"When I'm with Aoi-senpai, the world looks different."
Yuki said quietly.
Aoi looked at Yuki in surprise. "Suddenly, what?"
"No, I was thinking about joint entropy."
"H(X,Y)?"
"Yes. Entropy when looking at two variables together."
Aoi opened the notebook. "H(X,Y) ≤ H(X) + H(Y)"
"Equality only when X and Y are independent."
"With correlation, it becomes inequality," Yuki continued.
"Yes. With correlation, total entropy decreases."
"Why?"
"Because knowing one allows predicting the other."
Aoi gave an example.
"Weather and temperature. Not independent. If sunny, likely warm."
"So knowing both gives less information than knowing each separately."
Yuki understood. "There's overlap."
"Exactly. That overlap is mutual information I(X;Y)."
"I(X;Y) = H(X) + H(Y) - H(X,Y)"
Aoi drew a diagram. Venn diagram with two overlapping circles.
"The overlap is shared information."
Yuki looked out the window.
"Do senpai and I also have correlation?"
Aoi blushed slightly. "What do you mean?"
"Spending time together, our ways of thinking become similar."
"That's increasing correlation."
"H(Yuki, Aoi) < H(Yuki) + H(Aoi)"
Aoi laughed. "Expressing it in equations."
"But is that good?" Yuki asked worriedly.
"What do you mean?"
"If correlation is too strong, I feel like individuality is lost."
Aoi thought seriously.
"Sharp observation. H(X|Y) = H(X,Y) - H(Y)"
"Conditional entropy. X's uncertainty after knowing Y."
"With perfect correlation, H(X|Y) = 0."
"Knowing one, you completely know the other."
Yuki said anxiously. "Does that mean one person isn't needed?"
"No, different," Aoi said gently.
"Perfect correlation is a theoretical limit. In reality, independent parts always remain."
"Independent parts?"
"You have your experiences, I have mine."
"Becoming completely the same is impossible."
Yuki felt a bit relieved. "So moderate correlation is ideal?"
"I think so. I(X;Y) not too high, not too low."
"Can understand each other but maintain independence."
Aoi summarized. "Joint entropy shows the world seen by two isn't a simple sum."
"There's interaction."
"Yes. Mutual influence."
Yuki wrote in her notebook. "H(X,Y) < H(X) + H(Y). Being together reduces uncertainty."
"But doesn't become zero."
"That's beautiful," Aoi said.
"If completely predictable, no meaning in being together."
"Surprises make it interesting."
Yuki smiled. "Senpai always gives me surprises."
"You too."
The two sat quietly.
"Joint entropy," Yuki murmured. "Mathematics of shared future."
"Future might also be correlated," Aoi said.
"Being together changes future choices."
"H(future_Yuki, future_Aoi)"
"That's still undefined," Aoi laughed.
"But maybe a day will come when we can measure it."
Yuki nodded. "Until then, I want to keep learning together."
"While optimizing joint entropy."
"Moderate correlation and moderate independence."
Sunset illuminated the club room.
Their future is still high entropy. But correlation gradually increases.
That might be what nurturing a relationship means.
Today too, their joint entropy recorded a new value.