Fibered Relative Consistency: Certified Semantic Transport at the ZFC Incompleteness Boundary

This paper develops a preliminary fibered theory of relative consistency. Formal theories form a metatheory-indexed base, model categories form contravariant semantic fibers, and satisfaction proofs form dependent proof-relevant fibers over models. A relative-consistency witness is not a bare implication: it is a forward model construction equipped with axiomwise satisfaction, naturality, size, and metatheoretic certificates. Such witnesses compose, and their nonvacuity yields semantic relative-consistency implications. For set-sized first-order theories, the passage to syntactic consistency is made only through explicitly formalizable soundness and completeness theorems. A definitional extension, the constructible core \(M↦ L^M\), countable-transitive Cohen forcing, and Boolean-valued ultrafilter quotients supply closed cases. Noncanonical generics are represented by existential choice fibers rather than by an unjustified globally selected functor. Opposite certified branches give relative independence certificates for \(V=L\) and for the continuum hypothesis over ZFC. For the CH negative branch, the internal forcing \(Add(ω,ω_2)\) is accompanied by separate ccc, cardinal-preservation, distinct-real, Boolean-value, proper-ultrafilter, and quotient-transfer certificates. Metatheory changes are organized as certified base changes: syntax, proofs, models, satisfaction, and size data move only through declared comparison functors and coherence cells. A certified consistency spectrum separates external, internal, syntactic, semantic, model-type, interpretability, conservativity, and equivalence readings. An arithmetized layer represents relative consistency by primitive-recursive transformations from target contradiction proofs to source contradiction proofs, together with internal totality and correctness derivations. Certified natural transformations between semantic transports yield a bicategory with explicit satisfaction, size, and metatheory coherence. Local witnesses on a site of theory fragments glue only under effective model, morphism, satisfaction, and size descent. A finite verifier checks local proof obligations, while a dated ZFC atlas separates typed transport edges from an acyclic proof-dependency graph. These data form a semantically enhanced structural doctrine whose morphisms preserve language, derivation, model, satisfaction, size, metatheory, and forward-transport certificates through explicit comparison cells. Marked semantic towers are organized into fixed-length categories and bicategories. A flattening pseudofunctor sends a certified tower to its composite transport, while rebundling fibers retain the intermediate theories, models, transition kinds, forcing data, and proof certificates forgotten by endpoint transport. A typed calculus of set-model transformations distinguishes ordinary set models, transitive models, countable transitive models, well-founded extensional presentations, Boolean-valued models, and forcing extensions. The constructible application certifies \(M↦ L^M\) from set models of \(ZF\) to set models of \(ZFC+V=L+GCH\). Further applications treat inaccessible rank segments, Mostowski collapse, Löwenheim–Skolem compression, Henkin choice fibers, and finite-fragment compactness through ultraproduct descent. Finally, proof-relevant consistency carriers support three forms of consistency action: total functorial action, choice-dependent span action, and admissibility-restricted partial action. Marked fiber towers are compared with an independently generated grammar of certified classical proof trees. The resulting evaluation pseudofunctor is essentially surjective on the registered fragment. A complete ZFC execution sends one supplied syntactic-consistency certificate through a common Henkin choice fiber and then bifurcates into certified ordinary models of \(ZFC+CH\) and \(ZFC+\). The construction organizes relative consistency at the boundary imposed by Gödel’s second incompleteness theorem; it does not prove \(Con(ZFC)\) within \(ZFC\). ## Keywords Relative consistency; ZFC; model theory; Grothendieck fibration; satisfaction; interpretation; definitional extension; Gödel incompleteness; certified model transport; semantic fiber tower; continuum hypothesis; Boolean-valued model; forcing independence.