I've been applying myself really hard to trying to figure out the meaning behind Homestuck's liberally employed sudden scene transitions and psychs/fakeouts, and coming up with far, far emptier hands than I would like. but scrutinising the sequence starting on p. 003037, where Jade's birthday gift opening is interrupted by Dad walloping Jack in the face, has at least provided me with some insights:
Dad strikes Jack (p. 003038) and places emphasis on desecrating his hat (p. 003041). note that the harlequin hat originates from Nannasprite; which is to say, it is Dad's mother's hat.
in the background of the scene looms a great big vault safe, which has already long been established as a symbol of maturity and strength in the Egbert family (p. 002446), but in this sequence takes on additional significance as a signifier of scaling hierarchies of power: when attempts to stop Dad using a safe (bigger than the one he keeps at home, which John is expected to lift) fail dismally (p. 002851), the order is made to bring in an even bigger one (p. 002857) - which Dad again proves his superiority by making short work of (p. 002861)
in the subsequent flash (Enter, p. 003049), John homes in on an imp wearing his father's hat, and establishes dominance by socking it in the head and knocking the headwear loose. having already established the imps' tendency to appropriate the trappings of John's childhood, this reads straightforwardly as a symbolic walloping of his own Dad (if I may get way ahead of myself, it's significant that this same image is replicated in one of John's brawls with Caliborn, after the latter starts donning a fedora.)
(perhaps Act 4's pivot to focusing on the Prospitian monarchs, and especially their crowns as badges of office, plays into this motif of hat-as-authority?)
all of which is to say that by the time the Intermission starts, I'm well-primed to read Jack's adventure as a psychodramatisation of the Egbert family history. John's fedora'd father stands-in for Problem Sleuth as Homestuck's predecessor, and on its face the Intermission is of course a tribute to Problem Sleuth starring a dark mirror version of its protagonist (establishing a hierarchy of power by "punching [the other guy] in [the] snout" goes back to a recurring PS gag). but there are a few particulars from the very beginning of the Intermission that lead me to connect it to Dad more intimately:
Jack's Crosbytop (p. 003066) - which he pulls out of the same kind of chest both John and Jade use to honour their family legacies with pictures of their ancestors - is modelled after the same photograph Dad keeps hung above his bed (p. 002846; it should be noted that Jack's introduction comes only pages after John first enters his Dad's room and encounters this picture; p. 002853), and the portrait of Jeff Foxworthy he finds under the carpet (p. 003059) is one that will later come to be associated with Jane's home; which is to say, it is the kind of picture Dad's mother kept around. Jack's commentary on both images is especially telling: the difference between the two is that the latter has "a ghastly furred upper lip". not only is this Dad's obsession with shaving poking through, but, with retrospective context, it's also the tiniest glimpse at just how he might have come to develop this obsession.
the first little family of clocks Jack encounters - son, father and grandfather (p. 003056) - offers a quaint hint at the kind of escalation-backwards-through-time that the Intermission is actually about. but naturally it all comes to a head at the end, where Jack has to crack the biggest one - not just the biggest clock (p. 003118):
...but the biggest safe of them all (p. 003237).
in its highly abstract manner, Homestuck's first Intermission establishes the impetuous backbone for the narrative moving forward: it is by following John's family tree backwards through time, climbing a ladder of ever-bigger and stronger father figures (capable of lifting bigger and heavier safes) to the very highest rung, that we will find the source of all this family dysfunction in the man who is called Lord English.
