recognizer instrument

Definition	The instrument that decides whether a clustering certificate is right, wrong, or vacuous, by naming the manifold from the eigenvalue multiplets or certifying that none is present. Chapter 9.
Example	The ratios of the lowest eight eigenvalues matched the sphere template, and a shape matching no template was refused.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `recognizer`, kind instrument.
Status	no ledger row names this entry. Corrections: none recorded.
Defining equation	Book equation 9.2.
Assumptions and scope	The recognizer names a manifold from the low eigenvalue multiplets of a neighbourhood graph’s Laplacian by template match, or certifies that none is present, and it can only name shapes in its template set. It was held to a preregistered held-out battery, and the vacuity threshold below which it certifies nothing is derived from the problem rather than tuned. The limitation applies to every spectral reading in the book. A finite graph spectrum names a shape only against a finite list of templates, and the manifold, multiplet, and Weyl’s law entries carry the same scope.
Prior art	none recorded
Evidence	`geometric-observation/chapters/ch11_the_recognizer.md:1-95`, `lean/DataMiningAsObservation/Recognizer.lean`
Reviewed	semantic review 2026-09-06; generated 2026-09-10 from records at the commits on the provenance page.

The spectrum read against a finite list of templates, and a refusal when none matches.

Equation

Book equation 9.2.

\[N(\lambda)=\#\{k:\lambda_k\le\lambda\}\ \sim\ C_d\,\lambda^{d/2}\qquad\Rightarrow\qquad d=2\,\frac{d\log N}{d\log\lambda}.\]

Conditions

The recognizer names a manifold from the low eigenvalue multiplets of a neighbourhood graph’s Laplacian by template match, or certifies that none is present, and it can only name shapes in its template set.
It was held to a preregistered held-out battery, and the vacuity threshold below which it certifies nothing is derived from the problem rather than tuned.
The limitation applies to every spectral reading in the book. A finite graph spectrum names a shape only against a finite list of templates, and the manifold, multiplet, and Weyl’s law entries carry the same scope.

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

Ledger

none

First stated

Volume 14, chapter 11, geometric-observation/chapters/ch11_the_recognizer.md:1-95, DOI 10.5281/zenodo.21776291, and the-angular-observer’s recognizer battery, the-angular-observer/PREREG_RECOGNIZER_BATTERY.md:1-80.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 9 section 9.2	the recognizer’s mechanism, low multiplets and angular distances, dimension before shape by Weyl’s law, refusal, the growth gotcha	`geometric-observation/chapters/ch11_the_recognizer.md:1-95`

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/Recognizer.lean, theorems templateDist_self, templateDist_comm, templateDist_scale, templateDist_nonneg, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation chapters 9.

certificate; vacuity threshold; 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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