Why does pure water dry without leaving spots?

Follow one drop back to where it came from, and it has already picked something up.

The U.S. Geological Survey puts it plainly: whether water "comes out of your kitchen faucet, is in a swimming pool or dog dish, comes out of the ground or falls from the sky, the water will contain significant amounts of dissolved substances, minerals, and chemicals." USGS adds that the most common dissolved substances are salts. Part of that load is what USGS calls hard water, water high in dissolved minerals, largely calcium and magnesium, which groundwater gathers as it moves through soil and rock.

USGS calls hardness "not a health concern, but it can be a nuisance." One household example it gives is glasses taken out of the dishwasher with spots or film on them, which USGS calls hard-water residue.

The next step is what separates a pure water rinse from a tap.

NGA's rule for the pure water technique sets the number: the water used in both wash and rinse "must have a total dissolved solids content (TDS) of 20 parts per million (PPM) or less to prevent spotting and streaking of cleansed surfaces," and "The use of tap water is not acceptable." NGA also says effective treatment by ion exchange and/or reverse osmosis equipment should be used with the delivery and rinse methods at all times. What the TDS count measures is answered separately.

Twenty PPM is a ceiling, and a ceiling is not zero. USGS describes pure water, distilled or deionized, as an excellent insulator, "although even water of this purity can contain ions." The treated rinse has been brought down to NGA's limit, and nothing in NGA's paper says it has been brought to nothing.

A spot is not something water does to glass. It is something water leaves behind on its way out.

Liquid water evaporates and what was dissolved in it does not, so the solids stay on the glass where the drop last sat. That reasoning is built into NGA's own paragraph: the rinse water is generally allowed to evaporate, and therefore the water must be at 20 PPM or less "to prevent spotting and streaking."

By the same reasoning, a rinse at or under the limit still leaves its small load behind. NGA's paper does not say how much residue the eye can see; it sets 20 PPM or less as the level for the pure water technique and leaves the matter there.

The water a rinse starts from decides what it leaves, and NGA names the water it will not accept.

Tap water is not acceptable for the wash or the rinse, in NGA's words, and the glass figure NGA sets is its 20 PPM for the pure water rinse. EPA does publish a 500 mg/L secondary standard for total dissolved solids, but that is a drinking-water guideline, not a glass-cleaning figure, and neither number stands in for the other.

Whether a garden hose belongs on glass at all is answered on whether windows can be cleaned with a garden hose, and the TDS entry defines the reading a meter gives.