What is the meaning of this tempo marking?

Question

I came across this in a piece for piano four hands (title and composer are unknown; as pointed out in the comments, it might also be a part of a longer piece). Here's the beginning of the upper part (primo):

What does it mean?

• "♫=♫" - does this indicate straight (as opposed to swing) rhythm?
• "4T. = 4s." - I don't recognize this at all.
• "63 𝅗𝅥." - is this equivalent to "𝅗𝅥. = 63", i. e., 63 measures (in 3-4 time) per minute?

More information on the piece:

• It has 67 measures; measure numbering starts at 1
• It is printed on two pages per part (upper and lower)
• No title or composer is given on the first page
• It appears to be from a book, with the first page of the lower part starting on page number 128
• It has sections A, B, and C (circled letter, section A starting at measure 9 as shown)
• It has a coda (starting with a coda sign, 𝄌) that ends with a final bar line (thin and thick vertical line)
• The last measure before the final bar line is a whole rest in all four hands

Answer 1

Let's talk at least about what we can know. First of all: The standard way of indicating a tempo is in bpm, with an indication of what kind of note value that bpm pulse measures. For instance, ♩ = 120 m.m. (The "m.m." stands, not for "metronome marking," but for "Maelzel's metronome").

There seems to be some ambiguity about the time signature and the pulse. The mention of a dotted half note and 63 seems to mean that you might choose to feel the pulse "in 1," counting only the downbeats (thus, a dotted-half at a time), and that this pulse would be at 63 bpm. This seems reasonable, since a quarter note pulse at that speed is 189, which is not crazy but perhaps a bit hectic to "feel" as the main pulse.

Next: When we see "[note value] [equals sign] [some other note value]", that tends to tell us something about how to handle a transition in tempo or meter. For example:

... This is telling us that the quarter note pulse of the new 3/4 meter continues at the same tempo as the dotted quarter of the 6/8. So the "eighth notes equals eighth notes" in the mystery notation might relate to material that came before. Even if this is the start of a piece (and the measure number on the second system says it is), perhaps it's "piece number 41," and continued attacca from #40.

The music itself shows some metric ambiguity. It settled on 3/4, and the second measure reinforces that with three big quarter notes, but the first measure subverts it with an accent in the middle of the bar. Combined with the possibility of feeling "in 1," it might shift in and out of subdividing as either three groups of two or two groups of three. I have wondered whether the "s" in "4 s." is sextuplet, or whether " 4 t." is some kind of indication of "4-tuplet," but these theories don't add up to something clear. At any rate, the equals sign seems to suggest that "4t. = 4s." is continuing to talk about the relationship of this section to what came before.