Noether's Theorem and the Minimum Informational Boundary: Symmetry, Conservation, and the Unified Charge of Causal Consistency

Prior papers in this series have established that the Minimum Informational Boundary enforces probability conservation as a causality protection mechanism, and that the scalar field formalism, Lorentz invariance, and the Einstein field equations each independently encode this requirement. The present paper identifies the common mathematical origin of that convergence. Noether's theorem states that every continuous symmetry of the action of a physical system corresponds to a conserved quantity. This paper argues that the conservation requirement identified by the Minimum Informational Boundary framework — the prohibition against Deletion and Insertion Breaches — is best understood as an interpretive extension of the symmetry-conservation structure described by Noether's theorem, with global phase invariance providing the clearest internal-field example of that structure. This symmetry is not a new postulate. It is already present, implicitly, in every physical theory that admits a well-defined action principle. The Information Boundary Principle framework proposes to make this implicit structure explicit by identifying its physical content as a constraint on causally admissible evolution. The convergence of three independent mathematical formalisms on the same conservation requirement is therefore not coincidental. It is the expected consequence of a single symmetry operating at the foundational level of physical law. Noether's theorem is the mathematical bridge between that symmetry and the conservation it enforces. The Minimum Informational Boundary is the point in the present framework where that mathematical bridge is interpreted as a physical constraint on admissible evolution. This revised and expanded version adds a causality-protection analysis that was implicit but undeveloped in the original paper. The paper proposes that causal consistency is protected through three mutually reinforcing layers: first, the interlocking conservation structures generated by the major Noether symmetries considered here; second, irreversible registration once an event crosses the Minimum Informational Boundary; and third, the logical self-termination of any process that attempts to alter the causal conditions of its own existence. These claims are presented as consequences of the Information Boundary Principle framework, not as standard consequences of Noether's theorem alone.