Part XVI: The Presheaf Essence
Part XVI proved the Global Hartogs Extension Theorem: the limit determines the stages, and omega-tail data on 𝕃 uniquely extends to all of τ³. Part XVIII audited the proof-theoretic substrate and charted the enrichment frontier.
This two-chapter finale asks: what IS τ-holomorphy, structurally? Not what it does — extend, determine, decompose — but what it is as a categorical object.
The answer: a τ-holomorphic function is a natural endomorphism of the primorial presheaf
that respects the sector decomposition, captured in a single pasted commuting diagram — the holomorphy bi-square . The left square encodes tower coherence; the right square encodes spectral naturality. The Global Hartogs Theorem becomes the limit principle: the top row determines every bottom row.
No new machinery is introduced. The categorical language was earned in Part XIV; the holomorphic machinery in Parts XII–XIII; the spectral machinery in Part XI. This Part merely shows what they jointly say when expressed in the language they collectively earned. Five generators, seven axioms, one bi-square.