Q: Should authorship be acknowledged when publishing a preprint?

You have raised some interesting points. Let’s break down your query into parts.

1. Is it possible to acquire authorship for a theory that was submitted to a preprint server. It is possible for others to have public access and develop this theory, right? Then, can this preprint be the basis for claiming that I am the one who provided it in the first place?

Well, you have almost answered your own question! Indeed, the nature of preprints lends itself to allowing the establishment of priority. What you seem to be worried about is “scooping.” Suppose another researcher in the same field publishes a theory inspired by your preprint before you do. You will still be able to publish your paper, because your preprint will have a digital object identifier (DOI) and a date stamp, indicating that you came up with the theory first. In other words, your preprint will serve as the basis for claiming priority for the theory, and you can indicate this to the editor of your target journal. Many journals acknowledge the priority claims of preprints under “scoop protection” policies.

2. I also wanted to know how much a preprint publishing agency costs (When the goal is to publish in more than one place).

I assume that by “preprint publishing agency,” you mean preprint server. Note that there is no cost for posting a preprint. If you mean submitting your paper to an open access journal, you will need to check the article processing charges (APCs) of that journal (mentioned on the journal’s website). It is important to ensure that the journal you intend to submit to accepts preprints; while most journals accept preprint submissions, some may not. You can check the Sherpa Romeo database for this.

You state that you already have submitted the preprint to bioRxiv, Authorea, OSF, etc. Generally, posting on multiple preprint servers is discouraged because an author may find it difficult to keep all versions updated. While the aim for submitting to multiple preprint servers is to increase the exposure of your work, this can be easily done by sharing the preprint with colleagues via email and social media. In future, consider choosing a preprint server that best aligns with your discipline and target readership. Another point to consider is that the preprint server is indexed by major indexing services.

Further, since you mention OSF, we noticed that the OSF website states “Currently, you can only share your preprint on one community preprint server.” You might want to check this requirement when choosing a preprint server.

3. If I register a preprint on OSF, would this be evidence that the research was presented first by me in the future?

A preprint is a permanent and public record of the publication of your scientific work. The evidence of priority is the date of preprint posting, indicated as a timestamp on the preprint.