cosine concept

Definition	The dot product divided by the product of the two lengths, between minus one and one, unchanged by rescaling either vector. Its distance is the angle. Equation 0.1.
Example	Vectors (1, 0) and (1, 1) have cosine one over root 2, about 0.707, and doubling either vector leaves it unchanged.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `cosine`, kind concept.
Status	measures [predicted]; refutes or corrects [refuted]. Corrections: 1 item(s), see below.
Defining equation	Book equation 0.1.
Assumptions and scope	The dot product divided by the product of the two lengths. By Cauchy and Schwarz it lies between minus one and one, a vector has cosine one with itself, rescaling either vector by a positive factor does not change it, and it is the dot product of the two row-normalized vectors, so its distance is the angle on the quotient that discards length. A reconstruction cosine of 0.995 sat beside a consumer’s perplexity of ten thousand, so the cosine is never an acceptance metric for a code, and on a spectral embedding the cosine is the right similarity because the length has converged to degree noise.
Prior art	none recorded
Evidence	`geometric-observation/claims/LEDGER.md:119`, `geometric-observation/claims/LEDGER.md:95`, `geometric-observation/chapters/ch02_failure_of_observer_free_measurement.md:40-60`, `geometric-observation/chapters/ch16_honest_negatives.md`, `turboquant-pro/docs/KV_KEYS_FINDING.md:1-49`, `lean/DataMiningAsObservation/Cosine.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

The dot product over the two lengths, the cosine of the angle between the vectors.

Equation

Book equation 0.1.

\[x\cdot y=\sum_{i=1}^{d}x_i y_i,\qquad \|x\|=\sqrt{x\cdot x},\qquad \cos\theta=\frac{x\cdot y}{\|x\|\,\|y\|}.\]

Conditions

The dot product divided by the product of the two lengths. By Cauchy and Schwarz it lies between minus one and one, a vector has cosine one with itself, rescaling either vector by a positive factor does not change it, and it is the dot product of the two row-normalized vectors, so its distance is the angle on the quotient that discards length.
A reconstruction cosine of 0.995 sat beside a consumer’s perplexity of ten thousand, so the cosine is never an acceptance metric for a code, and on a spectral embedding the cosine is the right similarity because the length has converged to degree noise.

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

Ledger

measures. GO-B-legal (035→036) [predicted]. Legal-citation retrieval (CourtListener), cosine-ranking consumer, LaBSE embeddings — real large corpus, non-physical consumer geometric-observation/claims/LEDGER.md:119.
refutes or corrects. NEG-2 [refuted]. Reconstruction cosine as a proxy for key quality. geometric-observation/claims/LEDGER.md:95.

First stated

Chapter 0 section 0.1 of Data Mining as Observation, equation 0.1, with the program’s cosine-against-consumer negative in turboquant-pro/docs/KV_KEYS_FINDING.md:1-49.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 1 section 1.4	cosine 0.995 and perplexity of order ten thousand, the recalibration negative	`geometric-observation/chapters/ch02_failure_of_observer_free_measurement.md:40-60`; `geometric-observation/chapters/ch16_honest_negatives.md` NEG-2 and NEG-4; `turboquant-pro/docs/KV_KEYS_FINDING.md:1-49`
chapter 2 section 2.5	cosine 0.995 and the softmax reader	`turboquant-pro/docs/KV_KEYS_FINDING.md:1-49`
chapter 8 section 8.2	cosine 0.995, perplexity near 1e4	`turboquant-pro/docs/KV_KEYS_FINDING.md:1-49`; `geometric-observation/chapters/ch16_honest_negatives.md` NEG-2

Failures and corrections

NEG-2, [refuted]. Reconstruction cosine as a proxy for key quality. geometric-observation/claims/LEDGER.md:95.

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/Cosine.lean, theorems abs_dot_le, cosine_le_one, neg_one_le_cosine, cosine_self, cosine_smul, cosine_eq_dot_rowNormalize, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation primers L and S, chapters 0, 1, 2, 3, 6, 8, 10, 11, 12.

dot product; Euclidean distance; spectral embedding; quotient; identity reader.

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.

← correlationcoupling null →