multiplet concept
| Definition | A group of eigenvalues that are equal or nearly so, whose pattern names the shape a dataset lies on against a finite list of templates. On a finite graph the groups are approximate. Chapter 9. |
|---|---|
| Example | The cycle on 8 nodes has eigenvalues 0, 0.586, 0.586, 2, 2, 3.414, 3.414, and 4. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id multiplet, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | none |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | geometric-observation/chapters/ch11_the_recognizer.md:1-95, lean/DataMiningAsObservation/Multiplet.lean |
| Reviewed | semantic review 2026-09-06; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
none
Conditions
- A group of eigenvalues that are equal or nearly so. The cycle’s Laplacian eigenvalue at k equals the one at n minus k, so every eigenvalue of the discrete circle other than the constant mode and, for even n, the alternating mode appears twice.
- Symmetric shapes produce multiplets in the continuum, the sphere in groups of three, five, seven. On a finite neighbourhood graph the multiplets are approximate, their spacing depends on sampling and graph construction, and the recognizer reads them against a finite list of templates and refuses when none matches.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 0 section 0.15 and chapter 9 section 9.2 of Data Mining
as Observation, with the program’s recognizer in
geometric-observation/chapters/ch11_the_recognizer.md:1-95.
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/Multiplet.lean,
theorems cycleEig_zero, cycleEig_nonneg,
cycleEig_le_four, cycleEig_pair,
pair_distinct, 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, 9.
Related
Laplacian; recognizer; manifold; Weyl's law; eigenvalue, eigenvector.
See also
Book equations stated beside the entry’s terms, not defining it: 0.19, 0.31, 9.2.
Sources-table rows that share a record with the entry without naming it: chapter 9 section 9.2.
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.