Global Hartogs

The spectral character grammar, CRT reconstruction, tower coherence, and the τ-Identity Theorem now support Global Hartogs: τ-holomorphic data on L K, for primordially thin K, extends uniquely to L, so coherent ω-tail data determine the finite-stage values they project to. This is not an imported transfinite completion theorem; it is the first internal limit/stage discipline of τ. The construction can speak about its own infinite-stage extensions because boundary data, character grammar, and admissible ω-germ transformers have already been earned. This closes Book I’s mathematical layer by showing that the kernel has enough internal grammar to host limit constructions without leaving the system; proof-substrate audit remains part of the formal trust budget rather than a separate construction step.