app/src/content/chapters/04-bound.mdx

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# Why the Machine Stays Small

A small state count alone does not make a machine useful. An automaton can still be huge if the language is complex, and a huge automaton spreads its observations thinly, leaving every per-state statistic noisy. What makes the FSM a usable substrate is that it is small, stable, and converges fast, so each state pools enough traces to estimate from. That property, not the exact count, is what the prediction and monitoring chapters depend on.

## It converges on a few percent of the data

Structure stabilizes almost immediately. Across five random train/test splits the extracted state count is **identical every time**, zero variance, so the topology is a property of the agent, not of which traces you happened to sample. Replay fitness plateaus just as fast: within the first **1 to 10% of the training traces** every dataset clears 0.95 fitness, and SWE-smith holds 0.9996 from the first 1%.

<Sidenote>
A formal version: if traces are i.i.d. and each transition appears with probability at least $p_{\min}$, the extracted FSM equals the population FSM after $N \geq \frac{1}{p_{\min}} \ln(k/\delta)$ traces. For SWE-agent ($k=43$ transitions, $p_{\min} \approx 0.01$), that is at most 676 traces at $\delta=0.05$; in practice fitness stabilizes within the first 10% of the training set.
</Sidenote>

## Three different methods agree

The compactness is not an artifact of our particular merge rule. Three fundamentally different algorithms converge to nearly the same state count on every dataset:

- **Structural merging** (ours), a structural partition.
- **Alergia** [@carrasco1994alergia], a statistical merge, lands within 1.0 to 6.0$\times$.
- **[hidden Markov model](https://texonom.com/126c3c96247d80e0b880c38e5f3bb924)** (HMM) [@rabiner1989hmm], a probabilistic latent-state model, matches the count, though its states are not interpretable.

A structural, a statistical, and a probabilistic method agreeing rules out an algorithmic coincidence.