read distortion concept

Equation

none

Conditions

• Exact, with no centering, for a fixed read operator, which is the affine case. For a nonlinear consumer whose error depends on the row it is a first-order factorized surrogate, exact when the local operator and the error’s outer product are uncorrelated across rows.
• The second-moment matrix of the error, not its covariance, unless the error is centered by construction.
• The trace of the second moment is the mean squared vector norm, which is the dimension times the per-coordinate mean squared error.

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

Ledger

none

First stated

Volume 14, chapter 5, geometric-observation/chapters/ch05_the_read_metric_and_the_quotient.md:49-75, DOI 10.5281/zenodo.21776291, and chapter 2 of the same volume for the failure of observer-free measurement.

Measurements

Failures and corrections

none

Invariance envelope

Survived.

• OD:closure-geometry, . Claim: Third version: for a filtered two-dimensional flow with the spectral cutoff as consumer and budget, at closure rank 64 and above, the subfilter modes of largest read distortion (sensitivity times energy, the read operator probed blind on the solver) close the resolved tendency better than the same number of energy-ranked modes, pooled at every in-scope rank by a declared margin, behind in no cell by more than a declared tolerance, with a smaller margin below the scope and a margin stable between n = 128 and 192. experiments/DISCOVERY-TRACK.md, D7v3 record; experiments/OD/D7v3/grade.json. Witness: 48 in-scope rank cells on three fresh fields at n = 128 and 192: the read-distortion closure ahead of the energy closure pooled by 2.8 percent at rank 64 and 10.0 at rank 128 (bar 2), behind in 6 cells by at most 3.6 (bar 5), ahead by -0.2 and 0.4 percent below the scope; the in-scope margin 6.3 percent at n = 128 and 6.6 at n = 192 (bar 2.0 on the change); every structured closure below random; the eigen-direction closure behind energy in 40 of 48 cells; the read operator’s effective rank the same at both resolutions. Absorbed by: . Revision: none; the declared arm of D7 (the leading eigen-directions) is refuted in all three registrations and the read distortion is the ranking quantity inside the rank scope.

Boundary measured.

• OD:closure-geometry, . Claim: For a filtered two-dimensional flow with the spectral cutoff as consumer and budget, the read operator of the resolved tendency with respect to the subfilter state, probed blind on the solver, has leading eigen-directions on which a closure predicts the resolved tendency better than the same number of energy-ranked modes; and the modes of largest read distortion (sensitivity times energy) close better than the energy ranking by a pooled margin at every rank. experiments/DISCOVERY-TRACK.md, D7 record; experiments/OD/D7/grade.json. Boundary: 96 rank cells on three fields at n = 96 and 128, cutoffs 8 and 16, ranks 16 to 128: the eigen-direction closure behind the energy closure in 87 cells, median 15 percent, up to 53; the read-distortion ranking ahead of energy pooled by 0.4, 0.7, 1.4 and 6.1 percent at ranks 16, 32, 64 and 128 against a margin of 1, never behind by more than 7.1 against a tolerance of 13; the margin 0.016 at n = 96 and 0.026 at n = 128; every structured closure below random; the full-rank linear read leaving 3.2 percent (median) to 9.2 percent of the subgrid term at cutoff 8 and nothing at 16. Witness: the leading eigen-directions of the read operator rank subfilter modes by the resolved dynamics’ response alone, and the flow’s energy is elsewhere; the read distortion, response times energy, is the quantity that ranks, and it beats energy by an amount that grows with rank and cutoff, which is where a closure keeps enough modes for sensitivity to separate them. Absorbed by: declaration. Revision: none registered; the track’s declared order ends with this gate.
• OD:closure-geometry, . Claim: Second version, the rehabilitation of D7: for a filtered two-dimensional flow with the spectral cutoff as consumer and budget, at closure rank 64 and above, the subfilter modes of largest read distortion (sensitivity times energy, the read operator probed blind on the solver) close the resolved tendency better than the same number of energy-ranked modes, pooled at every in-scope rank by a declared margin, behind in no cell by more than a declared tolerance, with a smaller margin below the scope and a margin stable in resolution. experiments/DISCOVERY-TRACK.md, D7v2 record; experiments/OD/D7v2/grade.json. Boundary: 48 in-scope rank cells on three fresh fields at n = 96 and 128, cutoffs 8 and 16, ranks 64 and 128: the read-distortion closure ahead of the energy closure pooled by 2.1 and 8.1 percent (bar 2), behind in 9 cells by at most 14.1 (bar 15), ahead by 0.0 and -0.5 percent at ranks 16 and 32 below the scope; the in-scope margin 6.5 percent at n = 96 and 3.7 at n = 128 against an allowed change of 2.0; every structured closure below random; the eigen-direction closure behind energy in 43 of 48 cells by a median of 19 percent; the full-rank linear read leaving 7.6 percent (median) to 10.5 of the subgrid term at cutoff 8. Witness: sensitivity separates subfilter modes only where the closure has a choice among modes of comparable energy, at rank 64 and above, and there it is worth 2 to 8 percent of the closure error; below that rank both rankings keep the same modes; the margin’s size fell with resolution on this draw where it rose on D7’s, and at this ladder it is not a number. Absorbed by: declaration. Revision: a finer resolution ladder (n = 128 against 192) or the cutoff scaled with the resolution would say whether the margin settles; not registered.

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.

Used in

Data Mining as Observation primer L, chapters 1, 2, 4.

Related

the flip; alignment; identity reader; quotient.

See also

Book equations stated beside the entry’s terms, not defining it: 0.10, 1.2, 4.1.

Ledger rows that cite the entry’s records without naming it: GO-2 (pos. half: consumer-projected covariance controls).

Sources-table rows that share a record with the entry without naming it: chapter 1 section 1.4, chapter 2 section 2.2, chapter 4 section 4.3.

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.