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On the webpage about the projection EPSG:5071 (NAD83(HARN) / Conus Albers), it is said that the accuracy:

Remarks: Accuracy 0.1 to 0.2m in California, 0.05-0.11 in Oregon, elsewhere better than 0.05m.

The WKT says that it is a Albers_Conic_Equal_Area projection. But no projection can preserve distance everywhere. My questions are:

How can a projection achieve sub-meter accuracy over such a large geographical extent?

What exactly does the "accuracy" (of 1 meter or 0.1 meter) referred to in epsg.io mean? Do they mean that distance is only off by no more than a meter in the contiguous US? If not, how much is distance off here in the worst case?

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The accuracy values listed refer to the specified datum/coordinate transformation, not to the projection itself (the page you link to lists other transformations with differing accuracy). It's nothing to do with whether a certain projection is distance, shape or area preserving.

From the EPSG Registry:

Accuracy

Estimate(s) of the impact of this transformation on point accuracy. Gives position error estimates for target coordinates of this coordinate operation, assuming no errors in source coordinates. These accuracy estimates or position error estimates are given as plus or minus NN meters, where NN is the entered value.

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