Let's talk about whose handwriting this is:
As much as possible, we will be using handwriting samples from 1963-'65 since handwriting changes and we want samples from as close as possible to when the above dedication was written on the valentines card (i.e. February 1964, right before/otw to the Beatles' first Ed Sullivan appearance). I have done handwriting analysis as part of my job, but I am not a handwriting analysis EXPERT and as such take this all with a grain of salt.
Preliminarily, it bears noting that the word "love" appears to be in all caps. We can see that when one letter is upper case and the others are in lower case, as in "Paul," the upper case letter is written larger, whereas "with" which is all lower case, has same sized letters. Similarly, it is clear that L and E in "love" are both capitalized, and since O and V are written in the same size as them, we can assume that "LOVE" is in all caps.
With that established, let's start with John and whether we can really rule him out as some are suggesting.
The best frame of reference for John's 1964 handwriting is of course the If I Fell lyrics themselves, which were definitely written very close to when the dedication was written.
First thing of note is that the way "with" is written in the above lyrics and the dedication are not similar. For one thing, in the dedication the "t" is clearly crossed, while in the lyrics it is not. Here's a closer look:
Having said that, John is inconsistent with lower case Ts in the "If I Fell" lyrics. You can see it side by side here in "heart" and "to":
It would appear he does write curved and crossed lower case Ts (and does so several times in the lyrics). So the dissimilarities between how "with" is written in the If I Fell lyrics and the dedication does not really definitively establish that John did not write the dedication.
John's lower case Ls are also inconsistent, sometimes with large loops, sometimes thin as in the dedication. Thus the lower case L in "Paul" is not all that informative.
On the other hand, the lower case "au" in "cause" in the lyrics is similar to the lower case "au" in "Paul" in the dedication. Of particular note is that the "a" in both appearances of "cause" in the above lyrics has a faded top half as though the press of John's pen on the paper is consistently lighter when writing this part of the letter, or in the alternative when John's in a hurry his lower case cursive As look like lower case Us as he failes to connect the two ends. This same phenomenon can be observed in the dedication.
The "If I Fell" lyrics do not give us any frame of reference for the capital letters, but another set of lyrics John wrote in 1964 do:
This is from John's handwritten lyrics for "If You've Got Trouble." Here we can see John's capital "L" and "VE" in "LEAVE" are similar to the capital "L" and "VE" in the dedication. "LEAVE" here is written in a similar pseudo script style as "LOVE" so the slight curve connecting the V to the E is especially striking.
Given this, it appears that we CANNOT rule John out of contention. It's still possible he wrote the dedication. There are enough similarities between handwriting samples of his from 1964 and the dedication.
The trouble with Jane is our handwriting samples for her are limited, the best I could find was a note she wrote in 1966:
Her lower case L is thin and never has a big loop unlike John. This is similar to the L in "Paul" in the dedication. Her upper case P in the note also resembles the P in Paul in the dedication. Her lower case "au" in "autograph" in the above note is somewhat similar to the "au" in the dedication, but Jane's handwriting is curvier, the strokes in the dedication are sharper than the laxer strokes in the above note written by Jane. In contrast, the curve in Jane's lower case h are consistently sharper than the very curved h in the dedication.
Notice also that when she writes a lower case T, the next letter is connected not to the base of the letter (as it does in the dedication) but with the cross/horizontal line. Even when the base ot the "t" and the next letter connect, as in "that" this is clearly more accidental then intentional. Jane's lower case Ts consistently do not have a curved bottom. In contrast, the lower case T in the dedication not only has a curved bottom, it is clearly purposefully connected to the next letter (h).
I am inclined to think the dedication was not written by Jane, it's not completely impossible that it was her, but in my considered opinion it is unlikely.
My favorite wild card theory. For his handwriting sample, lets use George's 1964 letter to Astrid:
Like Jane, the biggest problem with the George theory is his lower case Ts. Both in "think" and in "anything" the bottom of his lower case T is not curved and it connects to the H through the cross, not the base. He sometimes curves his lower case Ts when theyre the last letter of a word, as in "put", but this is the single instance of a curved lower case t in the entire letter to Astrid. His cursive "V" also curves inwards (as in "everyone" and "love") while the V in the dedication curves slightly outwards. In other words the way George's hand moves when he writes a cursive V seems slightly different from the way the person who wrote the dedication does it. George's capital Es are also consistently angular in the above sample, though its worth noting that he sometimes has curved capital Es in later handwriting samples, such as the handwritten lyrics of Something from 1969. Still, in so far as 1964, that does not seem to be the usual case.
Moreover, while his lower case Ls are inconsistent (sometimes thin, sometimes with big loops), he seemed to write the lower case L in "Paul" with big loops. Here is a letter he wrote to an Atlanta DJ, Paul Drew in 1966:
Notice also the slightly curvier capital Es, and the inward curving capital V.
A point in favor of the George theory however can be seen in his letter to Astrid. That is, George ended some of his letters with "x x x", especially in '63-'64, and the dedication also ends with x x x. Here's another example from 1963 of George ending a letter in this manner.
But the thing is... George isnt the only one who signs off letters with "x x x," John did this on occassion in the late 50s, early 60s. He does it here in a letter to Cynthia (sent from Hamburg, 1959):
Another letter to Cynthia in 1962 also shows John signing off with multiple Xs, and he also signs off with "xxxx" in an alleged letter of his to George's mother, Louise from the early 60s. That said, George is more consistent with using just three Xs. So the sign off is still a point more in George's favor than John's, but it is definitely not a point AGAINST John.
If we are to rank our three main "suspects" based on this discussion, John would be the most likely writer of the dedication, followed by George. Personally I would have loved if it was Jane, and John wrote the lyrics on the card in a jealous frenzy lmao; but it does still seem more likely that it was John.
Side Note: I checked some Brian, Mal, and Ringo handwriting samples and they were all too dissimilar to the handwriting on the dedication to merit serious consideration. I don't think anyone else in their inner circle would send Paul a Valentines Card either, but feel free to correct me if I'm wrong.
