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I'm not sure why I'm still so hung up on this. For context, I kinda fell out of theorycrafting a while ago — at this point, I've seen every shred of evidence supporting or opposing every possible theory, and yet I feel no closer to any concrete conclusions on the game's lore. There are just too many questions, too many possibilities.
Then the secret Valentine dropped. This letter raised a lot of questions I was eager to discuss! Instead, I was disappointed to find the discussion dominated by something I didn't even think to question: the writer's identity. This frustrated me a bit, as I felt that, for once, the conclusion was actually pretty clear-cut; it didn't strike me as something Toby Fox even intended for us to debate.
At the time, this drove me to write a post discussing it (don't bother reading it, this post is better). I thought that would be the end of it for me, but somehow the debate has yet to leave my mind. Even now, from what I've seen, people are too busy arguing about who wrote the letter to discuss what's in it. All the while, I've not only grown more confident in the conclusion I've reached, but I feel more prepared to articulate why. I've also heard more counterarguments since then, which I will address in this post.
I'll start with what we can agree on, before addressing more significant counterarguments and delving into progressively deeper levels of conjecture. Should be fun!

Context and implications

Even before looking at the letter itself, we can see that this is a rare, cryptic secret, already giving us an idea of who might be involved. I'd go so far as to say that Gaster is "rare, cryptic secrets" personified. To me, this is the main thing setting him apart from other characters.
Obviously that's not to say that other characters can't be responsible for secrets like these, but giving the letter a cursory glance, the format doesn't fail us. All caps — this doesn't tell us much on its own, but when used alongside a very, very specific manner of double line spacing between and within sentences, compounded by the aforementioned association with secrets... Toby Fox is clearly trying to tell us something here, so I'm not a fan of any theory that completely discards that.
You'll probably agree with this point — even Gaster Valentine deniers admit that there is an intentional association being drawn. If you disagree... sorry, I don't know what to tell you. When it comes to Gaster, having a mysterious secret with all-caps weirdly-spaced text is basically equivalent to Susie walking onscreen and saying "Hi, I'm Susie". Sure, maybe a plot twist down the line will reveal that this isn't actually Susie... But you'd only suspect that if she says something that would imply it, and for now, we're only looking at the presentation.
There's another association I haven't seen as many people bring up: the fact that the letter is anonymous. Ironically, "not being confirmed to be Gaster" is one of Gaster's identifying characteristics. In every case, we're left to assume his involvement from cues such as those we see here: secrets, crypticism, capitalization, spacing, all that. Anonymity is his signature. By including these quirks and leaving it uncredited, he may as well be signing "GASTER" in flashing letters. And that's not even getting into the implication of Wingdings and the letter disappearing after being read, which are both Gastery as all get out.
A couple minor notes regarding the writer's anonymity:

• The lack of a telltale 666 motif or gratuitous "VERY, VERY" could be seen as a point of contention, but I think this naturally follows the pattern we've seen thus far: as we grow more familiar with Gaster, there's less need for these kinds of identifying motifs. In Undertale, the name Gaster is directly associated with 666 and Wingdings, but as we already know these connections going into Deltarune, the game's intro more-or-less drops the name and font associations.
• Notably, this is the only Valentine without a confirmed sender — if it was meant to tease a new character (à la Lanino and Elnina), why not include a visual or first initial to indicate that? Because it's supposed to seem like Gaster, only to end up a red herring...? Seems like a pretty cheap twist to me. Characters have deceived us in the games themselves, but we've generally been able to take supplemental content more-or-less at face value.

Of course, that anonymity is a double-edged sword. It implies Gaster's involvement just as much as it leaves room for doubt. If Toriel does something un-Toriel-like, we simply have to reconcile what we previously knew of her character with what we do now, whereas if Gaster does something un-Gaster-like, it calls his entire identity into question. I still don't think the Gaster associations can simply be handwaved away — again, even Gaster Valentine deniers agree that there's some significance to the similarities — but, by his very nature, it's nothing more than an implication.
If the voice from the vessel creation sequence says something to the effect of:
https://preview.redd.it/lsfu4bxrqn0d1.png?width=514&format=png&auto=webp&s=14867b9e9ea46dfc4f7c7fc91de69c574cab530d
I'll be the first to admit that it probably isn't Gaster. But I don't think that's the case here, and to discuss why, we'll have to move on from the context to the content.

Personality and mannerisms

The obvious problem with analyzing Gaster is that he has yet to be properly introduced, much less developed. We've only ever interacted with him outside Deltarune's story and world, and furthermore only briefly, within a very narrow range of contexts. Additionally, these interactions are written to provide us with as little characterization as possible — he is succinct and direct, never shifting the subject beyond what is relevant to us. This itself could be considered characterization, but without the "why", there's not much to glean from it.
Regardless of whether this trait is dependent on context (there's little room for conversation in a survey program, after all) or is simply an ever-present aspect of his personality, I'd argue it carries over to the letter. He tries to begin with polite small talk, but each topic he broaches is swiftly dismissed; once again, he can hardly bring himself to deviate from "the purpose of the message". Speaking of which, that fact — that this is the only Valentine with an explicit "purpose" — itself provides characterization. Almost as though this character would only contact us for an important reason (perhaps a character with a history of doing so, often outside the game). Almost as though we've met this character before (otherwise, I suspect Toby would've focused exclusively on characterization rather than motives, as with Lanino and Elnina).
I realize I'm getting into full speculation territory now, but hopefully I've made it clear why it's necessary to do so. Gaster's personality and motives are largely up to personal interpretation — I think we can agree that, were that not the case, the debate would be a lot more one-sided. Your interpretation of the character can't be used as conclusive evidence for your theory. Of course, I'm not exempt from this either! My interpretation could be completely off-base as well. That said, I'd like to at least explain why it all lines up in my head. (Feel free to compare my interpretation with his dialogue, which I've compiled here: https://pastebin.com/yR5Y8qhw)
Let's get the specific shared mannerisms out of the way before moving onto the more general similarities. Specifically:

• Gaster employs a specific kind of repetition, mostly limited to single words, but occasionally multiple ("OF COURSE", "SHALL WE", "THE SECOND"). We see this reflected in the letter ("BELIEVE IT SO", "WANT[ED] TO HELP", "SEEM TO HAVE FORGOTTEN").
• He alternates between present and past tense when describing what is currently happening (seen throughout the vessel creation and save menu text). The letter writer does this multiple times.
• Perhaps a generic word choice, but the writer says "HOW ABSURD", much like how Gaster has said "HOW WONDERFUL", "HOW INTERESTING", and "HOW LONG IT HAS BEEN".

There are a couple other things that struck me as extremely Gastery, but I couldn't find many examples for them. I might just be thinking of the stilted way Toby Fox writes in status updates and newsletters, which reminds me more of Gaster than any other character... but that's a tenuous connection at best, so take these for whatever they're worth:

• The letter writer puts "VALENTINE'S DAY" in quotes. I thought there were more examples of Gaster putting random terms in quotes (it suits his general robotic-yet-whimsical demeanor), but the only ones seem to be the names you enter in the vessel creation.
• I find the phrasing "THE PURPOSE OF THE MESSAGE" (as opposed to, say, "the reason I wrote this to you") very Gastery — something about the detached feeling and repeated use of "THE" — but the only instance I could find of this particular sentence construction was "CHOOSE THE TARGET FOR THE REFLECTION", so maybe it doesn't mean much.

Moving onto the general personality, the easiest similarity to argue is the metaknowledge. Gaster introduces you to the Deltarune program, and is the only character known to acknowledge it. The letter writer is clearly interested in your thoughts on Deltarune (albeit spelled "DELTA RUNE"; beyond Toby's general inconsistency with minor details like these, I'm not sure what to make of that). The writer also acknowledges the wait between chapters, much like Gaster does — contrasting this, everyone else simply acknowledges the time that's passed within the story, perceiving two real-world years as one night. It could also be argued that Gaster is the only character who has contacted us, the player, directly.
Gaster is also known to be polite and formal, always offering greetings ("WELCOME"), gratitude ("THANK YOU FOR WAITING SO LONG"), and compliments ("YOU HAVE CREATED A WONDERFUL FORM"). We see this reflected quite well in the letter — "AS YOU ARE WAITING PATIENTLY" and "DO YOU BELIEVE IT SO?" certainly don't feel like the words of someone speaking casually. Like Gaster, the writer also uses fewer exclamation marks and contractions than most. Granted, the fact that the writer uses contractions at all is cause for suspicion... but, since they're outnumbered by the uncontracted phrases, contraction usage remains closer to Gaster than anyone else (except Toriel, I guess). Although we haven't seen Gaster use contractions in his few prior instances of dialogue... evidently, he does use them, albeit infrequently.
The formality even goes beyond word choice — his repeated backtracking on the exact order of "PUT ON YOUR COAT AND WASH YOUR FACE" brings to mind his polite accommodation of our choices in the vessel creation or save menu. It doesn't seem like him to forcefully tell us what to do, or in what order to do it, and the letter's ending serves to exaggerate that quality. He may not use any highly technical language here, but that makes sense in this less scientific context (especially when you consider that he doesn't use scientific terminology all that often anyway).
Gaster's constant emphasis on the subject at hand, saying something odd or outright wrong whenever he alludes to anything else (listing the wrong options for whatever he means by "FAVORITE BLOOD TYPE", or listing such favorite flavors as "PAIN" and "COLD") — it's always given me the impression that Gaster's transcendent brilliance doesn't extend far beyond his own work. That's not to say he's stupid, far from it, but definitely a bit "out there" (I feel this much is apparent from the multiple typing quirks he uses). At the very least it feels inarguable to me that, while Gaster is smart, he doesn't know everything, and (like any Toby Fox character) he isn't always intended to be taken 100% seriously.
Casting our gaze over to the letter, this once again checks out. Confusing the new year with the old year, considering himself to be the person he's forgotten, mixing up the recipient's face and coat — these feel more-or-less like a natural development of Gaster's established eccentricity, like the kinds of weird things that would only make sense from his unique, potentially fractured, extradimensional perspective. Sure, "THE TIME IS GOING AROUND" may not make sense to us mortal beings, but neither does "DELTARUNE GLOWS BRIGHTLY FROM YOUR HOPE". Ending a message with the archaic "GOOD BY" feels roughly equivalent to randomly putting [24] in brackets.
The increased emphasis on these quirks makes sense in this more casual context, and adds tonal consistency with the other Valentines — being a spooky creepypasta character is hard work, and I think the guy's earned a day off. The friendliness also makes it out to seem like the writer has communicated with us in the past, which wouldn't make sense for anyone other than Gaster, and certainly wouldn't have made sense in something like the vessel creation sequence. I find that it also makes sense from an extradiagetic perspective — Toby Fox is just providing characterization here, not announcing a new chapter, so there's less reason to write Gaster with as much purpose and brevity as in previous appearances.
Basically, the idea that Gaster is completely serious and grounded while the letter writer is completely goofy... I don't get it. I really don't think either of those things are true. I'm not even sure what I would change about the letter to make it sound more like Gaster... Did you expect less exclamation marks? More line breaks (which would make it overly long and monotonous, might I add)? Less whimsical turns of phrase? At that point the letter wouldn't deepen our understanding of the character at all, and I fail to see what the point would be.
And the idea that the letter ruined Gaster's character makes even less sense to me — for one, what does this letter really establish about Gaster that wasn't already alluded to? I suppose the letter makes him out to be more whimsical than most of us thought, but how is that a downgrade from "static, one-dimensional robotic scientist"? Can "whimsical scatterbrain" and "robotic scientist" not coexist? Is the gradual reveal of depth and contrasts not at the heart of most Toby Fox characters?? The directness of his speech has made him feel more like a plot device up to this point, so this letter was the first time I truly felt invested in Gaster as a character outside of his unique presentation. I don't think "saying things in a casual context that could be construed as humorous" (again, especially when he already says things like "FAVORITE BLOOD TYPE") is enough to consider him "scrunkly goober #78" — and even if it was, Toby has a way of delivering the most emotionally impactful moments through the funniest characters, so I'm not particularly worried.
All that to say: it's my belief that, even looking past the context and format, the letter aligns more closely with Gaster than any other character. Who else has such a formal and cordial composure contrasted against an air of inscrutable strangeness?

Motives and memory

Gaster hasn't exactly been transparent about his goals, and the letter writer isn't giving us a whole lot to work with either, so we're going to have to get even more speculative here. That said, the conclusion I've arrived at makes a lot of sense to me, relates to established elements of Deltarune, and even explains away some people's reasons for this not being Gaster. Those being:

1. "Gaster asks us for help, even though we've already been helping him!" This is easily explained if we assume he's asking our help with something else this time. The more casual tone and secrecy of the letter supports the interpretation that this is a less generally important, more personal matter; it's the difference between "Hey, if you don't mind, would you lend me a hand with this thing that's been on my mind?" and "I'm subjecting you to an experiment I've spent years preparing, please follow these exact instructions." I suppose it's strange that he doesn't mention how we can help, but I imagine that's something that will become clear in the future.
2. "Gaster isn't forgetful!" While I'd say it's entirely possible that Gaster has memory problems that simply hadn't been alluded to yet, I find the more likely explanation to be that there's something supernatural at play, and once again I believe this is hinted at in the letter itself. Gaster says it's "IRONIC" that he forgot something — what could this imply, beyond the fact that he was forgotten himself? (To recap the theory, Goner Kid mentions a world where they don't exist, no one acknowledges Gaster outside Fun events, and while Asgore is said to have taken a long time replacing Gaster, it's unconfirmed whether he remembered who he was replacing.) I was 50/50 on this theory myself, but I take this letter to be more-or-less confirmation of it, further tying it to Gaster. The only other explanation I can think of for this "IRONIC" line is that perhaps Gaster makes people forget things...? For the purposes of this theory, it doesn't matter too much, as it establishes a precedence for supernatural memory loss either way. Furthermore, by suggesting the person he's forgotten may be himself, he draws a direct parallel between himself and the person he's helping, lending further credence to this interpretation.

This brings us to my theory. I posit that the forgotten character is someone in similar circumstances to Gaster himself; someone who cannot be found in the story, and is instead associated with secrets outside of it (much like this letter). As an added bonus, it would help strengthen the theory if it happened to be a character who is known to call out for help, and who we've previously been requested to find (particularly in secret material outside of the game, much like this letter). If only there was a character fitting all of those criteria...
But this post isn't about that. My point is, while there's not much to glean in the way of connections between the letter person's goals and Gaster's, I don't think there's anything contradicting such connections either.
https://preview.redd.it/gswysspqtn0d1.png?width=392&format=png&auto=webp&s=248273e7bacc131cf9e9422a0aebdaeeacc6c9b6
Oh. Right, I guess there's that. If this is the same Gaster we've worked with before, why doesn't he say "YOU HAVE PROVEN YOURSELF TO BE RELIABLE"? I feel like this minor word choice could be justified by any number of explanations. For one, the phrasing makes a bit more sense within the full context: basically "you're odd, but you seem reliable regardless". You could also argue that, since he's asking something different of us this time, we haven't proven ourselves to be reliable for this specific task. Or that since we've only been assisting in the "Deltarune" project for 2 of 7 chapters, that's not enough time for him to fully consider us reliable. Or, maybe the only reason we do seem reliable to him is because we've been assisting him — otherwise we wouldn't seem reliable at all. Heck, it could just be odd phrasing for a character who constantly uses odd phrasing, only misleading under a specific interpretation.

Translation

To recap, I think the context and format convey a very clear implication — one that could be a red herring, but that I believe is only supported by the writer's personality, and (at the very least) not contradicted by their goals. Here is where I believe we come to the first hole in my argument.
The Japanese translation doesn't sound particularly close to how Gaster speaks in Japanese. I don't speak Japanese, so there's not much I can do to back up or debunk this claim — perhaps people are overlooking some of the more minor similarities, as with the English version...? Or maybe there's more nuance to the translation process than most people think, and the writing style is highly adaptive to tone, mood, context, or method of delivery, or affected by cultural differences...? I don't know, so for now I can do nothing but take it at face value, and consider what this inconsistency might imply.
People seem divided into two camps with regards to the translation: "the English version very clearly sounds like Gaster, therefore the Japanese version doesn't matter" and "the Japanese version very clearly doesn't sound like Gaster, therefore the English version doesn't matter". Frustratingly, neither of these actually address the inconsistency. If we assume it isn't Gaster, the English version clearly goes out of its way to mislead us into thinking it is (as I've already justified extensively), so why not do the same for the Japanese version? If Toby wanted to clear up ambiguity by making the Japanese version distinct from Gaster, why not do the same for the English version? Whichever way you slice it, it's a contradiction.
That said, I think there are a few things working in my favor here. For one, English is Toby's native language, as well as the most common language spoken by his fanbase. Japanese translation is handled by a different team under Toby's supervision; they had a lot of Valentines to translate, and likely a pretty strict deadline. It's believable that time constraints forced the team to prioritize accuracy to the content and tone of the letter over consistency with established text quirks.
There's also the fact that the English version is the only one to imply a clear sender (unless you want to reach and say the secrecy and metaknowledge of the Japanese version imply Gaster as well); personally, I feel inclined to believe a deliberate implication over the lack of one. Adding weight to this point, this isn't generally the sort of thing casual fans are going to speculate about — anyone who knows about the letter has likely heard of both versions, which means most will gravitate toward whichever one implies a specific identity. Because of this, the Gaster interpretation seems to prevail even in the Japanese fanbase.
Basically, I don't like that making sense of the letter forces me to either make up an arbitrary narrative explanation for the discrepancy, or to write off either the English or Japanese version as unreliable... However, this decision is made much easier by the fact that, the way I see it, only one of them provides mounds of evidence (from the context to the specific format to the anonymity to the metaknowledge to the formality to the strange mannerisms and so on) pointing toward a single interpretation.

Alternative explanations

I've seen a few other theories regarding the writer's identity, the most common of which being the idea that Gaster was shattered into multiple personalities, and that this is a different "shard" from the one we've communicated in the past — I take issue with any theory that compartmentalizes one complicated character into multiple simple ones, but I suppose it doesn't necessarily have to be done that way. There are also theories that this is the Chapter 3/4 secret boss (associated with Gaster based on precedent), Mike (associated with Gaster through Spamton), or IMAGE_FRIEND (associated with Gaster via filename conventions), some of whom may or may not be the same person.
My main argument against these theories is that I simply don't think they're necessary. To reiterate, I think the whimsy Gaster displays here is consistent with his previous characterization (we've only seen him in scientific contexts until now, and even then a bit of whimsy manages to slip through), and the forgetfulness can easily be explained as well (since it only seems to apply to one subject). Again, the only hole I can find in my interpretation is the Japanese version, and none of these theories really explain the inconsistency there, leaving us back at square one — if it's merely a character associated with Gaster as opposed to the man himself, why aren't the similarities and differences roughly the same in both versions?
These theories don't make much sense to me when I attempt to look at them from Toby Fox's perspective, either. Getting a secret Valentine from Gaster is, technically speaking, like the coolest thing ever, and I think he realizes this. If he's willing to include a secret letter with this much lore in it, why leave out such an important character we've already communicated with? And furthermore, why give a different mysterious character so many of the same characteristics we use to identify Gaster in the absence of his appearance or name?
Also, isn't it telling that the debate seems split evenly between "Gaster" and "a theoretically infinite supply of basically made-up Gaster-adjacent characters"? While these theories could end up being accurate, I don't yet see any precedence for either Gaster having multiple personalities, nor any as-of-yet unseen/nonverbal characters sounding like the letter writer; these theories simply exist to explain a contradiction that I believe isn't truly there. They embody the principle of explosion, or "from contradiction, anything follows" — if you accept that it sounds both "like Gaster" and "unlike Gaster", you can make up anything in between these possibilities, an untouchable theory that can neither be proven by evidence or disproven by counterevidence. I personally don't find that line of thought very compelling.
In the most popular video on the topic, SpookyDood analyzes the letter through cadence, language, context, and function; however, I think something is lost from analyzing these aspects separately. Cadence, language, and function are largely dependent on context. If we accept the letter into that pool of Gaster characterization rather than nitpicking it into oblivion, we simply see new patterns emerge. We see that Gaster only uses particularly large words when he's referring to something scientific. We see that Gaster is slightly more enthusiastic when there's less pressure to be serious and professional. Basically, I feel that all of the inconsistencies SpookyDood brings up are easily explained by this letter being a different narrative context from the one Gaster has previously occupied (and I presume will largely continue to occupy), which is what I find so interesting about it.
As for cadence, SpookyDood says that "[Gaster's] pattern of each few words being broken up by a line break is no longer present", but the first 3 sentences alone are divided into 6 lines. He even calls out certain sentences in the vessel creation as going against this pattern, proving that it doesn't have to be consistent — while these line breaks do define the flow of Gaster's speech, I think the frequent use of ellipses here achieve the same effect without needlessly extending the length of the image.
The video ultimately posits that the writer is the man behind the tree, and honestly, I don't entirely disagree with that conclusion. The use of "well" checks out, as does the happy mood. In the case of both "DO YOU BELIEVE IT SO?" and "He might be happy to see you. What do you think?", the text moves on without directly responding to our answer. There's also a more loose connection to be made, in that both of these questions involve reality being dependent on our thoughts (whether each day is or is not a day of love, and whether there is or is not a man there), and likewise, dropping the egg causes the narration to act as though the egg was never there. (Obviously the man himself doesn't say any of this, but the association's still there through the narration.)
The connections don't strike me as plentiful or strong enough to stand toe-to-toe with the Gaster interpretation, but without much to go off of regarding the man, it doesn't have to deal with as much counterevidence either. I'd say I have the same problem with this theory as the others I've mentioned (an excessive avoidance of contradiction such that, instead of associating with said contradictions, the letter is attributed to a character we know so little about that it can't be argued against), but there's a fairly reasonable basis for argument here.
Thing is, I don't think these interpretations are mutually exclusive. I was ambivalent toward Gaster and the man (the mystery man, you could say) being one and the same, but the letter has done a lot to sway me toward that conclusion. Strengthening the connections between the letter, the man, and Gaster are the mutual association with forgetting (due to the blog post where Noelle can't remember the name of her egg), secrets, and disappearances. The letter's contrasting of contradictory statements ("NEW YEAR" vs. "OLD YEAR", "WASH YOUR FACE" vs. "WASH YOUR COAT") call to mind both the man ("a man" vs. "not a man", "not too important" vs. "not too unimportant") and the strange someone who corrupted Jevil ("didn't make sense" vs. "didn't not make sense").
If the letter was supposed to be written by the man, but not Gaster, I imagine the letter would have been formatted like this instead:
https://preview.redd.it/igudqtpisn0d1.png?width=2500&format=png&auto=webp&s=0dd63050999a033459f6255ac671a1ac4ad4fcd9

Conclusion

Sorry, that was long. I'm done!