The character sheet in Polaris works a little differently than other RPG character sheets. It does some of the things a standard character sheet does, like tracking specific stats, but it has two other important functions.
1: Tracking Expended Themes
When a player uses one of the more powerful ritual phrases in Conflict ("And Furthermore ..." or "You Ask Far Too Much ...") They must exhaust one of the four Themes on the character sheet of the current Heart player. The character sheet has a checkbox over each of the four Themes for the Heart, and another checkbox over each for the Mistaken. These don't refresh until the Heart gains experience and rolls above their current Zeal or Weariness. Having a written record of who has exhausted what Themes is extremely useful, especially between sessions.
2: Tracking Story Progression
Most RPGs don't directly reflect story progression on the character sheet. This is because most RPGs either have a GM to keep track of story progression, or are designed to last only one session. Polaris is in a unique position because it is a GMless RPG where players enter into binding negotiations over the fate of their characters. Sometimes the elements introduced in a conflict won't be directly related to what's happening in the scene, and this is what the character sheet is designed to handle.
From the rulebook, page 75-76:
Almost anything can be negotiated in a conflict, not just events which
have to do with the struggle at hand. Other events, seemingly
unconnected, can be introduced, although if the effects are lasting or
removed in time it is good to tie them to a change in the Cosmos or in
the Themes. Further, the Heart and the Mistaken can ask for Aspects to
be added, removed, or changed; Themes to be exhausted; and the Cosmos
to be shifted.
Basically, the character sheet serves as an immediate way to reflect the long-term changes mandated by a Conflict. I've been playing an ongoing game of Polaris for a few months now (once or twice a month) and it's incredibly useful to have this record to keep the story on track.
For example, two players in my game were negotiating a confrontation between the Protagonist and his Mistaken wife. The negotiation moved into the realm of prophecy. The outcome of the negotiation was something like "The Protagonist will be able to banish the wife now, but their half-Mistaken children are destined to return. All but one will die, and the one that does not die will survive by eating the protagonist's father." This basically amounts to a prophecy, something that will need to be played out in future scenes.
It would be very difficult to remember the specifics of this negotiation without noting them in the Cosmos, and so The Heart player reflected the outcome on his character sheet by noting it next to his half-mistaken children in his Cosmos, moving his wife to Mistaken, and adding an aspect under his Fate theme referencing the prophecy. Because of this, the consequences of the negotiation were not immediately mitigated or forgotten after the Conflict ended, and we were easily able to pick up the thread when we reconvened next month.
Since very early in the game, my Protagonist has also had a Fate aspect called "A Doom Only I Can Avert," and subsequent negotiations have altered it, bit by bit, until it has become "A Doom Only My Death Can Avert" and been joined by another aspect referencing the innocents I'll bring down with me, (who have also been added to my Cosmos).
As you can see, the complicated web that naturally grows throughout a game of Polaris would be impossible to keep straight without this particular kind of character sheet.