Generalized Fibered Spaces: A Two-Volume Treatise on Structural Doctrines, Transport, and Mathematical Applications

This two-volume treatise develops a typed theory of fibered spaces in which transport, rather than local triviality, is the primary additional structure. A fiber is treated as relative to a projection or, in a category with pullbacks, as the pullback of that projection along a generalized point. The resulting framework distinguishes covariant transport from contravariant restriction, strict functoriality from pseudofunctorial coherence, and geometric curvature from metric, numerical, or statistical path defects. Finite typed panels of such defects are normalized into defect signatures, whose exact profiles and resolution-dependent cells provide computable structural subtypes. The framework admits towers of inner and outer fibers, moving bases, conditional connections, filtrations of both bases and fibers, and descent data for sheaves and stacks of structured spaces. An information site and a typed extractor turn restricted structural fibers into local information presheaves; sheaf or stack descent then controls gluing, while derived global sections measure higher local-to-global obstructions. Spectral sequences enter only after a declared cohomological realization produces a filtered chain object; their existence, naturality, and convergence therefore remain separate from the existence of the underlying fiber system. The resulting Information Spectral Sequence carries a source certificate, while a separate convergence certificate records boundedness, completeness, derived-limit obstructions, and unresolved extensions. A pruning certificate records an actual filtered comparison, the future differential hull of its retained region, the stabilization range, and the extension mechanism lifting stable-page comparison to declared cohomological targets. In finite-dimensional normed models, page conditioning and extension mixing give an explicit quantitative error bound; statistical validity then requires a separate identifiable estimand, natural target reading, sampling-law comparison, and loss-specific risk certificate. A finite \(S\)-decomposition is defined as a subset-indexed interaction complex with an explicit reconstruction map; its spectral reduction theorem permits deletion only after page, future-differential, convergence, extension, and target-reading checks. Classical bundles, Grothendieck fibrations, probability-kernel fields, and filtered complexes serve as typed specializations. Volume I develops the foundations and structural theory: generalized points, indexed categories, transport, structural doctrines, dependent sums, finite towers, reconstruction, base change, and abstract directions. Volume II develops realizations and applications: filtrations, sheaves, descent, homological and spectral constructions, analytic and probabilistic models, arithmetic examples, and finite algorithms. The final interface packages a finite fiber tower as a node for a possible higher theory of dependent structural composition, reserved for a future higher-composition sequel. ## Keywords Generalized fiber; path lifting; Grothendieck construction; cartesian fibration; cocartesian fibration; information site; fiberwise information sheaf; derived global information; base change; descent; hypercohomology; spectral sequence; certified pruning; approximate pruning; statistical identifiability; risk preservation; stochastic transport; \(S\)-decomposition; interaction filtration; spectral reduction.