A Unified Metatheory of Structural Claim Transformation: Recoverability, Obstructions, and Reachability — Series Synthesis, Transformation Certificates, Semantic Invariants, Certification Gaps, Obstruction Spectra, Module–Obstruction Dual Bases, and

This paper is a purely theoretical study that combines a series-level synthesis with an independent metatheory. Taking seven theoretical works, 2026ay–2026be, as its input, it first reconstructs their logical lineage from local geometric, informational, and dynamical results, through statistical representation and stochastic lifting, to a relative modular basis. It then upgrades the question “can a structural claim change its theoretical role?” from a narrative judgment to a transformation problem with explicit semantic objects, role equivalence, generation–recovery relations, and failure boundaries. The paper does not attempt an exhaustive classification of transformations of arbitrary mathematical theories. Within the declared class of structural claims, it uses conditional reasoning (CR), structural grounding (SG), and framework unification (FU) as the principal mechanism language, while stochastic lifting and module-basis extraction are placed strictly at a subsequent organizational layer. First, a basic recoverability criterion on the role quotient is given, followed by a structural-invariant obstruction-screening theorem and a conditional completeness result. For a source instance space X_A and a target instance space X_R, role-equivalence classes are taken as the basic semantic objects. A situation in which nonequivalent source objects are folded into equivalent target representations becomes a falsifiable failure mode of recovery. This allows the relative observation quotient of 2026ay and the local statistical representation of 2026bc to enter a unified obstruction interface for non-transformability. Second, a typed calculus of transformation certificates is established. A certificate records additional conditions, an enlarged structure, a mechanism word, a generation–recovery correspondence, a semantic validity domain, a failure boundary, and non-circular dependence. Under explicit equivalence congruence and coherence conditions for gluing intermediate types, certificates can be composed, and a union–inverse-image propagation law for failure boundaries is obtained. Third, the paper strictly separates “the transformation does not exist mathematically” from “the current certificate language cannot cover the transformation.” For the semantic feasible domain Feas, the certificate-effective domain Eff*, and the certification gap Gap, it proves X_A = ObstSem ⊍ Gap_L ⊍ Eff*_L, thereby allowing genuine semantic obstructions to be distinguished from removable expressive gaps across certificate languages. Fourth, an obstruction-spectrum theory is developed. For objects coverable by a complete certificate language, a minimal deletion-obstruction family Cut_min and an obstruction order ord_obs are defined. Under module monotonicity, the minimal obstruction-deletion family and the minimal support family satisfy the classical blocker duality. A one-hot identity matrix is only a special coordinate realization of an obstruction of order 1; higher-order synergetic irreducibility is characterized by ord_obs > 1. Fifth, an expressive-capacity theory for transformation certificate languages is introduced: transformation simulation and semantic completeness are defined, and the known strict separation between tree-like and general Resolution is used as an external calibration, showing that the resulting hierarchy of transformation certificate languages is not vacuous. Sixth, the computability boundary is characterized: unbounded reachability in an effective certificate language is recursively enumerable, fixed-length truncations are decidable, and under an effective embedding of first-order theoremhood into the transformation-certification problem, transformation reachability is r.e.-complete. A further Rice-type argument shows that, for nontrivial extensional semantic transformation properties, no uniformly total computable scheme can simultaneously generate and verify certificates soundly and completely. Semantic completeness, effective certification, and uniformly computable generation are therefore sharply separated. Finally, the metatheory is instantiated on 2026ay–2026be. The paper gives a finite target registry and a unified interpretation of series-level role migration, dependency types, modular bases, visibility, and stochastic two-scale signatures, and establishes a checkable priority-evidence structure. Priority is not assigned to mature individual tools such as theory morphisms, proof certificates, hypergraph blockers, axiom weakening, structural reduction, or Rice’s theorem. Instead, the priority object is restricted to the organization of transformation, recoverability, obstructions, support, and computability boundaries into one provable object system under explicitly specified structural claims, role-preserving semantics, certificate languages, and a common background, together with its instantiation on 2026ay–2026be. To prevent the review function from absorbing the original mathematical contributions, the paper separately registers previously established series results, new 2026bf results, and cross-layer interface results, and records academic priority through the six-tuple “object–mechanism–premise–result–boundary–version.” All completeness, minimality, irreducibility, and priority statements are restricted to explicitly declared model classes, common backgrounds, role-preserving relations, certificate languages, and effective encodings. Research Paradigm Statement: The core methodology, research direction, mathematical structures, and final decisions were independently controlled by the author. Artificial-intelligence tools were used only for text organization, drafting, and program assistance. The author assumes academic responsibility for all mathematical statements, citations, arguments, and the final version.