1st-Order Transition Matrices
P(next category | current category). Each cell (i,j) = probability of transitioning from category i to j.
2nd-Order Transition Matrices
P(next | prev, current). Rows = context pairs (prev→cur), columns = next category.
Transition Differences
Element-wise difference between group transition matrices.
Transition Flow
Weighted flow between categories. Link thickness = transition probability × source weight.
Start Distribution
Probability of each category being the first step in a sequence.
Path Length
Number of top-level category steps per sequence.
Error Split
Incorrect sequences split by path length: long (≥ 100 steps) vs short (< 100 steps).
Stationary Dist.
Long-run proportion of time spent in each category (Markov chain π).
Expected Steps to FA
Expected number of transitions to reach Final Answer from each starting category.
Bottom-Level Metrics
Hidden-state regime analysis from the bottom GMMs. Regimes = mixture components learned within each category (jointly trained on all data).
2D PCA projection of per-step activations colored by category. Diamond markers = category centroids. Separation between clusters indicates distinct hidden-state representations per category.
Cosine similarity between mean transition direction vectors. Each direction = average activation displacement when the model transitions from category A to B. High cosine between two directions means the hidden-state movement is similar.
Cosine similarity between the mean transition direction vector of correct vs incorrect sequences. cos≈1 = both groups move in the same activation direction at this transition; cos≈0 = orthogonal movements; cos<0 = opposite directions. Low or negative cosine indicates a directional signature that distinguishes correct from incorrect reasoning.
Density of per-step activations along principal component 0. Separation between category curves indicates the primary axis of hidden-state differentiation.
Density along principal component 1, the second axis of variation. Captures secondary structure in category representations.
Density along principal component 2, the third axis of variation.
Regime Analysis
Regime Structure
PCA projection (axes capture max variance among regimes). Well-separated islands = regimes are geometrically distinct.
Soft Activation Profiles (Correct vs Incorrect)
Mean posterior probability γ(layer, regime) averaged across all steps. Left = correct, middle = incorrect, right = difference (blue = more in correct, red = more in incorrect). Rows = transformer layers (top=early, bottom=late), columns = regimes. Strong colored cells in the difference plot indicate (layer, regime) pairs that discriminate correct from incorrect reasoning.
Step Trajectory: Layer-by-Layer Computation Pattern
Each dot = one transformer layer in training-PCA space (dim 0 vs dim 1). Color & label = regime (MAP-decoded). ◆ = regime transition layer. ✦ faded star = centroid. Open circle = layer 0. By default, 2 random correct and 2 random incorrect samples are shown. Click a case chip to highlight an individual sample.
Explicit Bridge
P(target category | source category, exit regime).
Length = number of top-level category steps. Click a card to expand and see the timeline. Click timeline steps for hidden-state details.
Examples
Randomly sampled sequences from all groups. Click to expand. Click timeline steps for details.