Can SPV clients offer "proof of absence" of a transaction?

Can SPV clients offer "proof of absence" of a transaction?

So light SPV clients give the user the ability to obtain merkle proofs (from a full node) that a given transaction is included in a block. Is there some cryptographic mechanism by which an SPV client could obtain proof that a transaction (or type of transaction) is not included in a block, while still being relatively "light" (O(log(n), say)?

Asking for a layer 2 solution ;)

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