density concept

Definition	The local crowding of rows around a point, the coordinate the geodesic reader of a spectral embedding discards and DBSCAN and the density detector read. Chapter 3 section 3.3 and chapter 9.
Example	A point with 20 neighbours within radius 1 sits in a denser region than one with 2.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `density`, kind concept.
Status	refutes or corrects [refuted]. Corrections: 1 item(s), see below.
Defining equation	Book equation 3.3.
Assumptions and scope	The local crowding of rows around a point, which on a neighbourhood graph is the degree. The radius of the spectral embedding converges to one over the square root of the degree, so the geodesic reader discards the density and DBSCAN and the density detector read it. A wider radius or a smaller count makes more core points, and the sum of degrees is twice the edge count. The radius-against-degree correlation was 0.92 to 0.99 across substrates, and the quotient that removed density to restore invariant fidelity was refuted four times, NEG-11.
Prior art	none recorded
Evidence	`geometric-observation/claims/LEDGER.md:104`, `lean/DataMiningAsObservation/Dbscan.lean`, `lean/DataMiningAsObservation/Degree.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

The local crowding of rows, the coordinate the geodesic reader discards.

Equation

Book equation 3.3.

\[X_i=r_i\,\theta_i,\qquad r_i\ \to\ \frac1{\sqrt{d_i}},\qquad \theta_i=\frac{X_i}{\|X_i\|}\in S^{m-1}.\]

Conditions

The local crowding of rows around a point, which on a neighbourhood graph is the degree. The radius of the spectral embedding converges to one over the square root of the degree, so the geodesic reader discards the density and DBSCAN and the density detector read it. A wider radius or a smaller count makes more core points, and the sum of degrees is twice the edge count.
The radius-against-degree correlation was 0.92 to 0.99 across substrates, and the quotient that removed density to restore invariant fidelity was refuted four times, NEG-11.

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

Ledger

refutes or corrects. NEG-11 [refuted]. (GO-5, prospective ×4) The α=1 density/hubness quotient decisively and density-specifically restores invariant fidelity in a non-spectral domain. geometric-observation/claims/LEDGER.md:104.

First stated

Chapter 3 section 3.3 of Data Mining as Observation, with the radius-against-degree measurement in the-angular-observer/README.md:135-139.

Measurements

none

Failures and corrections

NEG-11, [refuted]. (GO-5, prospective ×4) The α=1 density/hubness quotient decisively and density-specifically restores invariant fidelity in a non-spectral domain. geometric-observation/claims/LEDGER.md:104.

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/Dbscan.lean, theorems ball_mono, core_mono, core_anti, reach_from_core, noise_unreachable, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

lean/DataMiningAsObservation/Degree.lean, theorems sum_degrees, degree_lt_card, sum_degrees_even, average_degree, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation primer S, chapters 3, 8, 9, 10, 11.

degree; spectral embedding; DBSCAN; geodesic distance; hubness.

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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