Faithful Representation of Finitely and Generatively Specified Discriminative Theories: An Evoluist Theorem of Structural Fidelity

Abstract

This publication is the sixth study in the formal Evoluist research programme. It continues the status-diagonal analysis developed in The Evoluist Theorem: Formal Core (I), the distributed and indexed coordination framework developed in Indexed Evaluative Architectures and Terminal Coordination (II), the conditional global architecture developed in The Strong Evoluist Structural Alternative (III), the audit-domain completeness result established in The Semantic Obligations of Strong Universal Claims: An Evoluist Theorem of Audit-Domain Completeness (IV), and the five-checkpoint completeness result established in The Structural Audit of Strong Universal Claims: An Evoluist Theorem of Five-Checkpoint Completeness (V). The article develops a formal theory of faithful representation for a precisely delimited class of finitely and generatively specified source theories. Its purpose is to close the representation gap between independently specified source theories and the finite generative discriminative architectures used by the preceding structural results. It introduces an explicit source-theory ontology, three independently source-specified representability tiers — R-A_exp, R-B_fps, and R-C_effgen — a typed source-to-target correspondence, the MetaScaffold / OperativeImage / StructuralSupport separation, the canonical Minimal-Image Constructor v0.4, and eight exact fidelity invariants F1–F8. Its central result, FTT1, proves that every theory in the exact applicable representability class admits an Article III-compatible FGSDA minimal image satisfying the complete FaithRep₈ package. The construction preserves theorem-relevant source structure without weakening it, reflects target fidelity-relevant structure without strengthening the source, and prevents metatheoretic code, certificates, parsers, selectors, proof machinery, or other representation scaffold from becoming operative target resources without an independent source licence. The effectively generative R-C_effgen route is controlled by source-only RE12 and provides finite source-anchored access to each positively activated constitutively relevant instance without constructing a completed infinite aggregate. The theorem chain additionally establishes exact forward and reverse fidelity, No-Scaffold-Leak, bidirectional counterexample transport for token, family, simultaneous SCC, and finite active generative-instance structures, and invariance of faithful representations up to source-licensed operative equivalence. A qualified back-reflection theorem returns positive target limitations L1–L5 to independently defined positive source limitations and permits collapse-resource reflection only when the instantiated target package satisfies the explicit CollapseImageReady source-grounding conditions. Cor-VI1 further transports any source ClaimCandidate q into a faithful target-side claim image q̂ without classifying the claim as strongly universal. The paper does not claim faithful representability for arbitrary theories or source-free infinite structures. It does not infer GSU, FSD, RSUC, unrestricted source coverage, literal Article III D120/D124 completeness, direct applicability of the original T5 antecedent, the final Strong Evoluist Structural Alternative, or a mechanisation result. In particular, Article VI preserves the provenance-quotient versus literal-token boundary inherited from Article V. Strong-universality regimentation remains the independent task of Article VII, while final composition with the audit and collapse architecture remains the task of Article VIII. The Zenodo record contains the Main Article and the accompanying Technical Supplement. The Supplement provides the complete D-VI1–D-VI36 source and representation interface, RE1–RE12 eligibility conditions, the GCI-1–GCI-16 and C0–C12 specification of the Minimal-Image Constructor v0.4, the full T-VI0–T-VI9 / FTT1 and Cor-VI1 proofs, the adversarial catalogue and countermodels, representation-equivalence and back-reflection proofs, theorem and dependency registries, cross-article interface audits, and the final publication-concordance record.