"Riku, you're quiet today."
Yuki said. Riku, usually noisy, was uncharacteristically silent.
"Are you feeling okay?"
"No... just thinking."
Aoi looked with interest. "Riku's silence carries high information content."
"Information content?" Yuki asked back.
"Information content is proportional to surprise. I(x) = -log₂(p)."
"Since Riku always talks, probability p(talks) is high. So p(silent) is low."
Yuki understood. "Low probability means high information content?"
"Right. The event of Riku being silent rarely happens. So when it does, information content is large."
Riku smiled wryly. "Is my silence that rare?"
"Statistically, quite rare."
Yuki thought. "Then Aoi-senpai not talking has low information content?"
"I'm naturally taciturn. Since p(silent) is high, my silence has small information content."
"Interesting. The same silence has different information content depending on the person."
Aoi continued. "In information theory, the more unpredictable something is, the more information it carries."
"Predictability and information content are inversely proportional."
Riku asked. "Then what's the information content of something completely predictable?"
"Zero. If probability is 1, then log(1) = 0."
"Certain things have no information."
Yuki wrote in her notebook. "That's why news conveys unexpected things."
"Correct. Routine events have low information content, so they don't make news."
"There's no news that the sun rose in the east," Aoi gave an example.
"But if it rose in the west, big news," Riku continued.
"Information content depends on context."
Yuki thought. "Silence also has context."
"Right. A normally quiet person being silent versus a normally talkative person being silent have different meanings."
Riku made a serious face. "So is my silence conveying something?"
"It might be. Information is born not only from existence but from absence."
"Absence?"
"Something expected but not appearing. That itself is information."
Aoi supplemented. "In communication theory, no signal is also treated as a signal."
"The value zero is also data."
Yuki thought of an example. "Like messages without replies?"
"Exactly. The fact of no reply conveys information."
Riku nodded. "Like being angry or busy."
"Right. Interpreted from context."
Aoi presented a new perspective. "In music, rests are important too."
"Rests?"
"Parts without sound. But they create rhythm."
"Silence composes music."
Yuki was moved. "Information is the same. Zero has meaning."
"In information theory, entropy considers all events. Both what happens and what doesn't."
"H(X) = -Σ p(x) log p(x). Even low-probability events contribute to entropy."
Riku suddenly laughed. "Man, my silence carries responsibility."
"But that just shows how large your presence is," Yuki said.
"Because your usual behavior is predictable, deviations become information."
Aoi added. "That's why the more established a character is, the more noticeable changes are."
"Like branding," Riku understood.
"Human relationships are a kind of protocol too."
Yuki asked. "What about a completely random person?"
"You'd be surprised every time, so average information content is high. But too unpredictable to trust."
"Moderate predictability stabilizes relationships."
Riku took a deep breath. "Okay. I'll talk normally now."
"Information content drops," Aoi laughed.
"That's fine," Yuki said. "Normal Riku is reassuring."
"Low information content has value too."
Aoi concluded. "Information content isn't everything. Stability and predictability matter too."
"It's about balance."
The silence ended and the usual liveliness returned.
But the three didn't forget the information content that silence held.