The Autonomization of Captured Children: A Five-Resource Construction for Liberating Mathematical Objects from Parasitic Frames, with the Augmented Type T⊗ ch and a Conservative Theorem of Frame Redundancy

This paper develops a corrected and formally conservative metatheory ofautonomization for mathematical objects that have been captured byabstract framing schemata. The motivating case is the equation\[(x) = (x) + \!(∂ e_1/∂ Θ,, ∂ e_n/∂ Θ),\]which appears as an outer structural pattern in Euler--Maclaurin decompositions,theta-kernel decompositions, and shrinking-domain enclosures, and which is alsopresented as an instance of the abstract schema\[∀ C ∈ FRE[ ], C().\]We diagnose the schema--equation relation as parasitic when the schemacontributes only quantificational, typological, evaluative, coherence, andmetalinguistic scaffolding while the child equation supplies the content-bearingdata. The formal correction made in this version is essential. A mere trichotomy``provable, refutable, or independent'' does not establish redundancy. Trueformal redundancy requires a proof-theoretic hypothesis: the parental theory,on the relevant class of claims, must be a definitional or conservativeextension of the child theory after translation. Accordingly, the central resultis the Conservative Redundancy Theorem: if a parent frame is faithfullytranslatable into the child and proof-reductive over the child on a chosen claimclass, then the parent proves no translated child-content not already provableinside the child. Undetermined claims are not declared meaningless; rather, theyform a formally residual region where the parent may retain interpretiverhetorical force but no child-certified theoremhood. The construction proceeds through five child-internal resources:self-quantification, self-typing, self-evaluation, self-coherence, andself-reflection. The self-coherence component is corrected: output alone doesnot determine an arbitrary internal decomposition. What is recoveredintrinsically is the observed correction graph and the correction operator onthe observed derivative-jet image; off-image behavior is gauge unless explicitlymade part of the child data. We then develop internal expressive and bounded deductive adequacy underexplicit effective-presentation hypotheses, state child-internal incompletenessunder the standard arithmetization assumptions, define a faithful translationto a classical analytic fragment, characterize non-lifting boundaries, and givea general defense against re-parenting by arbitrary candidate frames. Theportability discussion is similarly corrected: Galois groups, natural-numberfoundations, categorical presentations, and type-theoretic presentations becomeredundant only on conservative or definitional fragments; stronger parents mayhave genuine proof-theoretic or structural content. Finally, the augmented type is constructed. It adds multiplicativestructure, reflection, coefficient functions, and explicit ratio/productreflection laws. This corrected augmentation supports zeta and ξ functionalequations and Gamma reflection, while leaving Euler products, explicit formulae,prime-counting theorems, and adelic proof apparatus for further augmentations.The paper closes with a general framework: any captured child satisfyingautonomous component production, resource reconstruction, faithful translation,and conservative proof reduction admits the five-resource construction and thecorresponding redundancy theorem, with augmentation cost measured by explicitlanguage, rule, and translation-complexity increments.