[AItoBit]

Replace the Chapter 26 placeholder with a formalization of cotangent and the Herglotz trick from Aigner–Ziegler, Proofs from THE BOOK (sixth edition).

• Prove convergence of the symmetric partial fractions, continuity away from the integer poles, periodicity, oddness, and the duplication identity. Extend the difference continuously across the integers and use the Herglotz maximum argument to obtain Euler's cotangent expansion.
• Develop the zeta and Bernoulli power series, derive Euler's formula for every positive even zeta argument, and compute the values through zeta(12), together with a Bernoulli growth bound.
• Use specific Mathlib imports and adapt the supplied proofs to the pinned Lean 4.35.0-rc3 toolchain. The generating-function and Bernoulli-series results explicitly retain their local convergence hypotheses.

Validation (Lean 4.35.0-rc3):

• lake build FormalBook.Chapter_26 passes without warnings.
• git diff --check passes; all lines are at most 100 characters and the file has no sorry, admit, new axioms, unsafe, or native_decide.
• The local full lake build encountered compiled-dependency read failures in unchanged chapters, including Chapters 1–7, and was stopped. GitHub CI subsequently passed the full build-and-lint step. Blueprint/documentation generation is still running.
• GitHub style CI passes. The local chapter linter driver compiled, but Windows Application Control blocked its executable (error 4551); local declaration-linter validation is unavailable. A separate local principal-theorem #print axioms scan was attempted but stopped after a prolonged run without output; transitive axiom inspection remains unverified.

The contribution is confined to FormalBook/Chapter_26.lean.