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The "But Wait, We Need Filters Too" paragraph mentions "US" filter which is introduced only later on.
Maybe I'm wrong, but for this query:

SELECT * FROM benchmark_logs WHERE severity < 3 ORDER BY timestamp DESC LIMIT 10;

this index

CREATE INDEX ON benchmark_logs (severity, timestamp);

cannot be used as proposed: "Postgres can jump directly to the portion of the tree matching severity < 3 and then walk the timestamps in descending order to get the top K rows."

Postgres with this index can walk to a part of the tree with severity < 3, but timestamps are sorted only for the same severity.

Do a partial index on just timestamp where severity < 3 instead.
The issue here is the row based format. You simply can't filter on arbitrary columns with that. Either use an external warehouse or a columnar plug-in like Timescale.
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My apologies, the article was talking about 100M-rows datasets so I presumed that was the scenario.
Lucene really does feel like magic sometimes. It was designed expressly to solve the top K problem at hyper scale. It's incredibly mature technology. You can go from zero to a billion documents without thinking too much about anything other than the amount of mass storage you have available.

Every time I've used Lucene I have combined it with a SQL provider. It's not necessarily about one or the other. The FTS facilities within the various SQL providers are convenient, but not as capable by comparison. I don't think mixing these into the same thing makes sense. They are two very different animals that are better joined by way of the document ids.

Under the hood, ParadeDB is built by integrating Tantivy (a Lucene-inspired Rust search library) inside Postgres. This is to say: I agree with you -- We're not trying to claim that Postgres itself is an alternative to Lucene, but rather that something like Lucene can be integrated inside Postgres so that you can get the power of both in a single system (or in a cluster of Postgres instances)
Just in case, there is a btree_gin extension which can be used in queries combining gin-indexable column and btree-indexable column. It doesn’t solve top-K ordering problem though.
Postgres is really good at a lot of things, but it's very unfortunate that it's really bad at simple analytics. I wish there was a plugin instead of having to have N databases
Curious whether you benchmarked against a partial index on the sort column. For fixed-category top-K queries the planner sometimes picks it over a full index scan, though I've seen it regress on high-write tables due to index bloat. Did write volume factor into your test setup?