The whole thing is an explanation! Just the GD-SCHL part, then.
Gödel is based on the historical figure of Kurt Gödel, who was a famous mathematician and logician. IRL, he contributed to a wide variety of fields, including formal logic, cosmology, set theory, etc. His most celebrated effort was his Incompleteness Theorem(s), which in overly simple terms, states that for a given set of axiomatic assumptions, there will necessarily be true statements that can not be proven to be true, using that same system; that there will always be statements that are true, but that are unprovable. It also proves that such a system cannot independently demonstrate its own consistency. In this fiction, he is recruited by the Authority for his preeminence in his fields of academics. (That he ends up working on a mathematical proof of God is deliberate irony.)
He is recruited really for the sole purpose of helping address a mathematical anomaly, and this puts him in contact and side-by-side in the field with GD-SCHL, Father Gavrilo, who is also an apt mathematician. Given impressive work there, Gavrilo sets him up with other research projects with his letter of recommendation. He ends up being selected to work on antimemes and their mathematical representations.
The gist here is that while Gödel is doing research on antimemes, the Authority is conducting an unwitting drug trial on him & his research team; studying the first use of a proto-videric pharmaceutical formulation, called abcixicin ("ab · sik · suh · sin"). The Authority is following up on initial reports and documented cases of anomalies involving the raw ingredient and raw extract of videric-like compounds (the original chemical precursor; whatever that might be is not covered in this article). They are curious to see how it might augment a research effort that had been stagnated for many years. So, they covertly administered abcixicin to Gödel and his team by dissolving it in the local's village's water supply (which also supplied Site-074). (They mainly ingest this in the form of copious alcohol!)
They felt antimemetics would be a fitting topic to see what effect the proto-videric would have, given that antimemes are "hidden" from experience. (The Authority set up another group of non-videric-medicated antimemetic researchers at another site as a control for the experiment, but that is out of the scope of this article.) Gödel's success meant a success for the Authority too. They then move to fund the manufacture of abcixicin, given the promising results, but soon after, the delayed, slow-punch effects of mathematically stumbling upon God eventually surface in the team's psychological health. Abcixicin is too potent, they made it too strong, and the team saw too much.
The success of the proto-viderics study is called into question, because what is the point of the academic & research edge something like abcixicin can give if it ruins the minds of those doing the work? So the project is halted for these concerns. The team atrophies into either insanity or into rogue cells that attempt to continue the work despite the official cessation. Somewhere in it all, the formula for abcixicin is either lost, stolen, or fractured.
Just prior to the time of Budget Cuts, a discovery is made; a band of vagabond researchers who are working with a chemical formula that is reminiscent of abcixicin. (Whether these researchers are a descendant of the original group or an independent, parallel discovery, isn't known.) So someone, a GD, who was at Gödel's time eager to get the original pharmaceutical production off the ground appeals to GD-SCHL to help advocate for him on a philosophical or theological front, as well as an emotional front, given SCHL's history with Gödel (SCHL didn't know he was signing Gödel up to be a guinea pig).
With Budget Cuts, the GD's are essentially competing with one another for budget allocation, so this anonymous one (this is grounds for disciplinary action after all) redacts information that they believe will be the basis for a powerful argument in favor of funding viderics, and a lucratively-sourced attempt to reverse engineer the original abcixicin for more controlled use. The implication is that this project resulted in the various formulations of Viderics in use by the Authority. Whether abcixicin is recreated is unaddressed.
It should be noted that the author of the note at the end seems to be the person to get The Viderics Department off the ground and is the de facto "leader" of The Visionaries, despite them being established under the purview of The Research Division and GD-SCHL. They are motivated by the high they once experienced from the original abcixicin. More practically though, their intention is to make low-dose viderics a daily supplement to all Authority research staff, so as to give them and the Authority at large an intellectual and so technological edge over competition. However, this does not turn out well, as the addictive properties of the substances quickly become abused by a class of over-zealous researchers. The Authority pulls back from the distribution and prescription in order to help reverse the damage, not dissimilar to the opiod crisis. However, there is some dirt that won't wash off; and these die-hard users of viderics eventually become the self-proclaimed Visionaries — the practitioners of viderics and all that is beyond — how the Authority knows them as in the present day.
In order of appearance:
https://en.wikipedia.org/wiki/Olivier_Messiaen#/media/File:Classe-dukas.jpg (Public Domain)
https://www.flickr.com/photos/aldoaldoz/4815419616 (CC BY-NC-SA 2.0)
https://en.wikipedia.org/wiki/Pierre_Boulez#/media/File:Pierre_Boulez_(1968).jpg (CC BY-SA 3.0 NL)
https://en.wikipedia.org/wiki/Arnold_Schoenberg#/media/File:Arnold_Schoenberg_la_1948.jpg (Copyright Schoenberg Archives at USC, released as free to use if attributed)
https://commons.wikimedia.org/wiki/File:Stravinsky_Igor_Postcard-1910.jpg (Public Domain)
https://www.piqsels.com/en/public-domain-photo-jzxdv (Public Domain)
https://commons.wikimedia.org/wiki/File:BH_Accretion_Disk_Sim_360_Continuous.gif (Screencapture of, Public Domain)
https://en.wikipedia.org/wiki/Equilateral_triangle#/media/File:Triangle.Equilateral.svg (Public Domain)
https://www.piqsels.com/en/public-domain-photo-smwhi (Public Domain)
https://commons.wikimedia.org/wiki/File:SLNSW_11560_Military_officers_lecture_to_exservicemen.jpg (Public Domain)
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