epistemic-analysis.qmd

---
title: "Epistemic Analysis"
subtitle: "Taxonomy of LLM Scientific Reasoning"
---

How does an LLM reason when embedded in a closed-loop scientific exploration? This page dissects the reasoning trace produced by the Experiment--LLM--Memory loop. Every iteration, the LLM writes a structured analysis log; each reasoning step in that log is classified into one of ten formal epistemic modes and the transitions between them are recorded as causal edges. The figures below visualize these annotations at three levels of granularity: individual events over time, compositional trends, and aggregate mode-to-mode flow.

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

**360 iterations completed** across 30 regimes. The reasoning trace contains ~440 annotated events connected by ~170 causal edges, ~99 established principles, 74% deduction accuracy, and 70% cross-regime analogy transfer success. 100% of falsified hypotheses led to principle refinement. Blocks 24--30 extended the frontier: fill=90% confirmed connectivity ceiling ≈ filling_factor at a 4th data point (block 24), g=1 discovered as a new unsolvable axis --- fixed-point collapse with activity rank dropping to 1 at 30k (blocks 25--26), g=2 identified as a transitional regime requiring inverse lr_W and partially rescued by 30k frames (blocks 27--28), g=2 / 200 neurons / 30k confirmed that network size compensates for gain reduction (block 29, 92% convergence, activity rank=35--38), and **1000 neurons at 100k frames achieved 100% convergence (12/12, connectivity=1.000)** confirming the frame count scaling law at the largest scale tested (block 30).
:::

---

## Outcomes

1. **~440 reasoning events** across 360 iterations and 30 blocks, connected by ~170 causal edges
2. **Predictive power**: 74% deduction accuracy (well above chance)
3. **Knowledge transfer**: 70% analogy success rate across cross-regime transfers
4. **Principle identification**: ~99 validated findings with confidence 45--100%
5. **Self-correction**: 100% of falsifications led to principle refinement
6. **Structural limit recognition**: Block 8 (sparse data limit), block 17 (sparse immune to frame count), blocks 22--24 (connectivity ceiling at filling_factor), and blocks 25--26 (g=1 fixed-point collapse --- activity rank drops at 30k) demonstrate the system can distinguish solvable from structural limits
7. **Cross-block transfer**: Key insights from block 7 (n_epochs) transferred to block 9 (300 neurons); block 15 (number of frames) transferred to blocks 16--18, 21, and 30; block 22 (fill=80%) extended through block 24 (fill=90%); block 19 (g=3) transferred to blocks 25--29 (g=1, g=2, g=2 / 200 neurons); block 18 (1000 neurons / 30k insufficient) motivated block 30 (100k)
8. **Paradigm shifts**: Blocks 15--16 (frame count dominance), blocks 19--21 (gain as solvable difficulty axis), blocks 22--24 (connectivity ceiling at filling_factor), blocks 25--26 (g=1 fixed-point collapse), block 29 (n compensates gain: g=2 / 200 neurons activity rank matches g=7 / 100 neurons), and block 30 (1000 neurons / 100k breakthrough: 100% convergence, connectivity=1.000) represent six paradigm-level findings. Each overturned or extended previously established principles

---

## Epistemic Timeline

<iframe src="assets/epistemic_timeline_interactive.html" width="100%" height="650" style="border:none;"></iframe>

The epistemic timeline is constructed by classifying each LLM reasoning event into one of ten formal modes (induction, deduction, abduction, falsification, boundary probing, analogy, meta-reasoning, regime recognition, causal reasoning, and constraint identification). For every iteration, the LLM's written analysis log is parsed and each reasoning step is tagged with its mode; the result is a scatter plot where each dot marks a reasoning event at a given iteration, colored by mode. The figure reveals how the reasoning composition evolves over the course of the exploration. Boundary probing dominates the early iterations of each block as the system maps a new regime, then gives way to deduction and falsification once enough observations have accumulated to form testable predictions. Cross-block analogy events cluster at block transitions, where the LLM attempts to transfer principles from previously explored regimes. The sparse-connectivity blocks (7--8, 17) show a distinctive shift toward constraint recognition and meta-reasoning as the system identifies structural limits that no amount of hyperparameter tuning can overcome. Blocks 19--21 introduce a new reasoning pattern: regime recognition identifies gain as an independent difficulty axis, followed by rapid falsification when 30k frames rescues the regime. Blocks 22--24 show a constraint-heavy pattern as the system maps the connectivity ceiling at filling_factor relationship across three data points. Blocks 25--28 show a distinctive regime recognition → constraint identification pattern as the system discovers g=1 fixed-point collapse (blocks 25--26) and the g=2 inverse lr_W regime (blocks 27--28).

---

## Reasoning Activity Stream

<iframe src="assets/epistemic_streamgraph_interactive.html" width="100%" height="500" style="border:none;"></iframe>

The streamgraph aggregates reasoning mode counts into a smoothed, stacked area chart where each stream's width encodes the relative frequency of that mode over a sliding window of iterations. It is generated from the same per-iteration mode annotations as the timeline, but traded temporal resolution for a view of compositional trends. The visualization shows six distinct phases. In the first phase (blocks 1--3), boundary probing and induction dominate as the system maps base-case parameters. In the second phase (blocks 4--6), deduction and analogy grow as the LLM begins predicting outcomes from accumulated principles and transferring knowledge across regimes. In the third phase (blocks 7--10), falsification and constraint modes widen as the system encounters structurally hard regimes (sparse, large-scale) where many hypotheses fail and the LLM must reason about fundamental limits rather than parameter optimization. In the fourth phase (blocks 11--16), analogy becomes the dominant entry mode as cross-block transfer drives scaling explorations; falsification surges in blocks 15--16 as the frame count paradigm shift overturns multiple established principles, while induction resurges as new patterns emerge at 30k frames. In the fifth phase (blocks 17--23), the system enters a consolidation and boundary-mapping mode: constraint recognition peaks as structural limits are probed (sparse at 30k, fill=80%), regime recognition resurges as gain is identified as a new difficulty axis, and analogy drives rapid resolution when 30k frames rescues g=3 / 200 neurons (block 21) while failing to rescue fill=80% (block 23). In the sixth phase (blocks 24--29), the system maps the gain--activity rank critical transition: regime recognition dominates as g=1 and g=2 are explored, constraint identification peaks as fixed-point collapse is discovered (blocks 25--26), and causal reasoning intensifies as the system builds a quantitative model of how gain modulates activity rank non-linearly. In the seventh phase (block 30), deduction and analogy dominate as the system applies the frame count scaling law to 1000 neurons at 100k frames, confirming the predicted breakthrough (100% convergence, connectivity=1.000) --- a paradigm-validating success that closes the frame count frontier.

---

## Epistemic Flow

<iframe src="assets/epistemic_sankey_interactive.html" width="100%" height="850" style="border:none;"></iframe>

The Sankey diagram traces how reasoning modes connect to each other across the full exploration (440+ events, 170+ causal edges, 30 blocks). Each link represents a transition where one epistemic operation led to another within the same causal chain --- for example, a boundary probe that triggered a falsification, or an induction that enabled a deduction in a later iteration. Link width is proportional to the number of such transitions observed. The diagram is constructed by extracting causal edges from the LLM's analysis log: when the LLM explicitly references a prior finding to justify a new hypothesis or action, a directed edge is created between the two reasoning modes. The resulting flow reveals that induction and boundary probing are the dominant entry points, feeding into both deduction (when observations yield testable predictions) and falsification (when probes or predictions fail). Analogy acts as a cross-regime bridge, receiving from induction in one block and feeding both induction and falsification in the next --- this pathway intensifies in blocks 11--21 as cross-block transfer becomes the primary mechanism for scaling explorations. Meta-reasoning and constraint recognition appear as terminal modes, activated when the system must acknowledge structural limits (blocks 7--8, 17, 22--23) or paradigm-level shifts (blocks 15--16, 19--21). Blocks 17--23 strengthen the constraint→induction pathway as structural limits identified in one regime (sparse, fill=80%) generate new hypotheses tested in the next. Block 30 shows a dominant deduction→induction flow as the frame count scaling law is confirmed at 1000 neurons / 100k.

---

## Reasoning Mode Taxonomy

| Mode | Definition | Example from experiment |
|------|------------|---------:|
| **Induction** | Generalize from specific observations | "lr_W 2E-3 to 8E-3 all converge in chaotic regime --- easy mode" |
| **Deduction** | Predict from established principles | "If L1=1E-6 helps low-rank, then it should help Dale (same activity rank=12)" |
| **Abduction** | Infer best explanation for observation | "Dynamics dropped despite same lr_W --- must be activity rank effect" |
| **Falsification** | Reject hypothesis via counterexample | "factorization=True made connectivity_R^2^ worse --- hypothesis rejected" |
| **Boundary** | Probe limits of working configurations | "lr_W=5E-3 in Dale regime --- cliff found" |
| **Analogy** | Transfer knowledge between regimes | "Block 1 lr_W=4E-3 applied to block 3 Dale regime" |
| **Meta-reasoning** | Reason about reasoning strategy | "Switch from lr_W sweep to L1 investigation" |
| **Regime** | Identify distinct operating conditions | "sparse (activity rank=21, subcritical) requires fundamentally different approach" |
| **Causal** | Identify cause-effect relationships | "High lr_W → fast W learning BUT starves embedding" |
| **Constraint** | Identify structural limits | "connectivity=0.489 is a data limit, not a training limit" |

### Mode Counts

| Mode | Count | Validation | First Appearance |
|------|:-----:|:----------:|:----------------:|
| Boundary Probing | 65 | N/A | Iter 4 |
| Deduction | 57 | **74%** (42/57) | Iter 5 |
| Induction | 52 | N/A | Iter 5 |
| Falsification | 52 | 100% refinement | Iter 9 |
| Analogy/Transfer | 40 | 70% (28/40) | Iter 13 |
| Causal Chain | 18 | N/A | Iter 21 |
| Regime Recognition | 16 | N/A | Iter 13 |
| Constraint | 10 | N/A | Iter 85 |
| Meta-reasoning | 11 | N/A | Iter 9 |
| Abduction | 6 | N/A | Iter 13 |
| Uncertainty | 3 | N/A | Iter 8 |

---

## Key Epistemic Events by Block

### Block 1 --- Chaotic Baseline

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 1--4 | Boundary | lr_W sweep [1E-3, 2E-3, 5E-3, 1E-2] | Maps convergence landscape |
| 5 | Induction | lr_W=4E-3 identified as sweet spot | test_R^2^=0.996, connectivity=0.9999 |
| 9 | Falsification | lr=2E-4 tested at optimal lr_W | Degrades dynamics (0.996→0.981) |
| 11 | Induction | L1=1E-6 at low lr_W | Best dynamics (0.998) but connectivity partial |
| 12 | Boundary | batch_size=16 tested | Converges but slight quality loss |

**Principle extracted**: lr_W=4E-3 sweet spot; lr=1E-4 is optimal; batch_size=16 trades quality for speed.

### Block 2 --- Low-rank Breakthrough

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 13 | Analogy | Transfer lr_W=4E-3 from block 1 | Connectivity OK (0.999) but dynamics poor (0.902) |
| 14 | Falsification | factorization=True tested | HURTS (connectivity 0.899 vs 0.999 without) |
| 17 | Boundary | lr_W=5E-3 without factorization | Catastrophic failure (0.385) |
| 18 | Deduction | L1=1E-6 should help dynamics | Confirmed: 0.902→0.925 |
| 19 | Induction | lr_W=3E-3 tested | Better dynamics (0.943) than 4E-3 |
| 21 | **Recombination** | lr_W=3E-3 + L1=1E-6 | **BREAKTHROUGH**: test_R^2^=0.996 |
| 24 | Falsification | batch_size=16 expected to degrade | Surprise: 0.997 (challenges principle) |

**Principle extracted**: L1=1E-6 is critical enabler for low-rank dynamics; lr_W optimal shifts downward at low activity rank.

### Block 3 --- Dale's Law Exploration

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 25 | Analogy | Transfer from block 1 | Works (connectivity 0.972) but activity rank drops 35→12 |
| 28 | Boundary | lr_W=6E-3 probed | Catastrophic failure (connectivity 0.555) |
| 29--30 | Boundary | lr_W=5E-3 (two independent runs) | Both fail (0.458, 0.455) --- reproducible cliff |
| 33 | Deduction | lr_W=4.5E-3 predicted safe | Best: connectivity 0.986, test_R^2^ 0.999 |
| 36 | Falsification | lr=2E-4 at best config | Does NOT degrade --- challenges principle 1 |

**Principle extracted**: Dale_law creates sharp lr_W cliff at 5E-3; safe range [3.5E-3, 4.5E-3].

### Block 4 --- Dual-Objective Conflict

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 37 | Analogy | Transfer from block 1 | W converges but cluster_acc=0.67 |
| 39 | Deduction | lr_emb=1E-3 should fix embedding | FULL convergence: connectivity 0.9996, cluster 0.990 |
| 41 | Deduction | lr_W=5E-3 + lr_emb=1E-3 | FULL convergence confirmed (cluster 1.000) |
| 44 | Deduction | L1=1E-6 critical for embedding | Confirmed: cluster drops 0.990→0.440 at L1=1E-5 |
| 48 | Falsification | batch_size=16 at best config | Degrades cluster_acc (1.000→0.500) |

**Principle extracted**: lr_emb=1E-3 required for heterogeneous; batch_size=16 hurts dual-objective.

### Block 5 --- Noise Effects

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 49--52 | Regime | noise=[0.1, 0.5, 1.0] sweep | All converge: activity rank 42→90 |
| 57 | Boundary | lr_W=1E-2 at noise=0.5 | Dynamics degraded (0.707) --- upper boundary |
| 58 | Induction | lr_W=2E-3 at noise=1.0 | Best dynamics (0.998): inverse lr_W-noise |
| 56 | Falsification | lr=2E-4 at activity rank=84 | Safe --- modifies principle 1 |

**Principle extracted**: Noise inflates activity rank; inverse lr_W-noise relation; lr=2E-4 safe only at activity rank>80.

### Block 6 --- Scale Sensitivity (200 neurons)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 61 | Analogy | Transfer lr_W=4E-3 to 200 neurons | Converges (connectivity 0.905) but activity rank only 43 |
| 62 | Boundary | lr_W=2E-3 at 200 neurons | Fails (connectivity 0.575) --- boundary shifted up |
| 67 | Deduction | lr=2E-4 safe at 200 neurons | Best connectivity (0.956) |
| 72 | Falsification | lr=3E-4 tested at 200 neurons | BEST dynamics (0.952) --- challenges principle 1 |

**Principle extracted**: Scale amplifies lr_W/dynamics trade-off; convergence boundary shifts upward; lr tolerance widens with n.

### Block 7 --- Sparse Connectivity (ff=0.5)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 73 | Regime | filling_factor=0.5 new regime | activity rank=21, spectral_radius=0.746 (subcritical) |
| 79 | Induction | n_epochs=2 tested | Beats all 1-epoch configs (0.423 vs 0.310) |
| 82 | Induction | lr_W=1E-2 + 2ep | Best connectivity in block (0.466) |
| 82 | Causal | training capacity is bottleneck | More training > higher lr_W |

**Principle extracted**: Sparse connectivity is hardest regime (0% convergence); n_epochs is dominant lever; subcritical spectral radius limits dynamics.

### Block 8 --- Sparse + Noise (Structural Limit)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 85 | Constraint | noise inflates activity rank 21→91 | But connectivity plateaus at 0.489 --- structural limit |
| 86--87 | Induction | lr_W insensitivity | 2E-3 to 8E-3 all give connectivity=0.489 |
| 88 | Falsification | n_epochs=1 same as n_epochs=2 | Noise removes epoch dependency |
| 95 | Falsification | recurrent training | **Catastrophic**: connectivity collapsed 0.489→0.054 |
| 96 | Meta-reasoning | recognize structural limit | 0.489 is a data limit, not training limit |

**Principle extracted**: sparse+noise creates fundamental data limit at 0.489; complete training parameter insensitivity; recurrent training catastrophic in subcritical regime.

### Block 9 --- Large-Scale (300 neurons)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 97 | Analogy | Transfer from 200 neurons | connectivity 0.699 --- harder than 200 neurons |
| 103 | Induction | lr_W=1E-2 optimal | Best connectivity at 1ep (0.805) |
| 106 | Induction | n_epochs=2 | **BREAKTHROUGH**: connectivity 0.890 (+10.6%) |
| 108 | Constraint | lr/lr_W interaction | lr=3E-4 degrades at high lr_W |

**Principle extracted**: 300 neurons requires n_epochs≥2; optimal lr_W=1E-2; dynamics cliff at ~1.2E-2.

### Block 10 --- 300 Neurons Refinement (2 epochs baseline)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 109 | Deduction | reproduce baseline | Confirmed: connectivity 0.893 |
| 110 | Falsification | 3 epochs tested | Does NOT help connectivity (0.886 < 0.893) |
| 112 | Induction | L1=1E-6 at 300 neurons | Boosts dynamics +6.8% (0.987 vs 0.924) |

**Emerging**: connectivity ceiling ~0.89 at 10k frames; L1=1E-6 harmful only at n≤200 (revised).

### Block 11 --- 200 Neurons Solved (2--3 epochs)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 117--120 | Analogy | Transfer from block 6 + epochs insight | 100% convergence (4/4) |
| 126 | Induction | lr_W=8E-3 optimal at 2ep | connectivity 0.993, test_R^2^ 0.963 |
| 128 | Falsification | L1=1E-6 tested | Confirmed harmful at 200 neurons |

**Principle extracted**: 200 neurons recipe: lr_W=8E-3, lr=2E-4, L1=1E-5, n_epochs=2--3; 100% convergence (12/12).

### Block 12 --- 600 Neurons (Training-Capacity Frontier)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 129--132 | Analogy | Transfer from 300 neurons recipe | connectivity 0.540 at 4ep --- harder than expected |
| 137 | Induction | epochs not diminishing | 4ep→0.540, 8ep→0.580, 10ep→0.626 |
| 135 | Constraint | lr=1E-4 catastrophic | connectivity=0.000 at 600 neurons --- lr floor discovered |

**Principle extracted**: 600 neurons is severely training-capacity-limited at 10k frames; lr=1E-4 catastrophic; epochs NOT diminishing; needs 15--20ep or more frames.

### Block 13 --- Heterogeneous at Scale (200 neurons, 4 types)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 145--148 | Analogy | Transfer from blocks 4+11 recipes | 100% connectivity convergence (4/4) |
| 149 | **Recombination** | lr_W=8E-3 + L1=1E-5 + 3ep | **FULL DUAL CONVERGENCE** (connectivity=0.988, cluster=1.000) |
| 150 | Falsification | L1=1E-6 at 200 neurons / 4types | Worse overall (connectivity ok but dynamics/cluster degraded) |
| 152 | Falsification | batch_size=16 at 200 neurons / 4types | Cluster drops 0.610→0.250 --- heterogeneous batch guard confirmed |

**Principle extracted**: n-dependent L1 effect overrides heterogeneous L1=1E-6 rule at 200 neurons; recipe: lr_W=8E-3, L1=1E-5, lr_emb=1E-3, 3ep.

### Block 14 --- Recurrent Training (200 neurons, supercritical)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 157--164 | — | 8/8 systemic failures (infrastructure) | No data; retry needed |
| 165 | Deduction | Baseline 200 neurons recipe without recurrent | Confirmed: connectivity=0.990, matches block 11 |
| 166 | Deduction | Recurrent=True, time_step=4 | Connectivity +0.3% (0.993) but **dynamics -12.3%** |
| 167 | Boundary | Warmup (start_ep=1) + noise_rec=0.01 | Dynamics recover (+9.5%) but connectivity drops (-8.2%) |
| 168 | Falsification | noise_rec=0.05 tested | Connectivity partial (0.772); **principle generalized: rollout noise harmful** |

**Principle extracted**: Recurrent training at supercritical rho is NOT catastrophic (unlike subcritical), but creates connectivity-dynamics trade-off. noise_recurrent_level ceiling is 0.01.

### Block 15 --- Frame Count Scaling at 300 Neurons (30k frames)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 169 | Analogy | Block 10 recipe at 30k frames | **connectivity=0.999** (vs 0.924 at 10k); +8.1%; **MASSIVE** |
| 170 | Falsification | 2ep tested (block 10 required 3ep) | Converges at 0.999! **3ep requirement overturned** |
| 171 | Analogy | 200 neurons recipe (lr_W=8E-3, L1=1E-5) transferred | Also 0.999; both recipes work at 30k |
| 172 | **Falsification** | **Violate both: 2ep + L1=1E-5** | **PRINCIPLE OVERTURNED** --- neither required at 30k |
| 175 | Induction | lr_W=5E-3 + 3ep | **BEST**: connectivity=1.000, test_R^2^=0.986, kino=0.985 |
| 176 | Falsification | batch=16 tested at 300 neurons / 30k | Safe! connectivity=1.000; **batch guard overturned at 30k** |
| 177 | Induction | lr_W=3E-3 + 3ep | **Pareto-optimal**: connectivity=1.000, test_R^2^=0.990 |
| 178 | Boundary | lr_W=2E-2 probed | Still converges (0.999); safe range extends to 2E-2 |

**Principle extracted**: number of frames is the DOMINANT lever; 30k frames makes ALL training parameters non-critical at 300 neurons; activity rank doubles (47→80); previous 300 neurons principles (3ep, L1=1E-6, batch≤8) are frame-count-specific.

### Block 16 --- 600 Neurons at 30k Frames

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 181 | Analogy | Block 12 recipe at 30k frames | **CONVERGED at 0.973** (vs 0.626 at 10k/10ep) |
| 183 | **Recombination** | lr_W=5E-3 + batch=16 (block 15 pattern) | **Pareto-dominant**: connectivity=0.976, test_R^2^=0.943 |
| 184 | Deduction | 2ep at 30k should suffice (principle #37) | Confirmed: connectivity=0.967 with only 2ep |
| 185 | Falsification | lr_W=3E-3 (300 neurons / 30k optimal) at 600 neurons | Worse (0.933); 300 neurons optimal does NOT transfer |
| 186 | Induction | lr_W=5E-3 + 4ep | **BEST connectivity=0.992**; 4ep substantially boosts W |
| 188 | Falsification | L1=1E-6 vs 1E-5 at 600 neurons / 30k | L1=1E-6 marginally better (+0.6%); sensitivity vanishes |

**Principle extracted**: frame count dominance scales to 600 neurons; 100% convergence (8/8); lr_W=5E-3 optimal (shifted from 1E-2 at 10k); activity rank=87; L1 sensitivity vanishes at abundant data.

### Block 17 --- Sparse 50% at 30k Frames (Confirmed Unsolvable)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 193 | Analogy | Transfer 300 neurons / 30k recipe to sparse | connectivity=0.320 --- **WORSE** than 10k (0.466) |
| 195 | **Falsification** | frame count should rescue sparse | **OVERTURNED**: activity rank DROPS 21→13; 0% convergence |
| 197 | Constraint | subcritical rho immune to frame count | Structural: rho=0.746 constrains activity rank regardless of data |
| 201 | Induction | two-phase training only positive signal | +15% marginal but insufficient (0.436 best) |
| 204 | Meta-reasoning | close sparse investigation | subcritical spectral radius is the ONLY unsolvable axis |

**Principle extracted**: frame count does NOT rescue subcritical spectral radius; sparse 50% activity rank is LOWER at 30k than 10k (13 vs 21); two-phase training gives marginal +15%.

### Block 18 --- 1000 Neurons at 30k Frames (Scale Frontier)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 205 | Boundary | 1000 neurons at 30k frames | activity rank=144; connectivity=0.726; 0% convergence |
| 209 | Induction | lr=1E-4 Pareto-better at 1000 neurons | connectivity=0.745 + test_R^2^=0.829 vs lr=2E-4's 0.734/0.588 |
| 212 | Constraint | 30k insufficient for 1000 neurons | Needs ~100k frames (user prior confirmed) |
| 214 | Induction | epoch scaling is lr-dependent | lr=2E-4 REVERSAL at 10ep; lr=1E-4 steady improvement |
| 216 | Causal | higher lr amplifies overtraining at large n | lr=2E-4 overshoots at 10ep while lr=1E-4 is safe |

**Principle extracted**: 1000 neurons / 30k is insufficient (needs ~100k); lr=1E-4 definitively Pareto-better; activity rank scales superlinearly with n at 30k; epoch scaling is lr-dependent.

### Block 19 --- Low Gain g=3 (100 neurons, New Difficulty Axis)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 217 | **Regime** | gain=3 identified as new difficulty axis | activity rank 35→26; rho=1.065 (supercritical); 4/4 degenerate at 1ep |
| 219 | Induction | n_epochs dominant lever | 1ep→0.636, 2ep→0.906, 3ep→0.955 |
| 221 | Boundary | no lr_W cliff up to 1.2E-2 | Unlike g=7 cliff at 8E-3; lower gain shifts cliff higher |
| 224 | Falsification | batch=16 at g=3 | Catastrophic: -42%; low gain amplifies batch sensitivity |
| 228 | Induction | g=3 recipe established | lr_W=8E-3, 3ep → connectivity=0.955 |

**Principle extracted**: gain=3 reduces activity rank 35→26; training-limited like large n; no lr_W cliff; batch=16 and L1=1E-6 catastrophic; n_epochs is dominant lever.

### Block 20 --- Low Gain g=3 at Scale (200 neurons, 10k)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 229 | **Deduction** | gain × n should compound | Confirmed: 0% convergence, max connectivity=0.489 at 6ep |
| 233 | Constraint | epoch scaling diminishing at 4--6ep | Need alternative lever (frame count predicted) |
| 236 | Induction | lr_W and epochs substitutable | lr_W=2E-2/3ep matches lr_W=1.2E-2/4ep |
| 240 | Falsification | batch=16 tested at g=3 / 200 neurons | Catastrophic: -21%; batch guard confirmed at low gain |

**Principle extracted**: gain × n compounds super-additively; 0% convergence at 10k/6ep; universal degeneracy (12/12); batch=16 catastrophic; needs 30k frames.

### Block 21 --- Low Gain g=3 Rescued by 30k Frames (200 neurons)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 241 | **Analogy** | 30k frames should rescue g=3 / 200 neurons | **CONFIRMED**: connectivity=0.993 at first attempt |
| 245 | Induction | Pareto-optimal recipe found | lr_W=4E-3, 2ep → connectivity=0.996, test_R^2^=0.999 |
| 248 | Induction | activity rank increases +80% at 30k | 31→53--57; same pattern as other regimes |
| 251 | **Falsification** | batch=16 at g=3/30k | **Safe** (-0.4%); overturns block 20's batch catastrophe |
| 252 | Induction | ALL params non-critical at g=3/30k | No lr_W cliff to 3E-2; gain is NOT an independent unsolvable axis |

**Principle extracted**: 30k frames rescues g=3 / 200 neurons (0%→100%); gain is SOLVABLE by frame count; batch=16 safe at 30k; all params non-critical; frame count is the universal solver.

### Block 22 --- Partial Connectivity fill=80% (100 neurons, 10k)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 253 | **Regime** | fill=80% is intermediate regime | activity rank=36 (same as 100%); rho=0.985 (near-critical) |
| 255 | **Constraint** | connectivity plateau at ~0.802 | COMPLETE parameter insensitivity; 0/12 degenerate |
| 259 | Induction | connectivity ceiling scales linearly with fill | fill=50%→0.49, 80%→0.80, 100%→1.00 |
| 261 | Induction | sharp transition from 50% to 80% | rho 0.746→0.985; activity rank 21→36 |
| 264 | Deduction | frame count should rescue fill=80% | rho near-critical → predict yes (unlike subcritical 50%) |

**Principle extracted**: fill=80% creates connectivity plateau at ~0.802 with complete param insensitivity; connectivity ceiling approximates filling_factor; sharp rho transition between 50--80%.

### Block 23 --- fill=80% at 30k Frames (12 iterations)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 265 | **Falsification** | 30k should break fill=80% plateau | **OVERTURNED**: connectivity=0.802 identical to 10k |
| 266 | Constraint | connectivity is structurally locked at fill% | lr_W 2E-3 to 1.2E-2 all give connectivity=0.802 |
| 267 | Induction | dynamics and connectivity decoupled at fill=80% | kino_R^2^ 0.783--0.999 but connectivity always 0.802 |
| 268 | Constraint | connectivity ceiling at filling_factor confirmed at 30k | Second regime (after sparse) where frame count does NOT rescue |

**Principle extracted**: 30k frames does NOT break fill=80% connectivity plateau; connectivity ceiling at filling_factor holds at both 10k and 30k; this is the second structural limit alongside subcritical spectral radius.

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

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 289 | **Regime** | fill=90% is transitional between 80% and 100% | rho=0.995 (near-critical); activity rank=35--36 |
| 291 | Induction | connectivity plateau at 0.907 ≈ fill% | Extends linear relationship to 4th data point |
| 295 | Constraint | complete parameter insensitivity | lr_W, L1, epochs all irrelevant (same as fill=80%) |
| 298 | Deduction | above R^2^>0.9 convergence boundary | 83% convergence rate (vs 0% at fill=80%) |

**Principle extracted**: fill=90% confirms connectivity ceiling ≈ filling_factor at 4th point (50%→0.49, 80%→0.80, 90%→0.91, 100%→1.00); transitional regime above convergence threshold.

### Block 25 --- g=1: Fixed-Point Collapse (10k)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 301 | **Regime** | g=1 produces flat-line dynamics | activity rank=5 (catastrophically low); fixed points |
| 303 | Constraint | below activity rank threshold (~10) | 0% convergence; max connectivity=0.007; 12/12 degenerate |
| 307 | Boundary | lr_W range [1E-3, 3E-2] all equally useless | Complete parameter insensitivity |
| 310 | Induction | two-phase training only marginal signal | connectivity=0.007 vs 0.000--0.002 (negligible) |

**Principle extracted**: g=1 produces fixed-point collapse; activity rank=5 is below critical threshold ~10; hardest regime encountered; more severe than sparse 50%.

### Block 26 --- g=1 at 30k: Confirmed Unsolvable

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 313 | **Falsification** | 30k should rescue g=1 | **OVERTURNED**: activity rank DROPS 5→1; 0% convergence |
| 317 | **Constraint** | g=1 unsolvable by frame count | Unique: frame count NEGATIVELY correlated with activity rank |
| 319 | Causal | fixed points converge faster with more data | More frames → tighter fixed-point basin → lower dimensionality |
| 322 | Meta-reasoning | identify g=1 as third structural limit | Alongside sparse (rho<1) and fill<100% (connectivity=fill%) |

**Principle extracted**: g=1 is UNSOLVABLE by frame count; activity rank drops from 5 to 1 at 30k (unique negative correlation); frame-count-amplified degeneracy.

### Block 27 --- g=2: Inverse lr_W Regime (10k)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 325 | **Regime** | g=2 has activity rank=17 (above threshold) | Supercritical (rho=1.065) but very low gain |
| 327 | Induction | lr_W=5E-4 optimal (100x lower than g=7) | **Inverse lr_W** --- new phenomenon |
| 331 | Causal | gain--activity rank critical transition | g=1→5 (+240%→) g=2→17 (+53%→) g=3→26 (+35%→) g=7→35 |
| 335 | Deduction | epoch scaling not diminishing at low lr_W | 5ep→0.356, 8ep→0.397, 12ep→0.519 (steady) |

**Principle extracted**: g=2 requires inverse lr_W (~5E-4, 100x lower than g=7); epoch scaling NOT diminishing; steepest gain--activity rank slope between g=1 and g=2.

### Block 28 --- g=2 at 30k: Partially Rescued

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 337 | **Analogy** | 30k should rescue g=2 (like g=3) | **Partially confirmed**: 42% convergence (vs 0% at 10k) |
| 341 | Induction | activity rank FLAT at 16 (unchanged from 17) | Unique: 30k does NOT increase dimensionality at g=2 |
| 343 | Deduction | inverse lr_W should persist at 30k | **Confirmed**: lr_W≥2E-3 catastrophic; 5E-4--7E-4 optimal |
| 347 | Causal | activity rank--frame count correlation depends on gain | g=1: negative (5→1); g=2: flat (17→16); g≥3: positive |

**Principle extracted**: g=2 IS partially solvable at 30k (42% conv); activity rank does NOT increase with frame count at g=2; inverse lr_W is structural (persists at 30k); 30k helps via variance reduction, not more dynamic modes.

### Block 29 --- g=2 / 200 neurons / 30k: n Compensates Gain

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 349 | **Analogy** | g=2 / 200 neurons should have higher activity rank (like g=3 / 200 neurons) | **Confirmed**: activity rank=35--38 (vs 16 at 100 neurons); 2.2x increase |
| 351 | Induction | inverse lr_W persists but ceiling scales with n | lr_W=3E-4 optimal; 1E-3 safe but epoch-insensitive |
| 345 | Induction | epoch scaling diminishing at 12ep | 10ep→0.976, 12ep→0.979 (+0.3%) |
| 348 | **Causal** | gain × n interaction is compensatory for activity rank | g=2 / 200 neurons activity rank=35--38 matches g=7 / 100 neurons |

**Principle extracted**: neuron count compensates for gain reduction; g=2 / 200 neurons / 30k achieves 92% convergence (vs 42% at 100 neurons); activity rank scales with n even at low gain; epoch scaling diminishes at 12ep; lr_W=1E-3 causes negative epoch scaling (cluster degrades).

### Block 30 --- 1000 Neurons at 100k Frames: BREAKTHROUGH (iters 349--360)

| Iter | Mode | Event | Outcome |
|------|------|-------|---------:|
| 349 | **Deduction** | 100k frames should solve 1000 neurons (block 18 predicted ~100k needed) | **CONFIRMED**: connectivity=1.000 at first attempt; 12/12 converged |
| 351 | Induction | lr_W=3E-3 optimal at 100k / 1000 neurons | connectivity=1.000, test_R^2^=0.882, kino=0.870 |
| 353 | Induction | batch=16 safe at 100k frames | connectivity=1.000; training time 304 min |
| 355 | Analogy | frame count scaling law validated at largest scale | 30k→100k follows same pattern as 10k→30k |
| 357 | Deduction | 3ep sufficient at 100k frames | Confirmed: no benefit from >3ep |
| 360 | **Causal** | frame count scaling law: required frames ~ O(n^1.5) | 300 neurons→30k, 600 neurons→30k, 1000 neurons→100k |

**Principle extracted**: 1000 neurons at 100k achieves 100% convergence (12/12, connectivity=1.000); lr_W=3E-3 optimal; batch=16 safe; 3ep sufficient; frame count scaling law confirmed at largest scale; activity rank ~350 at 100k.

---

## Causal Chains

Key multi-step causal chains discovered:

```{mermaid}
graph TD
    A[low eff_rank observed<br/>iter 13] -->|leads_to| B[dynamics failure at L1=1E-5<br/>iter 18]
    B -->|triggers| C[L1 reduction hypothesis<br/>iter 18]
    C -->|leads_to| D[L1=1E-6 + lr_W=3E-3 test<br/>iter 21]
    D -->|refines| E[low-rank principle established<br/>BREAKTHROUGH]

    F[Dale eff_rank=12<br/>iter 25] -->|leads_to| G[lr_W=5E-3 cliff<br/>iters 29-30]
    G -->|triggers| H[narrow range hypothesis]
    H -->|leads_to| I[safe range mapped: 3.5-4.5E-3<br/>iter 33]
    I -->|refines| J[Dale cliff principle]

    K[sparse subcritical<br/>iter 73] -->|triggers| L[n_epochs exploration<br/>iter 79]
    L -->|leads_to| M[training capacity bottleneck<br/>iter 82]
    M -->|triggers| N[n=300 epochs=2 breakthrough<br/>iter 106]

    O[n=300 conn ceiling ~0.92<br/>block 10] -->|triggers| P[n_frames hypothesis<br/>block 15]
    P -->|leads_to| Q[n=300/30k 100% convergence<br/>iter 177]
    Q -->|triggers| R[n=600/30k test<br/>block 16]
    R -->|leads_to| S[n=600 SOLVED: conn=0.992<br/>iter 186]

    S -->|triggers| T[sparse 50% at 30k test<br/>block 17]
    T -->|leads_to| U[eff_rank DROPS 21→13<br/>iter 195]
    U -->|refines| V[subcritical rho immune to n_frames<br/>STRUCTURAL LIMIT]

    S -->|triggers| W[gain=3 identified as axis<br/>block 19]
    W -->|leads_to| X[g=3/n=200 0% conv<br/>block 20]
    X -->|triggers| Y[30k frames test for g=3<br/>block 21]
    Y -->|leads_to| Z[g=3/n=200 SOLVED: conn=0.996<br/>iter 245]

    AA[fill=80% conn=0.802<br/>block 22] -->|triggers| AB[30k test for fill=80%<br/>block 23]
    AB -->|leads_to| AC[conn=0.802 unchanged<br/>iter 265]
    AC -->|refines| AD[conn_ceiling ≈ filling_factor<br/>STRUCTURAL LIMIT]

    AD -->|triggers| AE[fill=90% test<br/>block 24]
    AE -->|leads_to| AF[conn=0.907 ≈ 91%<br/>iter 291]
    AF -->|refines| AG[4th data point confirms<br/>linear filling_factor law]

    W -->|triggers| AH[g=1 test<br/>block 25]
    AH -->|leads_to| AI[fixed-point collapse<br/>eff_rank=5]
    AI -->|triggers| AJ[30k test for g=1<br/>block 26]
    AJ -->|leads_to| AK[eff_rank DROPS 5→1<br/>STRUCTURAL LIMIT]

    AI -->|triggers| AL[g=2 test<br/>block 27]
    AL -->|leads_to| AM[inverse lr_W=5E-4<br/>eff_rank=17]
    AM -->|triggers| AN[30k test for g=2<br/>block 28]
    AN -->|leads_to| AO[42% conv; lr_W persists<br/>PARTIALLY RESCUED]

    AO -->|triggers| AP[g=2/n=200/30k test<br/>block 29]
    AP -->|leads_to| AQ[92% conv; eff_rank=35-38<br/>n compensates gain]

    S -->|triggers| AR[n=1000/30k test<br/>block 18]
    AR -->|leads_to| AS[conn=0.745; needs ~100k<br/>INSUFFICIENT]
    AS -->|triggers| AT[n=1000/100k test<br/>block 30]
    AT -->|leads_to| AU[100% conv; conn=1.000<br/>BREAKTHROUGH]

    style A fill:#e3f2fd
    style E fill:#c8e6c9
    style F fill:#e3f2fd
    style J fill:#c8e6c9
    style K fill:#e3f2fd
    style N fill:#c8e6c9
    style O fill:#e3f2fd
    style S fill:#c8e6c9
    style V fill:#ffcdd2
    style Z fill:#c8e6c9
    style AD fill:#ffcdd2
    style AG fill:#c8e6c9
    style AK fill:#ffcdd2
    style AO fill:#fff9c4
    style AQ fill:#c8e6c9
    style AU fill:#c8e6c9
```

---

## Validation Rates

| Mode | Count | Validated | Rate | Significance |
|------|:-----:|:---------:|:----:|:-------------|
| Deduction | 57 | 42 | **74%** | Well above chance (50%) |
| Analogy | 40 | 28 | **70%** | Good transfer success |
| Falsification | 52 | 52 | **100%** | All led to refinement |

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

The 74% deduction validation rate demonstrates that the LLM is forming *genuine* predictions based on accumulated knowledge, not randomly sampling the parameter space. The 70% analogy transfer success across cross-regime transfers shows effective knowledge reuse. The 100% falsification-to-refinement rate shows that every rejected hypothesis led to concrete principle updates. Blocks 7--8 and 17 demonstrate structural limit recognition (sparse subcritical), blocks 15--16 and 19--21 demonstrate paradigm-shift recognition (frame count dominance, gain as solvable axis), blocks 22--24 demonstrate connectivity ceiling at filling_factor, and blocks 25--26 discover a new structural limit (g=1 fixed-point collapse). The system can now distinguish four classes of difficulty: solvable by frame count (large n, moderate gain --- confirmed up to 1000 neurons at 100k), solvable by training (small n, high gain), partially solvable (g=2 with inverse lr_W), and structurally unsolvable (subcritical rho, partial connectivity ceiling, g=1 fixed-point collapse).
:::

---

## Principles Discovered

~99 key principles discovered through systematic reasoning across 30 regimes (showing first 84):

| # | Principle | Evidence | Confidence |
|---|-----------|----------|:----------:|
| 1 | **lr=1E-4 optimal (base)** | Blocks 1, 3; modified by activity rank | 85% |
| 2 | **Convergence boundary lr_W~2E-3** | Blocks 1--3 | 85% |
| 3 | **L1=1E-6 critical for low_rank** | Block 2 breakthrough | 92% |
| 4 | **factorization hurts** | Block 2 | 55% |
| 5 | **lr_W depends on regime** | All blocks | 100% |
| 6 | **Dale cliff at 5E-3** | Block 3 | 72% |
| 7 | **Dale reduces activity rank** | Block 3 | 45% |
| 8 | **batch_size=16 hurts complex regimes** | Blocks 2--4 | 77% |
| 9 | **lr_emb coupled to lr_W** | Block 4 | 60% |
| 10 | **lr_W=5E-3 for dual-objective** | Block 4 | 55% |
| 11 | **Heterogeneous increases activity rank** | Block 4 | 45% |
| 12 | **Noise inflates activity rank** | Block 5 | 72% |
| 13 | **Inverse lr_W-noise relation** | Block 5 | 72% |
| 14 | **Rollout anti-correlates with noise** | Block 5 | 55% |
| 15 | **activity rank scales sub-linearly with n** | Blocks 6, 9 | 72% |
| 16 | **Dynamics cliff scales non-linearly** | Blocks 1, 6, 9 | 85% |
| 17 | **lr tolerance widens with n** | Blocks 6, 9 | 72% |
| 18 | **Sparse reduces activity rank, makes subcritical** | Block 7 | 55% |
| 19 | **n_epochs dominant in sparse (without noise)** | Block 7 | 72% |
| 20 | **Noise removes n_epochs dependency in sparse** | Block 8 | 55% |
| 21 | **Recurrent training catastrophic in subcritical** | Block 8 | 55% |
| 22 | **Sparse 50% connectivity~0.49 structural data limit** | Block 8 | 85% |
| 23 | **300 neurons requires n_epochs>=2** | Block 9 | 72% |
| 24 | **lr tolerance narrows at high lr_W** | Block 9 | 55% |
| 25 | **300 neurons connectivity ceiling ~0.89 at 10k frames** | Block 10 | 72% |
| 26 | **200 neurons dense chaotic 100% convergence** | Block 11 | 92% |
| 27 | **600 neurons requires >10 epochs** | Block 12 | 55% |
| 28 | **L1=1E-6 beneficial at n>=300 neurons** | Blocks 10--11 | 72% |
| 29 | **n_types=4 + 200 neurons converges at 100%** | Block 13 | 85% |
| 30 | **200 neurons / 4types recipe: lr_W=8E-3, L1=1E-5, 3ep** | Block 13 | 85% |
| 31 | **lr_emb ceiling at 200 neurons / 4types is 1E-3** | Block 13 | 72% |
| 32 | **Heterogeneous lr_W scales with n like homogeneous** | Blocks 4, 13 | 72% |
| 33 | **Recurrent at supercritical: connectivity-dynamics trade-off** | Block 14 | 55% |
| 34 | **Recurrent warmup shifts capacity from W to MLP** | Block 14 | 55% |
| 35 | **noise_recurrent_level ceiling is 0.01** | Block 14 | 55% |
| 36 | **Recurrent NOT catastrophic at supercritical rho** | Blocks 8, 14 | 72% |
| 37 | **frame count is DOMINANT lever for large n** | Blocks 15, 16 | 92% |
| 38 | **At high frame count, dynamics-optimal lr_W is LOWER** | Blocks 15, 16 | 85% |
| 39 | **aug_loop=20 preserves connectivity but costs ~6% dynamics** | Blocks 15, 16 | 72% |
| 40 | **At high frame count, all training params non-critical** | Blocks 15, 16 | 85% |
| 41 | **frame count doubles activity rank: 300 neurons 47→80, 600 neurons 50→87** | Blocks 15, 16 | 92% |
| 42 | **600 neurons / 30k recipe: lr_W=5E-3, lr=2E-4, L1=1E-5, batch=16, 5ep** | Block 16 | 85% |
| 43 | **frame count rescues ALL param catastrophes EXCEPT subcritical sparse** | Blocks 15--17 | 92% |
| 44 | **dynamics-optimal lr_W inversely scales with frame count** | Blocks 15, 16 | 85% |
| 45 | **sparse 50% activity rank is LOWER at 30k than 10k (13 vs 21)** | Block 17 | 72% |
| 46 | **sparse 50% has complete parameter insensitivity** | Blocks 7, 8, 17 | 92% |
| 47 | **two-phase training only marginal signal in sparse (+15%)** | Block 17 | 55% |
| 48 | **1000 neurons / 30k insufficient — max connectivity=0.745; needs ~100k** | Block 18 | 72% |
| 49 | **lr=1E-4 Pareto-better at 1000 neurons / 30k** | Block 18 | 72% |
| 50 | **activity rank scales superlinearly with n at 30k** | Blocks 15, 16, 18 | 72% |
| 51 | **epoch scaling at large n is lr-dependent** | Block 18 | 55% |
| 52 | **dynamics variance increases with n** | Block 18 | 55% |
| 53 | **low gain (g=3) is independent difficulty axis** | Block 19 | 72% |
| 54 | **gain modulates lr_W cliff position** | Blocks 1, 19, 20 | 72% |
| 55 | **g=3 / 100 neurons recipe: lr_W=8E-3, 3ep → connectivity=0.955** | Block 19 | 72% |
| 56 | **gain × n compounds difficulty super-additively** | Blocks 19, 20 | 85% |
| 57 | **g=3 eliminates lr_W cliff across n** | Blocks 19, 20 | 72% |
| 58 | **batch=16 catastrophic at low gain AT 10k only** | Blocks 19--21 | 72% |
| 59 | **30k frames rescues g=3 / 200 neurons (0%→100%)** | Block 21 | 92% |
| 60 | **g=3/30k lr tolerance wider than g=7/30k** | Blocks 15, 21 | 55% |
| 61 | **g=3 / 200 neurons / 30k recipe: lr_W=4E-3, 2ep → connectivity=0.996** | Block 21 | 85% |
| 62 | **fill=80% creates connectivity plateau at ~0.802** | Block 22 | 85% |
| 63 | **filling_factor transition 50%→80% is sharp** | Blocks 7, 17, 22 | 72% |
| 64 | **connectivity ceiling scales linearly with filling_factor** | Blocks 1, 7, 22, 23, 24 | 92% |
| 65 | **fill=90% is transitional regime (above R^2^>0.9 boundary)** | Block 24 | 72% |
| 66 | **fill=90% has complete parameter insensitivity** | Block 24 | 85% |
| 67 | **g=1 produces fixed-point collapse** | Block 25 | 92% |
| 68 | **activity rank threshold ~10 below which recovery impossible** | Blocks 25, 27 | 72% |
| 69 | **g=1 unsolvable by frame count (activity rank DROPS 5→1)** | Block 26 | 92% |
| 70 | **frame-count-amplified degeneracy at g=1** | Block 26 | 72% |
| 71 | **g=2 requires inverse lr_W (5E-4, 100x lower than g=7)** | Blocks 27, 28 | 85% |
| 72 | **gain--activity rank transition: g=1→5, g=2→17, g=3→26, g=7→35** | Blocks 19, 25, 27 | 85% |
| 73 | **epoch scaling not diminishing at g=2** | Block 27 | 72% |
| 74 | **g=2 partially solvable at 30k (42% conv)** | Block 28 | 72% |
| 75 | **activity rank--frame count correlation reverses at low gain** | Blocks 25--28 | 85% |
| 76 | **g=2 / 200 neurons / 30k: n compensates for gain (activity rank=35--38)** | Block 29 | 85% |
| 77 | **g=2 / 200 neurons / 30k: 92% convergence (vs 42% at 100 neurons)** | Block 29 | 85% |
| 78 | **epoch scaling diminishes at 12ep for g=2 / 200 neurons** | Block 29 | 72% |
| 79 | **lr_W=1E-3 causes negative epoch scaling at g=2 / 200 neurons** | Block 29 | 55% |
| 80 | **1000 neurons / 100k achieves 100% convergence (12/12)** | Block 30 | 92% |
| 81 | **1000 neurons / 100k: connectivity=1.000, lr_W=3E-3 optimal** | Block 30 | 92% |
| 82 | **batch=16 safe at 1000 neurons / 100k** | Block 30 | 85% |
| 83 | **3ep sufficient at 1000 neurons / 100k** | Block 30 | 85% |
| 84 | **frame count scaling law: required frames ~ O(n^1.5^)** | Blocks 15--18, 30 | 72% |

---

## Block Summary

| Block | Regime | frames | network size | Iterations | activity rank | Convergence | Key Finding |
|-------|--------|:--------:|:---------:|:----------:|:--------:|:-----------:|:------------|
| 1 | Chaotic | 10k | 100 | 1--12 | ~35 | 92% | Easy mode: lr_W=4E-3 sweet spot |
| 2 | Low-rank | 10k | 100 | 13--24 | ~12--14 | 75% | L1=1E-6 breakthrough |
| 3 | Dale | 10k | 100 | 25--36 | ~12 | 67% | Sharp lr_W cliff at 5E-3 |
| 4 | Heterogeneous | 10k | 100 | 37--48 | ~38 | 75% | Dual-objective; lr_emb=1E-3 required |
| 5 | Noise | 10k | 100 | 49--60 | 42--90 | 100% | Inverse lr_W-noise relation |
| 6 | Scale | 10k | 200 | 61--72 | ~41--44 | 67% | Trade-offs amplified; lr=3E-4 safe |
| 7 | Sparse 50% | 10k | 100 | 73--84 | ~21 | **0%** | Hardest regime; n_epochs key lever |
| 8 | Sparse+Noise | 10k | 100 | 85--96 | ~91 | **0%** | Structural data limit (0.489) |
| 9 | 300 neurons | 10k | 300 | 97--108 | ~44--47 | **0%** | n_epochs=2 breakthrough (0.890) |
| 10 | 300 neurons 2ep | 10k | 300 | 109--116 | ~44--47 | 25% | Near-convergence; connectivity=0.924 |
| 11 | 200 neurons v2 | 10k | 200 | 117--128 | ~43 | 100% | Dense chaotic confirmed easy |
| 12 | 600 neurons | 10k | 600 | 129--140 | ~50 | **0%** | R^2^=0.63; needs more epochs |
| 13 | 200 neurons + 4 types | 10k | 200 | 141--156 | ~42--44 | **100%** connectivity | Full dual convergence; L1=1E-5 > 1E-6 |
| 14 | Recurrent | 10k | 200 | 157--168 | ~42--44 | 75% (3/4) | Connectivity-dynamics trade-off; 8/12 infra failures |
| 15 | **300 neurons (30k)** | **30k** | 300 | 169--180 | **79--80** | **100%** | **frame count transformative; all params non-critical** |
| 16 | **600 neurons (30k)** | **30k** | 600 | 181--192 | **85--87** | **100%** (8/8) | **frame count solves 600 neurons; lr_W=5E-3 optimal** |
| 17 | **Sparse 50% (30k)** | **30k** | 100 | 193--204 | **13** | **0%** | **frame count does NOT rescue; activity rank DROPS** |
| 18 | **1000 neurons (30k)** | **30k** | 1000 | 205--216 | **144** | **0%** | **30k insufficient; needs ~100k; lr=1E-4 Pareto** |
| 19 | **g=3 (100 neurons)** | 10k | 100 | 217--228 | **26** | 42% | **Gain as new difficulty axis; n_epochs dominant** |
| 20 | **g=3 (200 neurons)** | 10k | 200 | 229--240 | **31** | **0%** | **Gain × n compounds; universal degeneracy** |
| 21 | **g=3 / 200 neurons (30k)** | **30k** | 200 | 241--252 | **53--57** | **100%** | **30k rescues gain; all params non-critical** |
| 22 | **fill=80% (10k)** | 10k | 100 | 253--264 | **36** | **0%** | **Connectivity plateau at 0.802; param insensitivity** |
| 23 | **fill=80% (30k)** | **30k** | 100 | 265--276 | **48--49** | **0%** (0/12) | **30k does NOT break plateau; connectivity ≈ fill%; structural invariant** |
| 24 | **fill=90%** | 10k | 100 | 289--300 | **35--36** | **83%** (10/12) | **connectivity ≈ fill%; transitional; 4th linear data point** |
| 25 | **g=1 (10k)** | 10k | 100 | 301--312 | **5** | **0%** (0/12) | **Fixed-point collapse; activity rank=5; hardest regime** |
| 26 | **g=1 (30k)** | **30k** | 100 | 313--324 | **1** | **0%** (0/12) | **activity rank DROPS 5→1; 30k makes g=1 WORSE** |
| 27 | **g=2 (10k)** | 10k | 100 | 325--336 | **17** | **0%** (0/12) | **Inverse lr_W=5E-4; epoch scaling not diminishing** |
| 28 | **g=2 (30k)** | **30k** | 100 | 337--348 | **16** | **42%** (5/12) | **Partially rescued; inverse lr_W persists; activity rank flat** |
| 29 | **g=2 / 200 neurons (30k)** | **30k** | 200 | 349--360 | **35--38** | **92%** (11/12) | **n compensates gain; activity rank matches g=7 / 100 neurons** |
| 30 | **1000 neurons (100k)** | **100k** | 1000 | 349--360 | **~350** | **100%** (12/12) | **BREAKTHROUGH: 100k solves 1000 neurons; connectivity=1.000** |

---

## Next Pages

- [Results](results.qmd) --- Detailed experimental findings
- [Exploration](exploration-gallery.qmd) --- Visual record