reconstruction error concept

Definition	The squared length of the difference between a row and its approximation, the identity reader's distortion. Chapter 1 section 1.4. Also reconstruction.
Example	Approximating (3, 4) by (3, 0) costs reconstruction error 16.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `reconstruction-error`, kind concept.
Status	measures [demonstrated] ×2. Corrections: none recorded.
Defining equation	none
Assumptions and scope	The squared length of the difference between a row and its approximation, the identity reader’s distortion. It is nonnegative, zero exactly when the approximation is the row, a sum of per-coordinate squared errors, and for a projection onto a unit direction it is the squared length less the squared component along it. A reconstruction cosine of 0.995 sat beside a perplexity of ten thousand, recalibration improved reconstruction and worsened the consumer, and at matched reconstruction the consumer-aware code won in twelve of twelve domains.
Prior art	none recorded
Evidence	`geometric-observation/claims/LEDGER.md:63`, `geometric-observation/claims/LEDGER.md:68`, `geometric-observation/claims/LEDGER.md`, `geometric-observation/chapters/ch07_cost.md`, `lean/DataMiningAsObservation/ReconstructionError.lean`, `lean/DataMiningAsObservation/ReadDistortion.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

The squared length of the difference between a row and its approximation.

Equation

none

Conditions

The squared length of the difference between a row and its approximation, the identity reader’s distortion. It is nonnegative, zero exactly when the approximation is the row, a sum of per-coordinate squared errors, and for a projection onto a unit direction it is the squared length less the squared component along it.
A reconstruction cosine of 0.995 sat beside a perplexity of ten thousand, recalibration improved reconstruction and worsened the consumer, and at matched reconstruction the consumer-aware code won in twelve of twelve domains.

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

Ledger

measures. GO-2 (neg. half: not reconstruction) [demonstrated]. At matched bits, downstream preservation is not controlled by reconstruction error. geometric-observation/claims/LEDGER.md:63.
measures. GO-6 [demonstrated]. At matched rate, output coding ≤ surrogate ≤ reconstruction on the consumer metric; the output–reconstruction gap is governed by the \(\ker P_C\) entropy share, and the surrogate–output gap vanishes as rate grows. geometric-observation/claims/LEDGER.md:68.

First stated

Chapter 1 section 1.4 and chapter 4 section 4.1 of Data Mining as Observation, with the identity slice in geometric-observation/chapters/ch02_failure_of_observer_free_measurement.md:1-40.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 4 section 4.2	GO-6, d 8 and r 4, output at or below surrogate at or below reconstruction at every rate, about 500 times, gap 0.41 to 0.005, isotropic control collapses	`geometric-observation/claims/LEDGER.md` row GO-6; `geometric-observation/chapters/ch07_cost.md`

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/ReconstructionError.lean, theorems recon_nonneg, recon_eq_zero_iff, recon_sum, recon_proj, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

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.

Used in

Data Mining as Observation primer L, chapters 0, 1, 4, 6, 7, 8, 11, 12, 13.

identity reader; read distortion; distortion; principal component analysis.

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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