Chapter 56: Differential-Geometric Agenda for Book~III

Book II has earned interior holomorphy: the τ-manifold structure (the relevant definition, II.D63), the holomorphic atlas (the relevant definition, II.D64), the Central Theorem O(τ³) ≅ A_spec(𝕃) (II.T40), and the categoricity of τ³ (II.T42). Book III must now earn differential geometry: connections, curvature, cohomology, metrics, and Hodge theory—the infrastructure that carries physics and the Millennium Problems. This chapter is a preview: it declares what Book III must accomplish without proving anything. We inventory what Book III inherits , then present four previews—τ-connections , τ-sheaf cohomology , τ-metric geometry , and τ-Hodge theory —and close with the enrichment ladder that organizes the full seven-book programme . Remark [rem:diff-geo-agenda] (II.R21) records the agenda as a formal forward declaration. Remark [rem:enrichment-forward] (II.R22) records the enrichment ladder.