Every arrangement of holders sorts into two piles. There is no third pile and nothing sits between them.
| holders in the room | arrangements | outcome |
|---|
Every node above is one possible room: some subset of the holders, turning up. An edge joins two rooms that differ by exactly one person walking in. Read it bottom to top and you are watching a room fill.
The colour changes on one row and never changes back. That is the whole character of a threshold scheme: it has no notion of nearly, no partial credit for the ones who did turn up, and no way to reward four people who tried. The row where it flips is the only decision anyone made, and it was made once, at issue time, by choosing k.
Raising k does not make the vault safer in some general sense. It moves the flip upward, which is simultaneously harder to compromise and easier to lose forever. Those are not a trade-off between two properties. They are one property, described from two sides.
Nodes are drawn for every subset, so the count doubles with each holder added and the diagram stops being legible somewhere around eleven. Counts are binomial coefficients computed here, not stored. Nothing on this page knows who holds what, because nothing on this page knows anything at all.