docs/METRICS.md

Metric reference

LIMEN v0.1 treats the layer axis as an ordered sequence. For each token, the hidden state is an array (h_\ell), where (\ell) is the layer index.

This is a layer-wise geometric path. It must not be confused with a continuous-time physical trajectory or with the token-to-token generation trajectory.

Descriptive trajectory metrics

Metric	Definition	Useful for	Does not establish
Path length	(\sum_\ell \lVert h_{\ell+1}-h_\ell\rVert_2)	Total movement through layers	Reasoning effort
Displacement	(\lVert h_L-h_0\rVert_2)	Net first-to-last change	Semantic progress
Tortuosity	Path length / displacement	Route indirectness	Confusion or intelligence
Mean speed	Mean adjacent-layer distance	Typical layer-wise movement	Runtime latency
Speed CV	Standard deviation / mean speed	Movement variability	Instability in a control-theory sense
Mean acceleration	Mean norm of the second difference	Changes in movement	Physical acceleration
Turning angle	Mean angle between consecutive differences	Directional change	A cognitive transition

Tortuosity is undefined when displacement is zero and is reported as NaN in the per-token arrays. Summary statistics omit non-finite values.

Probability baselines

Entropy is reported in natural units (nats). The top-1/top-2 margin is the difference between the two largest softmax probabilities.

These baselines are included because many apparent geometric effects can be explained by ordinary changes in model confidence. A trajectory metric should not be presented as incrementally useful until it is compared against them on held-out data.

Required controls for stronger claims

• exact model and tokenizer revisions;
• fixed extraction location and layer indexing;
• response length and token-position controls;
• prompt-family controls;
• repeated seeds when sampling is enabled;
• shuffled layer and token-order baselines;
• held-out evaluation;
• cross-checks on more than one architecture.

No v0.1 metric is a validated universal invariant, functional locator or causal control signal.