[Boshusan]

I Just watched Decade lastweek Do you want more headache ?

There was not an AR World of Den-O, the cross-over was literally in the Cho Den-O continuation. And the messy rules about time travel in Den-O universe also include something about the timeline restored with memories. The Den-O universe can just restored itself (or rebranching to a working timeline) by the memories of the people in the "end of time" where the time travelling trains are.

From the DenLiner's crew and the Shinkenger point of view, Decade did happen. Decade was also restored from the memories of all the riders he met, so they really met him, their worlds were restored in a pre-fusion state, and a few casulaties were alive again.

Edit: add minor events and correct names and formatting

I have a few theories that make more or less everything work in Decade: it's all about the cards. DaiShocker was wrong assuming the Riders powers lure each others and fuse their worlds (as seen when the plan didn't work in "All Riders Vs DaiShocker", or when Apollo Geist was defeated). It's the cards that provoke the fusion: the cards contain not only the "memory" or the "idea" of a Rider, but also the "memory" of his whole world. The worlds are fusing because they're attracted to their copy and all copies are gathered in Decade's deck. He is the catalyst, because DaiShocker!Decade made "bridges" travelling the worlds.

When DaiShocker!Decade became amnesiac, Eijirou hid the Decadriver from DiEnd, meanwhile Natsuki warned at least the original Wataru/Kiva about DaiShocker!Decade messing with the fabric of the multiverse, and the way to restore balance: let him fuse all the worlds then kill him; Kiva-la joined Natsuki.

When the fusion started in the very first episode, Eijirou guided Natsumi to the DecaDriver by manipulating the space travel gates. Then Amnesic!Decade quickly used the cards power and empty them (because his memories are no longer tied to the ones in the card), stopping the fusion, and restoring the worlds to a state where he has never been there before.

Eijirou manipulate Amnesic!Decade to travel the newly restored worlds and refill the cards, but Natsuki went to those worlds and warned the Riders about Onore!Decade.
After Decade death and the destruction of the DecaDriver, the memories are released from the cards (again!), restoring the worlds in a state where Amnesic!Decade has been there (because of the photos ?). As a side effect, it also restores an Eijirou from Natsumi's world as the viewers know him in the TV show.

Then Composite!Decade is bring back by the memories of the others, at this point, he gots another DecaDriver and cards that just work as the people remembering Decade think it is supposed to works: copying a Rider.