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Every act of intentional design—whether in branding, product development, education, or organizational architecture—involves transferring meaning from intent to expression through an intermediate structure. This paper presents E:Stack Theory, a computable axiomatic framework for characterizing, measuring, and guaranteeing the integrity of this meaning transfer process. The theory introduces a tri-layer model (Intent → Architecture → Manifestation) grounded in set theory, morphisms, and information theory, formalized through five axioms: Order, Coherence, Uniqueness (Anti-Substitutability), Semantic Preservation, and Structural Completeness. From these axioms, we derive a Transitivity Guarantee providing computable lower bounds on design quality, and a Failure Coverage Theorem proving that eleven structural failure patterns exhaust all single-point axiom violations. We further define operational modes for incomplete intent (Reverse Structure Inference) and evolving intent (Exploration Mode), addressing conditions where classical design theory assumes fixed requirements. A measurement protocol operationalizes the theory through embedding-based semantic similarity (cosine similarity in quasi-metric spaces with relaxation constant κ) and LLM-as-a-judge structural evaluation with domain-specific rubrics, combined with cross-model invariance testing across multiple LLM implementations to address instrument dependence. The theory is illustrated through instantiation across seven domains—brand design, product design, organizational design, instructional design, architectural design, music production, and academic writing—demonstrating structural consistency across diverse design contexts. E:Stack Theory offers a computable axiomatic foundation for what we term Design Integrity: the structural guarantee that design outcomes faithfully preserve and express their originating intent.