is there a rule to determine whether a word is a closed compound word

Solution 1:

is there a specific rule for closed compound words?

Not a reliable one, and your question gives the reason:

Initially, they may not have been joined together, but over time, many of them have come to be accepted as “real words”

This indicates that the language is evolving and, at any given time, there will be compound nouns, hyphenated-nouns and, quite separately, two nouns.

The transition to universal acceptance of the final stage "hyphenatedword" is usually quite protracted.

We have examples in inkwell , ink well , ink-well - all accepted

*iceaxe , ice axe , ice-axe - iceaxe not accepted but may be in the future.

busstop , bus stop , bus-stop - all accepted.

This is helpful guidance although you will realise that there are exceptions:

Solution 2:

One feature that can help in recognizing "closed compounds" in print is that they sometimes create clusters of consonants you do not encounter, or rarely encounter, in a "single" word: downward, withdrawn, hardware, newsletter, overcome, overthrow

Solution 3:

Many of them are wrong. A few examples:

• The "ship" in hardship is a suffix, so that's not a compound word.
• "Inter" means "bury," and it has nothing to do with the words in its row. ("Inte" is even harder to explain, as it doesn't appear in my dictionaries.)
• Finally, the some of the words starting with "beau" originated as compounds of "beau" and something else, but they were formed as compounds in another language entirely. Furthermore, most of those words are proper nouns.

It's not enough to identify words that can be formed by concatenating the letters of two other words if your goal is to identify words that were actually formed by joining two other words. In fact, there are some compound words that aren't spelled by concatenating the letters of their components. An example is (one spelling of) donut.