direction-only quantizer instrument

Definition	A quantizer that stores each vector's length exactly and rounds its direction. Chapter 11. Also direction-only, PolarQuant, polar.
Example	A vector of length 5 at 30 degrees is stored as the length 5 exactly and the direction rounded to the nearest of 16 angles.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `direction-only-quantizer`, kind instrument.
Status	measures [predicted]. Corrections: none recorded.
Defining equation	Book equation 11.1.
Assumptions and scope	A quantizer that stores each vector’s length exactly and rounds its direction to a unit codeword. The quantized vector keeps the original’s length, its cosine with any query is the codeword’s cosine, and its score against a query is the length times the query’s dot product with the codeword, so the score error is the length times the query’s dot product with the direction error. A key’s cosine with its own quantized form can be near one while its score against a query moves by a multiple of its length, which is why keys at cosine 0.995 raised the perplexity by three orders of magnitude.
Prior art	none recorded
Evidence	`geometric-observation/claims/LEDGER.md:117`, `turboquant-pro/docs/KV_KEYS_FINDING.md:1-49`, `lean/DataMiningAsObservation/DirectionQuantizer.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

The length stored exactly and the direction rounded.

Equation

Book equation 11.1.

\[\cos\big(k,\hat k\big)=0.995\qquad\text{while}\qquad \mathrm{PPL}:\ 12.24\ \to\ 10643.\]

Conditions

A quantizer that stores each vector’s length exactly and rounds its direction to a unit codeword. The quantized vector keeps the original’s length, its cosine with any query is the codeword’s cosine, and its score against a query is the length times the query’s dot product with the codeword, so the score error is the length times the query’s dot product with the direction error.
A key’s cosine with its own quantized form can be near one while its score against a query moves by a multiple of its length, which is why keys at cosine 0.995 raised the perplexity by three orders of magnitude.

Conditions are curated in entries.toml rather than read from a record.

Ledger

measures. GO-B-LOCATA [predicted]. Real microphone-array recordings (LOCATA), DOA consumer — held-out confirmation with the PolarQuant compressor geometric-observation/claims/LEDGER.md:117.

First stated

Chapter 11 section 11.2 of Data Mining as Observation, with PolarQuant in turboquant-pro/docs/KV_KEYS_FINDING.md:1-49 and the LOCATA comparison of the ledger.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 11 section 11.2	fp16 12.24, values-only 13.12, PolarQuant K4 10643 and 0.095, per-channel uniform K4 14.91 and 0.062, per-channel NUQ K3 15.77 and 0.148, 2.4x and 670x, pre-rotary near 22000, 2 key heads serve 12 query heads	`turboquant-pro/docs/KV_KEYS_FINDING.md:1-49`

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/DirectionQuantizer.lean, theorems length_preserved, score_eq, score_error, cosine_eq, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation chapters 0, 3, 4, 11.

quantization; dot product; KV cache; attention; the flip.

Status

Generated 2026-09-10 by encyclopedia/generate.py; book at observation-data-mining f3914f0; the commit of every record is listed in the encyclopedia’s provenance.

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