Create a 75-100 second cinematic 3D explainer of Morse theory through a vertically oriented torus and a rising horizontal waterline. This topic is new to our repo.
Teach an ambitious undergraduate who knows derivatives but not topology. Define sublevel sets M_a={p in T^2:h(p)<=a}, critical points, index, and Euler characteristic. Use a genuine Morse height with FOUR isolated critical points, not the degenerate height of a horizontal donut. Mathematically verify a parametrization and the four critical heights and indices before implementation. Show disk -> cylinder -> punctured torus -> closed torus as successive sublevel surfaces. Explain that the topology changes at critical levels, with Euler characteristic 1-2+1=0. Distinguish the topology of the surface sublevel set from the liquid volume and from the number of boundary curves.
Art direction: midnight blue background, luminous teal and cyan surface mesh, warm gold scanning plane, coral critical points, violet saddles. Shape and motion carry the explanation. Use ambitious but purposeful camera orbits, slow push-ins at saddles, pullbacks to restore orientation, color transitions encoding the included surface, and beautifully legible fixed on-screen LaTeX formulas. Preserve a clean hierarchy and large negative space. One main formula at a time. Use actual MathTex, 3D geometry and evolving cross-sections; no walls of text or generic slides. End with a strong view of the entire torus and the Morse relation chi(T^2)=m_0-m_1+m_2=1-2+1=0. No narration required.
Use the local Python interpreter C:/Users/chris/Math-To-Manim/.venv/Scripts/python.exe for numerical verification and Manim API inspection if needed. Installed Manim CE is 0.20.1. Research primary references where necessary. Return a complete working scene for 1920x1080 at 60fps; use efficient surfaces to keep rendering practical.