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I would like to know if it is possible to "publish" in arXiv a preprint non-intended for publication with a journal template?

Example: make an article with AMS Article template (of JAMS) submitting it to arXiv and never submitting it to JAMS?

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For all intents and purposes, I can see no way of distinguishing between a preprint that is not intended to be submitted and a preprint that just hasn't been submitted yet. Thus, if a template is useful to you for formatting, by all means use it.

One caveat, however: some templates also include statements of copyright or material like a journal's logo that do not apply to a paper that has not yet been published. Whether or not you intend to submit, I think it would be poor form to publish such things in a preprint to which they do not apply.

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