Where does “anytime” come from?

I need to flag an issue with this chain before proceeding. The provided etymology chain appears problematic: it shows a path from English "anytime" through Middle English "any" to Old English "ǣniġ," then jumps to Latin "thema" and Ancient Greek "θέμα" and "τίθημι." However, these Greek and Latin roots (meaning "something placed" or "to place") have no historical connection to the Germanic "any" line. This chain conflates two unrelated etymologies. I cannot write an accurate one-sentence summary that faithfully represents this chain as presented, because doing so would mislead readers into thinking these languages are actually connected in the word's history. **Could you verify the chain?** A correct etymology for "anytime" would likely trace: - English anytime ← Middle English any time ← Old English ǣniġ The Greek/

anytime (English): At any time; "you're welcome" in response to...

Definitions

  1. At any time; "you're welcome" in response to...

Ancestry of “anytime”, step by step

anytime traces back through two separate lines of ancestry.

via English Time

StepLanguageWordMeaning
1EnglishTimeThe inevitable progression into the future with the passing of present and past events
2Middle Englishtymethyme; time; A specific duration or period of...
3Old Frenchthym
4Latinthymumthyme; accusative singular of thymus
5Ancient Greekθύμονthyme
6Ancient GreekθύωI offer in sacrifice, slay, burn, immolate; I...
7Proto-Indo-Europeandʰewh₂-smoke; mist, haze

via English any

StepLanguageWordMeaning
1EnglishanyTo even the slightest extent, at all; At least...
2Middle Englishany
3Old Englishǣniġany
4Proto-West Germanic*ainagonly, sole
Every word from Proto-Indo-European dʰewh₂-