Unbounded Godbillon-Vey numbers on a fixed hyperbolic manifold

Classical compact-support suspension theory gives a foliated surface-times-interval block with product endpoint collars and nonzero ordinary Godbillon-Vey integral. Placing finite stacks in one fixed pseudo-Anosov mapping torus yields smooth cooriented taut foliations with GV=Na, a>0, and the same embedded positive closed transversal meeting every leaf. This gives a negative answer to the literal unrestricted volume-bound Question 10.6 in Calegari's arXiv:math/0209081v1 (2002). Compact leaves are permitted; no minimal-foliation variant is claimed. This is a credited consequence of classical Mather-Thurston/Tsuboi suspension theory, interval-bundle replacement documented by Gabai, and Thurston hyperbolization. A bounded primary-source audit did not authenticate an earlier explicit full consequence, but historical firstness, guaranteed novelty and present-day open status are not established. Relevant full texts remain unread, and the final 2003 target chapter was not compared. Version and access limits are documented in the supplement. The deposit includes the research note, standalone LaTeX source, repaired exact-rational and symbolic local controls with actual ordinary/optimized reproduction, independent mathematical audits, and bounded priority ledgers. Finite controls do not replace the global existence proof. AI tools were used extensively in development, drafting, source checking, reproduction and adversarial review. This preprint is unrefereed and has not undergone conventional human peer review.