distortion concept

Definition	The error a code costs a reader, the trace of the read operator against the error covariance. For the identity reader it is the reconstruction error. Chapter 4 section 4.2.
Example	Two codes with the same total error 2.0 cost a reader at 15 degrees 0.394 and 1.606, four to one.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `distortion`, kind concept.
Status	no ledger row names this entry. Corrections: none recorded.
Defining equation	Book equation 4.2.
Assumptions and scope	The error a code costs a reader, the trace of the read operator against the error covariance. For the identity reader it is the reconstruction error, and for a reader with one direction it is the error along that direction alone, so two codes with equal reconstruction error can differ four to one for a reader at fifteen degrees. At matched bits the downstream preservation is not controlled by reconstruction and is controlled by the consumer-projected error, twelve of twelve, and read distortion controls but is not a complete rank statistic.
Prior art	none recorded
Evidence	`geometric-observation/chapters/ch05_the_read_metric_and_the_quotient.md:49-75`, `lean/DataMiningAsObservation/ReadDistortion.lean`, `lean/DataMiningAsObservation/Quantization.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

The error a code costs a reader, which two readers rank differently.

Equation

Book equation 4.2.

\[D(b)=\sum_i s_i\sigma_i^{2}\,2^{-2b_i},\qquad s_i=v_i^{\top}P_C\,v_i,\qquad b_i^{\star}=\max\!\Big(0,\ \tfrac12\log_2\frac{s_i\sigma_i^{2}}{\theta}\Big),\qquad \sum_i b_i^{\star}=B,\]

Conditions

The error a code costs a reader, the trace of the read operator against the error covariance. For the identity reader it is the reconstruction error, and for a reader with one direction it is the error along that direction alone, so two codes with equal reconstruction error can differ four to one for a reader at fifteen degrees.
At matched bits the downstream preservation is not controlled by reconstruction and is controlled by the consumer-projected error, twelve of twelve, and read distortion controls but is not a complete rank statistic.

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

Ledger

none

First stated

Chapter 1 section 1.4 and chapter 4 section 4.2 of Data Mining as Observation, with the read distortion in geometric-observation/chapters/ch05_the_read_metric_and_the_quotient.md:49.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 2 section 2.2	read distortion controls but is not a complete rank statistic, twelve of twelve, one middle pair misordered, NEG-9	`geometric-observation/chapters/ch05_the_read_metric_and_the_quotient.md:49-75`
chapter 3 section 3.2	traces 2.0, diagonals 0.3 and 1.7, consumers at 15 and 75 degrees, distortion 0.39 vs 1.61, 4.1 to one, computed not drawn	`observation-theory\assets\make_flip_figure.py:4-24,107-119`

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/ReadDistortion.lean, theorems read_distortion, identity_reader, quad_one, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

lean/DataMiningAsObservation/Quantization.lean, theorems error_le_half_step, quantize_level, half_step_bound, sq_error_le, finer_step, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation primer L, chapters 0, 1, 2, 3, 4, 7, 8, 9, 11, 13.

read distortion; reconstruction error; identity reader; quantization.

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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