Complex analysis

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• Book II — Categorical Holomorphy Part 0
  Chapter The Boundary-First Paradigm

  Classical SCV: Interior to Boundary In the orthodox tradition of several complex variables, the interior comes first

• Book II — Categorical Holomorphy Part 0
  Chapter Roadmap and Inverted Dependency

  cylinder compat.) , continuity & topology & (Part III: unique topology making NF continuous) , topology & geometry & (Part IV: ultrametric ⇒ Tarski axioms) , geometry & transcendentals & (Part V: π, e, j, ι_τ from earned geometry) , In orthodox analysis the arrows run in the opposite direction: transcendentals (ℝ assumed) ⇒ geometry (ℝ^n) ⇒ topology (open sets) ⇒ continuity (-δ) ⇒ holomorphy (Cauchy–Riemann )

• Book II — Categorical Holomorphy Part 6
  Chapter Laurent Series and Residues

  Laurent Series and Residues Classical Laurent theory expands a holomorphic function in an annulus r_1 < |z| < r_2 as a doubly-infinite power series _n=-∞^∞ a_n z^n, with the residue a_-1 computed by contour integration around the singularity

• Book II — Categorical Holomorphy Part 9
  Chapter Hartogs Extension in H_τ

  The classical proof relies on the Cauchy integral or the ∂-equation — tools that are not available in Category τ

• Book II — Categorical Holomorphy Part 9
  Chapter Liouville Categorical Dodge and Categoricity

  The classical Liouville theorem states: f is constant

• Book II — Categorical Holomorphy Part 11
  Chapter What τ Earns

  Hartogs Extension (Earned 7) The Global Hartogs extension theorem is Mode E in the strongest sense: the orthodox proof uses the Cauchy integral (an analytic, non-constructive tool), while τ's proof uses the Chinese Remainder Theorem (an arithmetic, constructive tool)

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