quotient concept

Definition	The space of data with a set of transformations declared not to matter, so that two rows differing only by such a transformation are the same point. Every distance is a distance on some quotient. Equation 0.12.
Example	The cosine reader cannot tell (1, 1) from (2, 2), so the two are one row on its quotient.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `quotient`, kind concept.
Status	refutes or corrects [refuted] ×2. Corrections: 2 item(s), see below.
Defining equation	Book equation 0.12a.
Assumptions and scope	The equivalence x related to x’ when their difference lies in the kernel of the read operator is exact for an affine consumer with a fixed output geometry. Three things are kept apart. Exact equivalence, that the consumer’s outputs agree under its output metric. A local null direction, in the kernel of the operator at one row. A workload-common null direction, in the kernel of the workload average, unread at almost every row. For a nonlinear consumer the third does not deliver the first, and cosine similarity is the case, whose radial direction turns with the row. An invertible transform identifies no two rows and forms no quotient. It changes the geometry a reader sees. Discretization changes the quotient.
Prior art	The information bottleneck's view of relevant information, what the consumer does not read may be discarded, stated for a fixed consumer without a variational objective.
Evidence	`geometric-observation/claims/LEDGER.md:67`, `geometric-observation/claims/LEDGER.md:99`, `lean/DataMiningAsObservation/ReadOperator.lean`
Reviewed	semantic review 2026-09-06; generated 2026-09-10 from records at the commits on the provenance page.

What the reader cannot tell apart.

Equation

Book equation 0.12a.

\[x\sim_C x'\quad\Longleftrightarrow\quad d_G\big(C(x),C(x')\big)=0.\]

Conditions

The equivalence x related to x’ when their difference lies in the kernel of the read operator is exact for an affine consumer with a fixed output geometry.
Three things are kept apart. Exact equivalence, that the consumer’s outputs agree under its output metric. A local null direction, in the kernel of the operator at one row. A workload-common null direction, in the kernel of the workload average, unread at almost every row. For a nonlinear consumer the third does not deliver the first, and cosine similarity is the case, whose radial direction turns with the row.
An invertible transform identifies no two rows and forms no quotient. It changes the geometry a reader sees. Discretization changes the quotient.

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

Ledger

refutes or corrects. GO-5 [refuted]. An α=1 density/hubness quotient restores invariant fidelity in ≥1 non-spectral domain. geometric-observation/claims/LEDGER.md:67.
refutes or corrects. NEG-6 [refuted]. Relative per-channel-demeaned error norm is the quotient-tangential quantity that controls softmax-KL. geometric-observation/claims/LEDGER.md:99.

First stated

Volume 14, chapter 5, geometric-observation/chapters/ch05_the_read_metric_and_the_quotient.md:78-90, DOI 10.5281/zenodo.21776291, and chapter 3 section 3.1 of Data Mining as Observation, where every similarity measure induces a distance on some quotient.

Measurements

none

Failures and corrections

GO-5, [refuted]. An α=1 density/hubness quotient restores invariant fidelity in ≥1 non-spectral domain. geometric-observation/claims/LEDGER.md:67.
NEG-6, [refuted]. Relative per-channel-demeaned error norm is the quotient-tangential quantity that controls softmax-KL. geometric-observation/claims/LEDGER.md:99.

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/ReadOperator.lean, theorems rank_one_reads_one_direction, readOp_mulVec, quad_readOp, quad_readOp_nonneg, readOp_mulVec_eq_zero_iff, readOp_diag, readOp_offdiag, readOp_symm, readOp_neg, affine_const_along_nuisance, readOp_affine, readOp_sqLength_basis, 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, 5, 6, 8, 9, 11, 12.

read operator; identity reader; the flip; 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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