤3), partial connectivity (fill<100%), and fixed-point collapse (g=1). Four are solvable by increasing frames; two are structural limits; one is a new unsolvable axis.

3. **Three structural limits confirmed**: Sparse 50% (rho=0.746, connectivity~0.44 at both 10k and 30k), fill<100% (connectivity ceiling $\approx$ filling_factor --- confirmed across fill=50%, 80%, 90%, and 100%), and g=1 fixed-point collapse (activity rank=5 at 10k, drops to 1 at 30k --- more frames makes it worse).

4. **Activity rank is necessary but not sufficient**: High activity rank does not guarantee recovery if spectral radius is subcritical (sparse+noise: activity rank=91 but 0% convergence). More frames doubles activity rank for dense networks but has minimal effect on sparse and negative effect at g=1.

5. **L1 effect is n-dependent, non-monotonic, and vanishes at high frame counts**: Critical at 1E-6 for low-rank/heterogeneous at 100 neurons; harmful at $\leq$200 neurons; beneficial at 300 neurons/10k; harmful at $\geq$600 neurons/10k. At 30k frames, L1 sensitivity disappears entirely.

6. **Scale shifts boundaries**: Convergence boundary, optimal lr_W, and lr tolerance all change non-linearly with network size. At 30k frames, optimal lr_W shifts **lower** (5E-3 vs 1E-2 at 10k) because abundant data handles W while lower lr_W preserves MLP capacity.

7. **Low gain compounds with scale but is solvable (g≥2)**: g=3 reduces activity rank from 35 to 26 (100 neurons) and eliminates the lr_W cliff. g=3 / 200 neurons at 10k is universally degenerate (0% convergence), but 30k frames fully rescues it (100% convergence). g=2 requires inverse lr_W (5E-4, 100x lower than g=7) and is partially rescued at 30k (42% convergence).

8. **Gain modulates activity rank non-linearly, but neuron count compensates**: At 100 neurons the gain--activity rank relationship shows a sharp transition: g=1→5, g=2→17 (+240%), g=3→26, g=7→35. A critical threshold exists at activity rank ~10 (between g=1 and g=2). However, at 200 neurons/30k, g=2 achieves activity rank=35--38 --- matching g=7 / 100 neurons. Doubling the neuron count effectively compensates for the gain reduction, making g=2 / 200 neurons/30k much easier (88% convergence) than g=2 / 100 neurons/30k (42%).

9. **Partial connectivity creates a structural ceiling**: connectivity ceiling $\approx$ filling_factor, confirmed at four points: fill=50%→0.49, fill=80%→0.80, fill=90%→0.91, fill=100%→1.00. Complete parameter insensitivity at fill<100%, at both 10k and 30k frames.

10. **g=1 is a new unsolvable axis (fixed-point collapse)**: At g=1, tanh saturates weak-gain dynamics into stable fixed points. Activity rank=5 at 10k drops to 1 at 30k --- the only regime where frame count is anti-correlated with activity rank. More severe than sparse 50%. Connectivity_R^2^<0.02 regardless of training.

11. **Degeneracy is a critical diagnostic**: Two mechanisms: structural (subcritical spectral radius, g=1 fixed-point) and training-limited (fixable with more data/epochs). Abundant data (30k) eliminates training-limited degeneracy but amplifies g=1 degeneracy.

12. **1000 neurons at 100k frames is SOLVED (block 30 breakthrough)**: 100% convergence (12/12), connectivity=1.000, test_R^2^=0.882. lr_W=3E-3 optimal; batch=16 safe; 3 epochs sufficient. Confirms the frame-count scaling law: 1000 neurons needs ~100k frames, matching the pattern established at smaller n.

---

## Regime Landscape — Partitioned by Neuron Count

Three panels separate the landscape by network scale (100, 200--600, and 1000 neurons). Each data point shows the key mutation (parameter change) on hover.

```{=html}
<iframe src="assets/landscape_partitioned_interactive.html" width="100%" height="1450" style="border:none; border-radius:8px;"></iframe>
```

## Regime Summary

### Performance by Regime

| Block | Regime | frames | neurons | activity rank | Convergence | Best connectivity_R^2^ | Key Finding |
|:-----:|--------|:--------:|:---------:|----------|:-----------:|:---------------:|:------------|
| 1 | Chaotic | 10k | 100 | ~35 | **92%** (11/12) | 0.9999 | lr_W=4E-3 sweet spot; lr=1E-4 optimal |
| 2 | Low-rank (r=20) | 10k | 100 | ~12--14 | 75% (9/12) | 0.9997 | L1=1E-6 critical; lr_W=3E-3 |
| 3 | Dale (50/50 E/I) | 10k | 100 | ~12 | 67% (8/12) | 0.986 | Sharp lr_W cliff at 5E-3 |
| 4 | Heterogeneous (4 types) | 10k | 100 | ~38 | 17% FULL | 0.992 | Dual-objective; lr_emb=1E-3 critical |
| 5 | Noise (0.1--1.0) | 10k | 100 | 42--90 | **100%** (12/12) | 1.000 | Noise inflates activity rank |
| 6 | Scale | 10k | 200 | ~41--43 | 67% (8/12) | 0.956 | Boundary shifts up; lr=3E-4 safe |
| 7 | Sparse 50% | 10k | 100 | ~21 | **0%** (0/12) | 0.466 | Subcritical rho=0.746 |
| 8 | Sparse+Noise | 10k | 100 | ~91 | **0%** (0/12) | 0.490 | Structural data limit |
| 9 | 300 neurons (1--2ep) | 10k | 300 | ~44--47 | **0%** (0/12) | 0.890 | n_epochs=2 breakthrough |
| 10 | 300 neurons (2ep base) | 10k | 300 | ~44--47 | 25% (2/8) | 0.924 | L1=1E-6+3ep best |
| 11 | 200 neurons v2 | 10k | 200 | ~40--43 | **100%** (12/12) | 0.994 | lr_W=8E-3 optimal |
| 12 | 600 neurons | 10k | 600 | ~50 | **0%** (0/12) | 0.626 | Training-capacity-limited |
| 13 | 200 neurons + 4 types | 10k | 200 | ~42--44 | **100%** connectivity | 0.991 | Full dual convergence |
| 14 | Recurrent test | 10k | 200 | ~42--44 | 75% (3/4) | 0.993 | Connectivity-dynamics trade-off |
| 15 | **300 neurons (30k)** | **30k** | 300 | **79--80** | **100%** (12/12) | **1.000** | **Frames: 25%-->100%** |
| 16 | **600 neurons (30k)** | **30k** | 600 | **85--87** | **100%** (8/8) | **0.992** | **Frames: 0%-->100%** |
| 17 | **Sparse 50% (30k)** | **30k** | 100 | **~13** | **0%** (0/12) | **0.436** | **30k FAILS; activity rank DROPS** |
| 18 | **1000 neurons (30k)** | **30k** | 1000 | **~144** | **0%** (0/12) | **0.745** | **Needs ~100k frames** |
| 19 | **g=3 100 neurons** | 10k | 100 | **~26** | 42% (5/12) | **0.955** | **Low gain: new difficulty axis** |
| 20 | **g=3 200 neurons** | 10k | 200 | **~31** | **0%** (0/12) | **0.489** | **Gain x n compounds; universal degeneracy** |
| 21 | **g=3 200 neurons (30k)** | **30k** | 200 | **~53--57** | **100%** (12/12) | **0.996** | **30k rescues g=3; all params non-critical** |
| 22 | **fill=80%** | 10k | 100 | **~36** | **0%** (0/12) | **0.802** | **Connectivity plateau = filling_factor** |
| 23 | **fill=80% (30k)** | **30k** | 100 | **~48--49** | **0%** (0/12) | **~0.802** | **30k FAILS; structural ceiling; 12/12 at 0.802** |
| 24 | **fill=90%** | 10k | 100 | **~35--36** | **83%** (10/12) | **0.907** | **connectivity $\approx$ fill%; transitional regime; param insensitive** |
| 25 | **g=1 (10k)** | 10k | 100 | **~5** | **0%** (0/12) | **0.007** | **Fixed-point collapse; hardest regime; activity rank=5** |
| 26 | **g=1 (30k)** | **30k** | 100 | **~1** | **0%** (0/12) | **0.018** | **Activity rank DROPS 5→1; 30k makes g=1 WORSE** |
| 27 | **g=2 (10k)** | 10k | 100 | **~17** | **0%** (0/12) | **0.519** | **Inverse lr_W=5E-4; epoch scaling not diminishing** |
| 28 | **g=2 (30k)** | **30k** | 100 | **~16** | **42%** (5/12) | **0.997** | **30k partially rescues; inverse lr_W persists** |
| 29 | **g=2 200 neurons (30k)** | **30k** | 200 | **~35--38** | **92%** (11/12) | **0.979** | **n compensates gain: activity rank matches g=7 / 100 neurons** |
| 30 | **1000 neurons (100k)** | **100k** | 1000 | **~350** | **100%** (12/12) | **1.000** | **BREAKTHROUGH: 100k solves 1000 neurons; lr_W=3E-3** |

## Key Findings

### 1. Six Independent Difficulty Axes

::: {.callout-important}
## Key Insight

Six independent axes determine regime difficulty. Three are solvable by data (more frames); three are structural limits:

| Axis | Example | Solvable? | Mechanism |
|------|---------|:---------:|-----------|
| Subcritical spectral radius | Sparse 50% (rho=0.746) | **No** | rho < 1 limits information flow; activity rank drops at 30k |
| Partial connectivity ceiling | fill=80%/90% (rho~0.99) | **No** | connectivity ceiling $\approx$ filling_factor; 30k has no effect |
| Fixed-point collapse | g=1 (activity rank=5→1) | **No** | tanh saturates; activity rank drops at 30k; hardest regime |
| Parameter count scaling | 300, 600 neurons | **Yes** (30k) | More data reveals more signal dimensions |
| Low gain | g=2--3 (activity rank 17--26) | **Yes** (30k) | g=3: 100% at 30k; g=2: 42% at 30k |
| Training capacity | n_epochs, lr_W tuning | **Yes** (30k) | At 30k all training params become non-critical |
:::

### 2. Effective Rank Is Necessary but Not Sufficient

| neurons | frames | Regime | activity rank | Spectral radius | Convergence | Interpretation |
|:---------:|:--------:|--------|:--------:|:---------------:|:-----------:|:---------------|
| 100 | 10k | Noise=1.0 | 90 | >1.0 | **100%** | High activity rank + supercritical = easy |
| 100 | 10k | Sparse+Noise | 91 | <1.0 | **0%** | High activity rank + subcritical = hard |
| 100 | 10k | Chaotic | 35 | >1.0 | **92%** | Medium activity rank + supercritical = easy |
| 100 | 10k | fill=80% | 36 | 0.985 | **0%** | Medium activity rank + near-critical = plateau |
| 100 | 10k | Low-rank | 12 | ~1.0 | 75% | Low activity rank + critical = recoverable |
| 100 | 10k | g=3 | 26 | >1.0 | 42% | Reduced activity rank + supercritical = harder |
| 200 | 10k | g=3 | 31 | >1.0 | **0%** | Low activity rank + scale = training-limited |
| 200 | **30k** | g=3 | **55** | >1.0 | **100%** | **More frames restores activity rank** |
| 300 | **30k** | Chaotic | **80** | 1.03 | **100%** | **More frames doubles activity rank** |
| 600 | **30k** | Chaotic | **87** | 1.03 | **100%** | **More frames transforms 600 neurons** |
| 1000 | **30k** | Chaotic | **144** | ~1.0 | **0%** | **Highest activity rank but still insufficient** |
| 100 | **30k** | Sparse | **13** | 0.746 | **0%** | **Activity rank DROPS; rho controls activity rank** |
| 100 | **30k** | fill=80% | **49** | 0.985 | **0%** | **Activity rank rises but connectivity stuck** |
| 200 | **30k** | g=2 | **35--38** | >1.0 | **92%** | **n compensates: activity rank matches g=7 / 100 neurons** |

### 3. "Easy Mode" --- Chaotic Baseline (100 neurons)

- **Neuron count**: 100
- **Activity rank**: ~35
- **Tolerance**: lr_W range 1.5E-3 to 8E-3 all converge
- **Sweet spot**: lr_W=4E-3 (connectivity_R^2^=0.9999, test_R^2^=0.996)
- **lr=1E-4 is optimal**: increasing to 2E-4 or 3E-4 degrades dynamics

### 4. Low-Rank Breakthrough (100 neurons)

Low-rank connectivity (activity rank ~12) initially appeared much harder than chaotic, but a specific intervention unlocked near-chaotic performance:

```
Block 2 Progress (n=100):
Iter 13: lr_W=4E-3, L1=1E-5      → conn=0.999, test_R2=0.902 (dynamics poor)
Iter 18: lr_W=4E-3, L1=1E-6      → conn=1.000, test_R2=0.925 (improved!)
Iter 19: lr_W=3E-3, L1=1E-5      → conn=0.999, test_R2=0.943 (better lr_W)
Iter 21: lr_W=3E-3, L1=1E-6      → conn=0.993, test_R2=0.996 (BREAKTHROUGH)
```

::: {.callout-tip}
## Key Insight

For low-rank regimes (100 neurons), **reducing L1 from 1E-5 to 1E-6** is the critical enabler for dynamics recovery. Combined with lr_W=3E-3, this achieves chaotic-baseline-level performance (test_R^2^=0.996) despite activity rank=12.
:::

### 5. Dale's Law Creates Sharp Cliff (100 neurons)

Dale's law (excitatory/inhibitory constraint) reduces activity rank from 35 to 12 and introduces a sharp lr_W failure boundary:

| neurons | frames | lr_W | connectivity_R^2^ | Status |
|:---------:|:--------:|------|-----------|--------|
| 100 | 10k | 3.5E-3 | 0.958 | Converged |
| 100 | 10k | 4E-3 | 0.974 | Converged |
| 100 | 10k | 4.5E-3 | 0.986 | **Best** |
| 100 | 10k | 5E-3 | 0.458 | **FAILED** |
| 100 | 10k | 6E-3 | 0.555 | **FAILED** |

### 6. Noise Is Data Augmentation (100 neurons)

| neurons | frames | noise_level | activity rank | connectivity_R^2^ | Convergence |
|:---------:|:--------:|-------------|:--------:|-----------|:-----------:|
| 100 | 10k | 0 | 35 | 0.999 | 92% |
| 100 | 10k | 0.1 | 42 | 1.000 | **100%** |
| 100 | 10k | 0.5 | 84 | 1.000 | **100%** |
| 100 | 10k | 1.0 | 90 | 1.000 | **100%** |

### 7. Low Gain --- Independent Difficulty Axis (Blocks 19--21)

| neurons | frames | gain | activity rank | Convergence | Best connectivity | Key |
|:---------:|:--------:|:----:|:--------:|:-----------:|:---------:|:----|
| 100 | 10k | 3 | 26 | 42% (5/12) | 0.955 | New axis; no lr_W cliff; 3ep minimum |
| 200 | 10k | 3 | 31 | **0%** (0/12) | 0.489 | Gain x n compounds; universal degeneracy |
| 200 | **30k** | 3 | **55** | **100%** (12/12) | **0.996** | **30k rescues; all params non-critical** |

::: {.callout-tip}
## Key Insight

Low gain (g=3) eliminates the lr_W cliff seen at g=7, but compounds with network size to create severe difficulty (g=3 / 200 neurons/10k: 0% convergence with universal degeneracy). Unlike sparse connectivity, **30k frames fully rescues low gain** (0%-->100% convergence), with activity rank doubling from 31 to 55.
:::

### 8. Partial Connectivity --- Structural Ceiling (Blocks 22--23)

| neurons | frames | fill | activity rank | rho | Best connectivity | Key |
|:---------:|:--------:|:----:|:--------:|:---:|:---------:|:----|
| 100 | 10k | 50% | 21 | 0.746 | 0.466 | Subcritical; degenerate |
| 100 | 10k | 80% | 36 | 0.985 | 0.802 | Near-critical; no degeneracy |
| 100 | 10k | 100% | 35 | 1.065 | 0.999 | Supercritical; easy |
| 100 | **30k** | 50% | **13** | 0.746 | 0.436 | **Activity rank DROPS; not rescued** |
| 100 | **30k** | 80% | **49** | 0.985 | **0.802** | **Activity rank rises; connectivity stuck** |

::: {.callout-warning}
## Structural Limit

Connectivity ceiling $\approx$ filling_factor: fill=50%-->connectivity~0.49, fill=80%-->connectivity~0.80, fill=100%-->connectivity~1.00. At fill=80%, complete parameter insensitivity (lr_W, lr, L1, n_epochs, batch_size all irrelevant) at both 10k and 30k. The missing 20% of connections cannot be inferred from dynamics data regardless of volume.

![Connectivity ceiling vs filling factor: slope=1.027, R²=0.9995 --- an almost exact linear law](assets/connectivity_vs_filling_factor.png){.lightbox}
:::

### 9. Fixed-Point Collapse --- g=1 Unsolvable (Blocks 25--26)

| frames | activity rank | Best connectivity | Degeneracy | Key |
|:--------:|:--------:|:---------:|:----------:|:----|
| 10k | 5 | 0.007 | 12/12 | Flat-line dynamics; universal degeneracy |
| **30k** | **1** | **0.018** | **12/12** | **Activity rank DROPS (5→1); 30k makes it WORSE** |

::: {.callout-warning}
## Structural Limit

g=1 produces fixed-point collapse: tanh saturates weak-gain dynamics into stable fixed points (flat lines). Activity rank=5 at 10k (catastrophically low) drops to 1 at 30k --- the only regime where frame count is negatively correlated with activity rank. Below the critical activity rank threshold (~10), no amount of training or data helps. More severe than sparse 50%.
:::

### 10. g=2 --- Inverse lr_W Regime (Blocks 27--28)

| frames | activity rank | Best connectivity | Convergence | Key |
|:--------:|:--------:|:---------:|:-----------:|:----|
| 10k | 17 | 0.519 | 0% (0/12) | Inverse lr_W=5E-4 (100x lower than g=7) |
| **30k** | **16** | **0.997** | **42%** (5/12) | **Partially rescued; inverse lr_W persists; activity rank flat** |

::: {.callout-tip}
## Key Insight

g=2 sits above the critical activity rank threshold (17 > 10) and is partially rescued by 30k frames (0%→42% convergence at 100 neurons). However, it requires **inverse lr_W** (~5E-4, 100x lower than g=7's 4E-3) --- a structural property that persists at 30k. Epoch scaling is not diminishing at g=2 (5ep→0.356, 8ep→0.397, 12ep→0.519 at 10k). At 30k/8ep, the Pareto-optimal config reaches connectivity=0.997.

**Block 29 update (g=2 / 200 neurons/30k)**: Scaling to 200 neurons dramatically changes the picture. Activity rank jumps from 16--17 (100 neurons) to 35--38 (200 neurons), matching g=7 / 100 neurons. Convergence rate rises to 92% (11/12), with best connectivity=0.979 at lr_W=3E-4/12ep. The inverse lr_W pattern persists but the ceiling is higher (1E-3 safe at 200 neurons vs catastrophic at 100 neurons). Epoch scaling remains strong but diminishes at 12ep (+0.3%). This confirms that neuron count compensates for gain reduction: g=2 / 200 neurons/30k is comparable in difficulty to g=7 / 200 neurons/10k.
:::

### 11. fill=90% --- Transitional Regime (Block 24)

| frames | activity rank | rho | Best connectivity | Convergence | Key |
|:--------:|:--------:|:---:|:---------:|:-----------:|:----|
| 10k | 35--36 | 0.995 | 0.907 | 83% (10/12) | connectivity $\approx$ fill%; above R^2^=0.9 convergence boundary |

The fill=90% regime extends the connectivity ceiling $\approx$ filling_factor relationship to a fourth data point: fill=50%→0.49, fill=80%→0.80, fill=90%→0.91, fill=100%→1.00. Unlike fill=80%, the 90% regime crosses the R^2^>0.9 convergence threshold, making it a transitional regime.

### 12. Sparse Connectivity --- Confirmed Unsolvable (Blocks 7, 8, 17)

| frames | activity rank | Best connectivity | Degeneracy | Key |
|:--------:|:--------:|:---------:|:----------:|:----|
| 10k | 21 | 0.466 | 12/12 | Universal degeneracy; rho=0.746 |
| 10k (noise) | 91 | 0.490 | 0/12 | Noise inflates activity rank; no rescue |
| **30k** | **13** | **0.436** | **12/12** | **Activity rank DROPS (21-->13); 30k useless** |

::: {.callout-warning}
## Structural Limit

Sparse 50% at 30k is the ONLY regime where activity rank **decreases** with more data (21-->13). Subcritical spectral radius (rho=0.746) determines activity rank, not data volume. Two-phase training provides marginal +15% but is insufficient. This regime requires architectural intervention.
:::

### 13. 1000 Neurons --- Scale Frontier (Blocks 18, 30)

**Block 18 (30k frames)**: Insufficient --- max connectivity=0.745 (0% convergence). lr=1E-4 Pareto-optimal; lr=2E-4 causes overtraining.

| frames | n_epochs | lr_W | connectivity | test_R^2^ | Key |
|:--------:|:--------:|:----:|:----:|:---------:|:----|
| 30k | 3 | 5E-3 | 0.666 | 0.795 | Baseline |
| 30k | 8 | 5E-3 | 0.745 | 0.829 | Best at 30k |
| **100k** | **3** | **3E-3** | **1.000** | **0.882** | **BREAKTHROUGH** |
| **100k** | **3** | **5E-3** | **1.000** | **0.836** | **Also converged** |

::: {.callout-tip}
## Block 30 Breakthrough

1000 neurons at 100k frames achieves **100% convergence** (12/12), connectivity=1.000. This confirms the frame-count scaling law: like 300 neurons (needs 30k) and 600 neurons (needs 30k), 1000 neurons needs ~100k frames. lr_W=3E-3 is optimal; batch=16 is safe; 3 epochs suffice. Activity rank ~350 at 100k frames. The 1000 neurons regime is now a **solved regime** --- joining 300 neurons/30k, 600 neurons/30k, and g=3 / 200 neurons/30k.
:::

### 14. Degeneracy --- When Dynamics Quality Misleads

| Block | Regime | frames | Degenerate iters | Max gap | Mechanism |
|:-----:|--------|:--------:|:----------------:|:-------:|-----------|
| 1 | Chaotic 100 neurons | 10k | 0/12 | 0.15 | Healthy |
| 2 | Low-rank 100 neurons | 10k | 1/12 | 0.45 | Stochastic at lr_W=5E-3 |
| 3 | Dale law 100 neurons | 10k | 4/12 | 0.53 | lr_W above Dale cliff |
| 7 | **Sparse 50%** | 10k | **12/12** | **0.82** | **Universal --- subcritical rho** |
| 9 | 300 neurons 1ep | 10k | 2/12 | 0.38 | Training-limited |
| 17 | **Sparse 50%** | **30k** | **12/12** | **~0.80** | **Still universal at 30k** |
| 19 | **g=3 100 neurons** | 10k | **4/12** | **0.75** | **Low gain at 1ep** |
| 20 | **g=3 200 neurons** | 10k | **12/12** | **~0.70** | **Gain x n = universal degeneracy** |
| 21 | **g=3 200 neurons (30k)** | **30k** | **0/12** | **0.01** | **30k eliminates degeneracy** |
| 22 | fill=80% | 10k | 0/12 | 0.20 | **Not degenerate** (connectivity stuck) |
| 24 | fill=90% | 10k | 0/12 | 0.09 | **Not degenerate** (connectivity stuck at 0.907) |
| 25 | **g=1 100 neurons** | 10k | **12/12** | **~0.99** | **Fixed-point collapse; flat-line dynamics** |
| 26 | **g=1 100 neurons (30k)** | **30k** | **12/12** | **~0.99** | **30k amplifies degeneracy; activity rank→1** |
| 27 | **g=2 100 neurons** | 10k | **12/12** | **0.48--0.90** | **Low gain; inverse lr_W needed** |
| 28 | **g=2 100 neurons (30k)** | **30k** | **5/12** | **0.003--0.80** | **Partially rescued; 42% convergence** |
| 29 | **g=2 200 neurons (30k)** | **30k** | **0/12** | **0.12** | **Healthy --- n compensates; activity rank=35--38** |
| 30 | **1000 neurons (100k)** | **100k** | **0/12** | **0.12** | **Healthy --- 100k eliminates degeneracy** |
| 15--16 | 300/600 neurons (30k) | 30k | 0/20 | -0.01 | Abundant data eliminates |

### 15. Frame Count Is the Dominant Lever (Blocks 15--16, 21)

| neurons | gain | frames | activity rank | Convergence | Best connectivity | Key |
|:---------:|:----:|:--------:|:--------:|:-----------:|:---------:|:----|
| 300 | 7 | 10k | 47 | 25% (3--4ep) | 0.924 | Training-sensitive |
| 300 | 7 | **30k** | **80** | **100%** (even 1ep) | **1.000** | All params non-critical |
| 600 | 7 | 10k | 50 | 0% (10ep) | 0.626 | Data-limited |
| 600 | 7 | **30k** | **87** | **100%** (2--4ep) | **0.992** | Solved |
| 200 | 3 | 10k | 31 | 0% | 0.489 | Gain x n |
| 200 | 3 | **30k** | **55** | **100%** (12/12) | **0.996** | **Gain rescued** |
| 100 | 2 | 10k | 17 | 0% (10k) | 0.519 | Inverse lr_W |
| 100 | 2 | **30k** | **16** | **42%** (5/12) | **0.997** | **Partially rescued** |
| 200 | 2 | **30k** | **35--38** | **92%** (11/12) | **0.979** | **n compensates gain** |
| **1000** | **7** | **30k** | **144** | **0%** (0/12) | **0.745** | **30k insufficient** |
| **1000** | **7** | **100k** | **~350** | **100%** (12/12) | **1.000** | **100k SOLVES 1000 neurons** |

::: {.callout-important}
## Key Finding

Frame count transforms the training landscape for dense supercritical networks:

1. **Activity rank scales with frame count**: 300 neurons: 47-->80 (30k); 600 neurons: 50-->87 (30k); 1000 neurons: 144-->~350 (100k); g=3 / 200 neurons: 31-->55 (30k)
2. **Convergence universalizes**: 300 neurons (25%-->100%), 600 neurons (0%-->100%), g=3 / 200 neurons (0%-->100%), **1000 neurons (0%-->100% at 100k)**
3. **Training params become non-critical**: lr_W safe range widens; batch_size=16 safe; 1--3 epochs suffice
4. **Optimal lr_W shifts lower**: 3--5E-3 at 30k vs 1E-2 at 10k; 3E-3 at 100k for 1000 neurons

**Exceptions**: sparse 50% (rho<1), fill=80%/90%, g=1 (fixed-point collapse). g=2 / 100 neurons is partially rescued (42% at 30k); g=2 / 200 neurons is mostly rescued (92% at 30k, activity rank=35--38)
:::

## Summary Statistics

### Overall Performance

::: {.feature-grid}

::: {.feature-card}
### 360 Iterations
Across 30 simulation regimes
:::

::: {.feature-card}
### ~99 Principles
Novel findings with 45--100% confidence
:::

::: {.feature-card}
### 10 Solved Regimes
100% convergence (chaotic, noise, 200 neurons, 200 neurons+4types, 300 neurons/30k, 600 neurons/30k, g=3 / 200 neurons/30k, **1000 neurons/100k**)
:::

::: {.feature-card}
### 3 Structural Limits
Sparse (rho<1), fill<100% (connectivity $\approx$ fill%), g=1 (fixed-point collapse)
:::

:::

## Next Steps

- [Epistemic Analysis](epistemic-analysis.qmd) --- How these findings were discovered
- [Exploration](exploration-gallery.qmd) --- Visual record per block
- [Case Study: Low-Rank](case-low-rank.qmd) --- Dedicated low-rank exploration
- [Case Study: Sparse](case-sparse.qmd) --- Dedicated sparse exploration with code modifications