Budget and threshold
Both are asked once, up front, when discovery is started. The walk itself is unattended, and a budget agreed after the fact is not a budget.

The threshold is materiality
A branch is followed automatically when its effective stake in the subject is at or above the threshold. Effective means the product along the chain, so at a 25% threshold:
| Chain | Effective | Followed? |
|---|---|---|
| Subject ← A at 60% | 60% | Yes |
| Subject ← A at 60% ← B at 50% | 30% | Yes |
| Subject ← A at 60% ← B at 50% ← C at 40% | 12% | No, but expandable, with its price shown |
| Subject ← D, no percentage stated | Unknown | No, because unknown is not small |
Twenty-five percent is the common default and matches the usual beneficial-ownership test. Lowering it follows more branches and costs more; raising it costs less and leaves more chips on the canvas for you to decide about one at a time.
The threshold is also what the report's ownership part states it was traversed to.
The budget bounds the walk, not you
The budget answers how far may this run on its own, a question asked before anybody has seen the structure. It applies to unattended spending only.
- Each automatic step is quoted before it runs and charged against the budget. Steps in flight are held at their quote, so a burst cannot collectively overrun.
- When the budget is gone the walk stops and says so.
- Raise budget gives it more room, and is recorded in the case's activity.

Opening a branch by hand is neither charged against the budget nor blocked by it. Every branch is clickable at any budget, with its price on the pill: you are looking at one branch, with the number in front of you, and a figure typed earlier has no standing to overrule that. Blocking it would only teach people to inflate the figure.
Automatic spend and manual spend are tracked and shown separately.

What a step actually costs
A company already paid for on this case is not billed again, which is why an already-fetched branch can quote near zero. The quote on a pill is what expanding that branch costs now, given what the case already holds.
Running it unattended
Deployments can switch on a background walk that does what the open page does, settling the step in flight and then taking the next one, without anybody watching.
It ships off, and in dry-run when first switched on. Dry-run is not a stub: it chooses the branch, quotes the step and computes the cap exactly as it would if it were dispatching, then reports what it would have spent. Running it for a few days and reading its log against what manual expansions actually cost is how the quotes earn trust.
When it is live, several guards bound it, and your case's budget is only one of them:
| Guard | What it bounds |
|---|---|
| One live step per case | Two steps can never run against one case at once |
| One company per automatic step | A mis-quote costs one company, not twenty-five |
| The case budget | The walk never picks a step it cannot afford |
| A cap on the request itself | Anything over it is refused, free, and the real price is reported |
| A daily cap across all cases | Which no individual case can raise |
| A limit on steps per case | |
| A kill switch |
A branch is never retried automatically once it has failed, because a budget refusal recurs while the budget does not change, so retrying it automatically would mean retrying it for ever.
Two dispatchers would contend for the same lock and achieve nothing. Your walk still advances; it is just not your browser doing it.