The search that leaves no one out.

Picture a library whose card catalog is organized by subject, one drawer per topic. You ask for "everything similar to this book". A hurried clerk opens only the two most likely drawers. Most of the time that works. Sometimes a book that fit was in the next drawer, and nobody ever finds out.

SIEVE knows two things about each drawer: where its center is and which book lies farthest from that center. With that, it can say without opening it: "nothing in this drawer can possibly fit". It rules those out and opens the rest. Sometimes it opens more drawers than the hurried clerk, but at the end it can state that none was left out.

That is what market search indexes do not give you: they return results without saying how many true ones were left behind.

The similarity search indexes on the market (the ones behind Qdrant, Weaviate, pgvector, Milvus and Pinecone) are approximate and silent. We measured the most widely used one with 40,000 items: depending on the speed setting, it finds 56% to 97% of the 10 true neighbors, and does not warn you when it misses.

For recommending a movie, that is fine. For checking whether a customer is on a sanctions list, finding frauds similar to a known one, deduplicating records, searching prior patents or auditing what an AI consulted, a forgotten result is a loss nobody can sign off on.

The new methods of 2026 already promise an accuracy level, but by estimate. One of them (DARTH, 2026) reports that about 13% of searches fall below the promised level, and the index does not know which ones.

1. Drawers. Items are grouped into regions, and SIEVE keeps the center and size of each. A whole region leaves the count if even its closest point is outside the radius. A whole region enters without item-by-item checking if even its farthest point is inside.

2. Shadow. For the items that remain, SIEVE first looks at a "shadow" of them, a version with far fewer numbers. The distance in the shadow is never larger than the real one. So if even the shadow is far, the item is far, and the full computation is skipped.

3. Full computation. Only what passed both sieves gets its full distance computed. That is why SIEVE's worst case is the full search, never an incomplete answer.

None of this uses estimates, samples or confidence levels. It is geometry, and it is checked in tests against the full search in every configuration.

On the public SIFT1M set (1 million image descriptors, 128 numbers each), the full search takes 109 ms per query. SIEVE in its fastest configuration takes 16.4 ms, 6.3× less, without leaving any result out. Reading from disk instead of memory, the advantage was 7.2×.

We also publish what did not work. The first version, with 50,000 items in memory, only tied with the full search: the initial promise of "4 to 8× faster" did not hold at that size.

And we measured how cost grows with the size of the collection. From 100,000 to 1 million items, full computations grow slowly (proportional to N^0.41, similar to approximate indexes), but the cheap work of the sieves grows almost in proportion to the collection (N^0.94). In practice: the advantage shrinks on very large collections, and today SIEVE does not have the same cost as approximate indexes at scale. The next step is a layered sieve.

State absence. "No record crossed the limit" becomes a proven answer, not a "didn't find any". It is the sentence compliance and forensics need.

Catch AI agents going in circles. Reasoning models sometimes spiral, repeating ideas until they run out of memory. On 26 real labeled traces, SIEVE's search caught all three kinds of repetition (literal, alternating and reworded) with no false alarm on the 8 normal traces. The cheap detectors each missed one kind. See the full analysis.

Summarize the neighborhood with proof. Combined with CLAMP, you can ask "what is the average temperature of the cases similar to this one?" and get a guaranteed range: on real data, 327 of 327 answers contained the right value.