[Scene: Kenta's Room - Early Morning]
(Kenta's eyes snap open. He stares at the ceiling. His head throbs slightly.)
Kenta: (groaning) "What... happened last night?"
(He sits up slowly. Memories are fuzzy. Festival. Dancing. Juice. Then... nothing.)
Kenta: (thinking) "I remember the fireworks. And then... blank."
(His phone buzzes. And buzzes. And buzzes.)
Kenta: (picking it up) "29 messages?!"
(He scrolls through:)
Takahashi Kenji (12 messages):
· "DUDE!!!!!"
· "I HEARD EVERYTHING!"
· "YOU KISSED THEM! BOTH OF THEM!"
· "ON THE CHEEK! INITIATED BY YOU!"
· "KENTA MORIK IS A LEGEND!"
· "WAKE UP AND FACE YOUR GLORY!"
· (8 more messages of varying levels of caps lock)
Sui Makoto (8 messages):
· "Kenta-kun... last night... "
· "I can't sleep..."
· "My cheek still feels warm..."
· "Did you mean it?!"
· "Maybe don't answer that I'm scared"
· "Good morning! "
· "Or good night? I don't know anymore"
· "I'm never recovering from this"
Misu Yui (9 messages):
· "Kenta-kun~ "
· "Last night was magic..."
· "You kissed me! YOU! KISSED! ME!"
· "I'm writing this in my diary forever"
· "And burning the diary so no one else reads it"
· "But I'll remember forever"
· "Good morning my prince~ "
· "Are you awake?!"
· "Please don't forget "
Kenta: (staring at the phone) "I... kissed them? Both of them?"
(He touches his own cheek. Feels nothing. Remembers nothing.)
Kenta: (panicking internally) "I DON'T REMEMBER KISSING ANYONE!"
---
[Scene: Class 1-A - Morning]
(Kenta walks through the hallway. Immediately, whispers start.)
Student 1: "That's him! The guy who kissed both girls at the festival!"
Student 2: "Behind the stage! Someone saw everything!"
Student 3: "The ice prince melted!"
Kenta: (ears red, walking faster) (thinking) "Why does everyone know?! I don't even know!"
(He reaches his classroom. Takahashi is waiting. Grinning like a maniac.)
Takahashi Kenji: "THERE HE IS! THE LEGEND HIMSELF!"
Kenta: (sitting down quickly) "I don't remember anything."
Takahashi Kenji: (freezing) "What?"
Kenta: "The festival. After the dancing. I don't remember."
Takahashi Kenji: (grabbing his shoulders) "YOU DON'T REMEMBER KISSING THEM?!"
Kenta: (wincing) "Apparently I did. But no. I don't."
Takahashi Kenji: (shaking him) "KENTA! This is the greatest moment of your life! How could you FORGET?!"
Kenta: "I was tired. And that juice... had something in it."
Takahashi Kenji: (realizing) "Oh no. The festival special. It has a little alcohol for flavor."
Kenta: "I drank the whole thing."
Takahashi Kenji: (facepalming) "You lightweight! One sip of alcohol and you become a KISSING MACHINE!"
Kenta: "I'm not a kissing machine."
Takahashi Kenji: "You kissed TWO girls! Voluntarily! That's MACHINE LEVEL output!"
---
[Scene: Classroom Door - Sui and Yui Appear]
(The door slides open. Sui and Yui stand there, both bright red, both avoiding eye contact with Kenta.)
Sui: (mumbling) "Kenta-kun... can we talk?"
Yui: (also mumbling) "About last night..."
Takahashi Kenji: (whispering to Kenta) "This is it! The confrontation! The romance! The—"
Kenta: (standing) "I don't remember last night."
Sui & Yui: (in unison, freezing) "WHAT?!"
Kenta: "The juice. It had alcohol. I don't remember anything after the dancing."
Sui: (face cycling through emotions) "You... don't remember... kissing us?!"
Yui: (tearing up) "On the cheek?! Both of us?!"
Kenta: (guilty) "I'm sorry. I really don't."
Sui: (grabbing Yui's arm) "He doesn't REMEMBER?!"
Yui: (grabbing Sui back) "Our first kisses from him and he FORGOT?!"
Takahashi Kenji: (watching) (thinking) "This is better than TV."
Kenta: (thinking) "I'm in trouble. I don't know why, but I'm in trouble."
---
[Scene: Empty Classroom 3C - Again]
(Yes, again. Sui dragged him there. Yui followed. Takahashi tried to follow but was locked out.)
Sui: (pacing) "So let me get this straight. You kissed us. On the cheek. Voluntarily. For the first time ever. And you don't remember."
Kenta: "Correct."
Yui: (also pacing) "You said thank you. You said we were important. You said—" (she stops, blushing) "—you said we made you feel things."
Kenta: (ears red) "I said that?"
Sui: "YES! And then you kissed us! Each! On the cheek! Soft! Warm! Perfect!"
Kenta: (covering his face) "I don't remember any of this."
Yui: (stopping) "Kenta-kun... look at me."
(He looks. She's red but determined.)
Yui: "Do you... regret it?"
Kenta: (thinking) "I don't know. I can't regret something I don't remember. But..." (he pauses) "...if I did it, I probably meant it."
Sui: (stopping too) "Probably?!"
Kenta: "I don't do things without reason. Even drunk. If I kissed you, it's because I wanted to."
Yui: (softening) "You wanted to?"
Kenta: (looking away) "Apparently."
Sui: (also softening) "Kenta-kun..."
Kenta: "I'm sorry I don't remember. That's not fair to you. But I'm not sorry it happened. I think."
Yui: (grabbing his hand) "You THINK?!"
Kenta: "I'm still processing."
Sui: (grabbing his other hand) "Process faster! Our hearts can't take this!"
---
[Scene: Back to Class - Kenta's Solution]
(Kenta returns to his desk. Takahashi pounces immediately.)
Takahashi Kenji: "WHAT HAPPENED?! DID THEY KILL YOU?! ARE YOU ALIVE?!"
Kenta: (sitting down) "They're embarrassed. I'm embarrassed. We're all embarrassed."
Takahashi Kenji: "That's it?!"
Kenta: "I told them I don't regret it. Even if I don't remember."
Takahashi Kenji: (eyes widening) "Dude. That's actually... really sweet."
Kenta: "It's honest."
Takahashi Kenji: "So what now?"
Kenta: (pulling out a massive stack of papers) "Now I solve problems."
Takahashi Kenji: (staring) "That's your coping mechanism?!"
Kenta: "It's my only mechanism."
---
[Scene: The Problems - Professor Hayashi's Ultimate Challenge]
(Kenta spreads the papers across his desk. Takahashi reads the titles and pales.)
---
[Problem Set: Advanced Astronomy & Geometry]
From: Professor Hayashi, Tokyo Institute of Technology
Time Allotted: 1 week
Note: These are doctoral qualifying exam questions. Good luck.
---
[Question 1: General Relativity & Orbital Mechanics]
Consider a test particle in orbit around a Schwarzschild black hole of mass M = 10 M☉.
a) Derive the effective potential for radial motion and find the radius of the innermost stable circular orbit (ISCO).
b) Calculate the periastron shift per orbit for a particle in a slightly eccentric orbit with semi-major axis a = 10 GM/c² and eccentricity e = 0.1.
c) Determine the gravitational wave frequency at the ISCO.
d) If the black hole is spinning with a = 0.8 (Kerr metric), how do your answers change? Derive the new ISCO radius.
---
[Question 2: Celestial Mechanics - Three-Body Problem]
A binary star system with masses M₁ = 2 M☉ and M₂ = 1 M☉ has a separation of 1 AU. A small planetoid orbits this binary in a stable configuration.
a) Find the five Lagrange points and determine which are stable.
b) Calculate the orbital period of the planetoid at L4.
c) If the planetoid is perturbed, derive the frequency of small oscillations around L4.
d) Show that the condition for stability of L4 and L5 is M₁M₂/(M₁+M₂)² < 1/27. Prove this using linear stability analysis.
---
[Question 3: Differential Geometry - Riemannian Manifolds]
Let M be a compact Riemannian manifold of dimension n with sectional curvature K ≥ 1.
a) Prove Myers' theorem: The diameter of M is ≤ π/√K.
b) Using the Bonnet-Myers theorem, show that the fundamental group π₁(M) is finite.
c) If M is also simply connected and n is even, prove that the Euler characteristic χ(M) ≥ 2.
d) Construct an example where equality holds in Myers' theorem and compute its diameter.
---
[Question 4: Algebraic Topology - Homology Groups]
Consider the 2-torus T² = S¹ × S¹.
a) Compute all homology groups Hₖ(T²; ℤ) using cellular homology.
b) Calculate the cohomology ring H*(T²; ℤ) and identify the cup product structure.
c) Let f: T² → T² be a continuous map of degree d. Determine the induced maps on homology and cohomology.
d) Prove that any map f: T² → T² with degree |d| > 1 has at least one fixed point. (Hint: Use the Lefschetz fixed point theorem.)
---
[Question 5: Mathematical Physics - Quantum Gravity Interface]
In loop quantum gravity, area is quantized. The area operator has eigenvalues: A = 8πγ lₚ² Σ √[j(j+1)], where γ is the Immirzi parameter and lₚ is the Planck length.
a) Derive the area spectrum for a surface punctured by N edges with spins jᵢ.
b) Calculate the entropy of a black hole horizon using this area quantization and show that S = A/(4lₚ²) for an appropriate choice of γ.
c) Determine γ by matching with the Bekenstein-Hawking entropy.
d) Discuss the physical implications of area quantization for black hole thermodynamics.
---
[Question 6: Geometric Analysis - Minimal Surfaces]
Let Σ be a minimal surface in ℝ³ with Gauss curvature K.
a) Derive the relationship between K and the mean curvature H for a minimal surface.
b) Prove that the only complete, stable minimal surfaces in ℝ³ are planes.
c) Find all minimal surfaces of revolution and classify them.
d) Show that the catenoid and helicoid are locally isometric but not globally. Compute their Gauss maps.
---
[Question 7: Cosmology - Inflationary Perturbations]
During cosmic inflation, quantum fluctuations are stretched to superhorizon scales.
a) Derive the Mukhanov-Sasaki equation for scalar perturbations in a Friedmann-Robertson-Walker background.
b) Solve for the power spectrum of curvature perturbations in the slow-roll approximation.
c) Calculate the spectral index nₛ and the tensor-to-scalar ratio r in terms of slow-roll parameters.
d) If the potential is V(φ) = ½ m² φ², predict nₛ and r and compare with current CMB observations.
---
[Question 8: Symplectic Geometry - Hamiltonian Systems]
Consider a Hamiltonian system on a symplectic manifold (M, ω) with Hamiltonian H.
a) State and prove Liouville's theorem on the conservation of phase space volume.
b) Define Poisson brackets and show they satisfy the Jacobi identity.
c) If the system has a symmetry generated by a vector field X, prove Noether's theorem: there exists a conserved quantity.
d) Apply this to the Kepler problem and derive the conservation of angular momentum and the Laplace-Runge-Lenz vector.
---
Takahashi Kenji: (after reading five pages) (hyperventilating) "Kenta. This isn't math. This is TORTURE."
Kenta: (already writing) "It's interesting."
Takahashi Kenji: "INTERESTING?! I don't even understand the WORDS! 'Symplectic geometry'?! 'Mukhanov-Sasaki equation'?! 'Loop quantum gravity'?!"
Kenta: (still writing) "Mm-hmm."
Takahashi Kenji: (grabbing his hair) "You're not human!"
---
[Scene: Class 1-A - Six Hours Later]
(The school day has ended. Students have gone home. But Kenta is still at his desk, surrounded by papers, writing furiously.)
Takahashi Kenji: (sitting nearby, watching) "Dude. It's been six hours. You haven't moved."
Kenta: (not looking up) "Finished question 4. Starting question 5."
Takahashi Kenji: "You've done FOUR of those?! In SIX HOURS?!"
Kenta: "Question 3 took longer than expected. The Myers theorem proof needed careful handling."
Takahashi Kenji: (banging head on desk) "You're a monster. A beautiful, terrifying monster."
(The door opens. Sui and Yui peek in.)
Sui: (quietly) "Kenta-kun?"
Yui: "We brought food..."
(Kenta looks up. His eyes are tired but focused.)
Kenta: "Oh. You're still here."
Sui: (entering) "We've been waiting outside for hours! Takahashi told us you were solving problems!"
Yui: (placing a bento on his desk) "You need to eat!"
Kenta: (looking at the bento) "You made more food?"
Sui: (also placing a bento) "Of course! We always make food!"
Yui: "Even if you forgot about last night..."
Sui: (elbowing her) "YUI!"
Yui: "What?! It's true!"
Kenta: (looking at both bentos, then at them) "I'm sorry I don't remember."
Sui: (softening) "It's okay. Takahashi explained about the juice."
Yui: (nodding) "You're a lightweight. Good to know for future festivals."
Kenta: "There will be future festivals?"
Sui & Yui: (in unison) "YES!"
Takahashi Kenji: (grinning) "And future kisses!"
Kenta: (ears red) "Can we not."
Sui: (giggling) "His ears!"
Yui: (also giggling) "They're so red!"
Kenta: (looking back at his papers) "I have work."
Sui: (sitting next to him) "Then work! We'll watch!"
Yui: (sitting on his other side) "And feed you!"
Kenta: "I can feed myself."
Sui: "Not anymore! You have us now!"
Yui: (holding up a piece of tamagoyaki) "Say ahhh~"
Kenta: (sighing) "This is inefficient."
Sui: "Say it!"
Kenta: "...Ah."
(Yui places the food in his mouth. Sui follows with another piece. Kenta chews, still writing with his free hand.)
Takahashi Kenji: (watching) (whispering) "This is the most domestic thing I've ever seen. He's solving PhD problems while being fed by two girls. Legend."
---
[Scene: Library - Evening]
(They've moved to the library. Kenta is still working. Sui and Yui are "studying" next to him, which means occasionally peeking at his papers and making confused faces.)
Sui: (whispering) "Yui... do you understand any of this?"
Yui: (whispering back) "Not a single word."
Sui: "Same. But look at him go!"
Yui: "He's so focused. It's attractive."
Sui: (blushing) "Right?!"
Kenta: (without looking up) "I can hear you."
Sui & Yui: (in unison) "WE KNOW!"
Kenta: (almost smiling) "Question 7 done. Two more to go."
Yui: (awed) "You're almost done?! In one day?!"
Kenta: "Professor Hayashi said one week. This is inefficient pacing."
Sui: (facepalming) "Finishing a week's worth of PhD work in one day is 'inefficient pacing'?!"
Kenta: "I could have finished faster if I didn't have classes."
Yui: (staring) "Kenta-kun. You're a genius. But also insane."
Kenta: "Those are the same thing."
Sui: (laughing) "He makes jokes now! Progress!"
---
[Scene: Library - Night]
(The library is empty except for them. Kenta writes the final line on the last page.)
Kenta: (stretching) "Done."
Sui: (perking up) "FINISHED?!"
Yui: (also perking up) "ALL OF THEM?!"
Kenta: "All eight. Took 7 hours total. Question 3 was the bottleneck."
Sui: (hugging his arm) "MY GENIUS KENTA-KUN!"
Yui: (hugging his other arm) "OUR GENIUS KENTA-KUN!"
Kenta: (sandwiched again) "This is... warm."
Sui: (looking up at him) "Kenta-kun... about last night..."
Kenta: (stiffening) "I still don't remember."
Yui: "We know. But... can we tell you what happened?"
Kenta: (pause) "...Okay."
Sui: (taking a breath) "You took us behind the stage. You said you were frustrated because we always kiss you and you never kiss back."
Yui: "Then you said thank you. To me for always being there. To Sui for finding you again."
Sui: (blushing) "And then you kissed Yui. On the cheek. Soft. Warm."
Yui: (also blushing) "And then you kissed Sui. Same way. Same softness."
Sui: "We kissed you back. On the cheeks too."
Yui: "And then we watched fireworks together. Holding hands."
Sui: (quietly) "It was perfect, Kenta-kun."
Kenta: (listening, ears red) "I... wish I remembered."
Yui: (squeezing his arm) "It's okay. You were tired. And the juice..."
Sui: "But now you know. And maybe... next time... you'll remember?"
Kenta: (looking at both of them) "Next time?"
Sui & Yui: (in unison, blushing) "IF THERE IS A NEXT TIME!"
Kenta: (almost smiling) "There might be."
(Their faces explode in red.)
Sui: (sputtering) "HE—HE SAID—"
Yui: (also sputtering) "MIGHT BE—NEXT TIME—"
Kenta: (picking up his papers) "Let's go home. It's late."
Sui: (still recovering) "Y-yes. Home. Walking. Together."
Yui: (nodding) "Together. Good."
(They pack up and leave, Sui and Yui still holding his arms, faces still red, hearts still pounding.)
---
[Scene: School Gate - Night]
(They stand under the streetlight. Crickets chirp.)
Sui: "Kenta-kun... today was good."
Yui: "Even with the awkward morning."
Kenta: "I'm sorry again. For forgetting."
Sui: "Stop apologizing! We forgive you!"
Yui: "Yeah! Just... next time you drink juice, we'll monitor you!"
Kenta: "Monitor?"
Sui: (grinning) "To make sure you remember the important parts!"
Yui: (nodding) "Like kisses!"
Kenta: (ears red) "Goodnight, Sui. Goodnight, Yui."
Sui: (standing on tiptoes) "Goodnight, Kenta-kun!"
Yui: (also on tiptoes) "Sweet dreams!"
(They both kiss his cheeks simultaneously. Both cheeks. At the same time.)
Kenta: (frozen) "..."
Sui: (pulling back, red) "There! Practice for next time!"
Yui: (also pulling back) "So you remember this one!"
Kenta: (touching both cheeks) "I... will remember."
Sui: (beaming) "Good!"
Yui: (beaming) "Perfect!"
(They run off together, laughing.)
Kenta: (standing there) (muttering) "Two kisses. Both cheeks. Simultaneous."
(He touches his cheeks again.)
Kenta: "...I'll definitely remember this."
(He walks home, a small smile on his face.)
---
[Scene: Kenta's Room - Night]
(He lies in bed. His phone buzzes. Two messages:)
Sui's Text: "Kenta-kun! Today was perfect! Even with the awkwardness! We're not going anywhere! Sleep well and dream of us! (But not TOO wild dreams... or maybe yes?) "
Yui's Text: "My prince~ today reminded me why I love spending time with you! Even when you forget things, you're still the best! Goodnight! Don't let the geometry problems bite! "
Kenta: (smiling) (typing back to both): "Goodnight. Thank you for today. And for the kisses. I remember this time."
(Immediate replies:)
Sui: "AHHH HE REMEMBERS! "
Yui: "KENTA-KUN ACKNOWLEDGED OUR KISSES AGAIN! "
Sui: "THIS IS PROGRESS!"
Yui: "HA REMEMBERING PROGRESS!"
Kenta: (putting down the phone) (muttering) "Chaos."
(He closes his eyes, still smiling.)
Kenta: (thinking) "But my chaos. And I remember."
End of Chapter 7
Ready for Chapter 8?