Stabilizer Quantum Gravity III: Logical Composition, Norm Preservation, and the Hurwitz Ceiling

This paper constitutes Part III of the Stabilizer Quantum Gravity (SQG) research program. We investigate the algebraic constraints governing the logical layer of recoverable stabilized sectors. The central question addressed is whether a logically persistent recoverable sector—modeled as a finite-dimensional real algebra with a bilinear composition law, a unit, and a positive-definite multiplicative norm—is subject to an intrinsic dimensional ceiling. We prove a conditional structural theorem: if the relevant logical sector satisfies a norm-preserving bilinear composition law, contains no nontrivial zero divisors, and belongs to the class of recoverably persistent finite-depth sectors, then its real dimension is strictly restricted to the Hurwitz values 1, 2, 4, or 8. The paper carefully distinguishes between this rigorous algebraic ceiling and the stronger physical claim of dimensional selection, explicitly identifying the physical conditions and potential failure modes (such as the failure of the multiplicative norm) required to link this algebraic limit to the emergent universe.