Notes

Notes are split into short, printable pages. Some are standard “toolkit” pages; others are problem-driven.

Toolkit notes (standard)

• Laplacians on cochains: definitions and identities
• Rank-one PSD updates and interlacing
• Variational principles: Rayleigh quotients and Courant–Fischer
• What “spectral gap” means here
• Linear-algebra lemmas for PSD operators

Worked theorem–proof pages (complete)

• Triangle addition gives a rank-one PSD update (complete proof)
• Codimension-1 Courant–Fischer trick (complete proof)
• A simple improvement bound under rank-one update (complete proof)

Problem-driven notes

• Triangle selection as a combinatorial optimization problem
• Kernel bookkeeping: why $\lambda_{\min}^+$ is subtle

Reading order

1. Foundations
2. Laplacians + Variational principles
3. Rank-one updates + Kernel bookkeeping
4. Triangle selection

These pages mix standard material with your framing. Replace or tighten statements as you turn notes into paper sections.