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Only Genji and Kyrie still wish to count days and hours, even as they are now. And so, Battler kindly gifted them tiny timekeeping devices, visible only to their eyes. The counters of days on them have impressive digit counts, but still will eventually be forced to use scientific notation, and then hyperexponentiation, and eventually reach numbers even Kyrie's sharp mind cannot hope to really wrap itself around.
All other calendars are unmarked in the Golden Land, and all physical clocks are broken. Except for one.
That one single clock is always ticking. It never stops. It is certainly not allowed to stop, as guaranteed by the great Lambdadelta.
And on every even-numbered day, a tiny switch on that clock certainly flips. That is the solemn duty of BATTLER, the Territory Lord - it doesn't have to be his personally present body, of course, it could be any of the residents, it could be an unfortunate coincidence, a tiny seismic tremor - a stray golden butterfly - it certainly doesn't have to be Beatrice, ever, ever, ever again. But it happens, without fail, every time.
There's a wire that goes from this clock to a warehouse. No matter what happens in the Golden Land, what kind of a wild witch party or an unusual event, that wire is certainly never cut. It goes into a massive warehouse that looks to be full of sorcerous tools - mystic telescopes and astrolabes, crates and vats of alchemical ingredients, crystals of pure magic, gearworks that perpetually move on their own...
It isn't. And it's not in Lord Battler's intentions to force Golden Truths upon anyone, definitely not after the 8th game and mostly not before. So if you, standing in that magic warehouse, close your eyes and firmly tell yourself that you want a peek behind the embellishment, you will faintly smell the volatile nitrous compounds, trinitrololuene and trimethylenetrinitramine and ammonium picrate and nitrocellulose and a few others.
And so, on every even-numbered day, when that clock certainly strikes midnight, it is the duty of the Endless Sorcerer to perform a feat of Endless Magic, of death and resurrection. It's instant and unnoticeable for anyone - unless they firmly concentrate on wanting to notice it, of course. It's smooth like butter, and people in the middle of some conversation or activity don't feel a thing. And of course, after a few awkward first days, it became Battler's second nature. It isn't hard for a witch to perform magic, after all, and naturally, the magic of an Endless Witch is endless too. And this, too, is guaranteed by Lambdadelta's own seal. This endless potential for resurrection will certainly be used every single time.
Thus, the contraption that brought death in the human world brings endless life in this one, and every time, a new happy tale can be written.
It's technically possible, though inconvenient, to leave the island for a bit. The usual window of opportunity for that is the morning of odd-numbered days. It's impossible to be noticed by anyone outside the airport or plane, though. With the sole exception of Ange, no one notices anyone of the Ushiromiya family outside the island on those two once short and now endless days. You can walk around a bit, sneak a look at the books in a local library, or even abuse your position to spy on pedestrians or steal a dango stick without ever risking being caught even a little bit. It's impossible for a member of the Ushiromiya family to be caught stealing street food on October 4th or 5th.
And of course, you are compelled to return, before long. You can just jump into the ocean, and the waves will themselves carry you where they need to, while you peacefully pass out and wake up on a shore, or in your bed. Or you can find some other way of making sure you are never found. There's a factory on one of the nearby islands, and theoretically, jumping into the furnace is another way to get back. The workers will not notice you, and the steel will be off in its carbon content, but that is nothing outside of usual.
It's a little different for Eva. She leaves. Every two days, she leaves the Golden Land for 12 years. But, of course, she doesn't care to remember any of it. The gentle touch of magic seamlessly compresses that terrible, false mockery of life into a single instant, every time. In her capacity as EVA-Beatrice, Eva excises this instant of awful nightmare from herself, to return to the pleasant, golden dream. After all, those years were nothing more than the march of a black witch, running on autopilot. Eva, too, can be said to live only on this island.
Once, she lived through the nightmare again, so that in her spare time from being a new head, at night, utterly unobserved by Ange or any other soul, she could memorize some books written during the near future. Battler and Beatrice assured her that she really needn't go through that horror just to bring them a few poems and detective stories, but Eva insisted to not mind it. As the Territory Lord, Battler have of course heard the steel-hearted woman cry in her bed, even in this place utterly free from bonds and obligations.
That excursion will get easier to do when Ange will, at last, join them.
I think you can see why most residents of the Golden Land don't care for acknowledging that anything outside of the island exists.
There's a lot to do on the island, surprisingly. One can build or destroy, draw or play games, chat, serve, drink, eat, laugh, cry, walk, run, sleep, and love, and be loved...
You could do things that seemed impossible before, as well. You could learn magic. When the Golden Land officially came into full and continued existence, many former humans within other than Battler were already capable of it, after all. Maria and Beatrice were, of course, already powerful witches, and Kinzo a respectable sorcerer. Kumasawa and Eva had their witch selves - separate from their usual selves most of the time, but not nearly to the extent of Beatrice's separation from her other aspects. Though, funnily enough, Kanon and Shannon lost their magic sword and shield, after transitioning to humans, and had absolutely no interest in any magic since.
As noted above, it is not in Battler's intent to force Golden Truths upon anyone, so there is a bit of anti-magic toxin still present, for those who reach for it hard enough. But even it is gentle, only affecting the ones who want a break from the illusion. So it wasn't that hard for Rudolf and Jessica to learn to do some impossible things, or for Rosa to fulfill her secret and embarrassing fantasy of being a magical girl. Outside of some regular guests like Erika, few people ever want to break free of silly and benign illusions.
After all, most of the time, behind the cover of the illusion, there's heavy rain and terrible wind outside - constantly on all even-numbered days, and every evening on even-numbered ones. But if you don't question it, the sea is always gentle, and swimming is always pleasant, as you don't pay close attention to the effort it really takes to stay afloat. The time on the island always a bright twilight that never gets in the way of sleep or forces it - and damn astronomy. A human can function for two days without water, food or sleep, after all, if they simply happen to happily never notice their hunger and exhaustion and pain.
Kinzo was no longer a shut-in in his study and secret mansion. At one point, Battler and Beatrice got together with Kumasawa and Genji to resurrect Beatrice in a ritual - the very first of them. Here, with no bonds over him, there wasn't any point to hide the love of his life from his children. Kinzo's mad affection for his muse painted him in a much different light. So did the knowledge of their situation, of course, which, if you thought really hard about, made him look a bit like a preserved corpse, and his miraculous wife (formerly mistress) weirdly skeletal.
Kyrie and Genji are the only ones who keep magic at an arm's distance from themselves, never fully dissolving in it. Genji, however, spends most of his time simply peacefully sleeping and walking in silence, free of any obligation. Kyrie, on the contrary, in contrast to her relaxed exterior, is always secretly alert, just as she used to be before. She knows there is almost no risk of her ever getting cornered by anything. She knows that flipping chessboards isn't necessary any longer outside of things like literal games ot chess. And yet, she is the human watchwoman of this place, a silent assistant to her son. In the long, long line of people who scheduled the Ushiromiya Family's rapid immigration into this realm, she was the one to truly buy the tickets for the ride. And so, in this place devoid of any duties, she took on an informal duty to watch over the dream of others, her chessboard counterspell and her rifle always at her side. After all, there is a world full of witches lurking just outside of this place, many of them wicked and cruel. Can even the Witch of Certainty be truly trusted to guard the dream of Rudolf and Battler and the rest of her family for all time, the dream in which nothing ever reminds anyone of THAT iteration of the 4th and 5th of October?
Of course, no competent strategist would ever answer this with a confident "yes".