Realizability of a Law's Symmetry and the Limits of Forced Local Heterogeneity in Three-Role Root Models

This paper investigates when a homogeneous local law on a finite graph admits a history realizing its global symmetries and when local homogeneity is possible without global transitive symmetry. The problem is motivated by the three-role root framework of the Wave Theory of the Universe (VTV), but the results are mathematical and do not establish a physical pre-vacuum or a mechanism of cosmological defect formation. For finite cycles with a prescribed alphabet-recoding group, we prove an exact criterion for the existence of a globally vertex-transitively symmetric admissible history. Although reflections can be essential to the symmetry of an individual history, they do not enlarge the class of laws admitting at least one globally transitive realization. We distinguish three regimes: (I) symmetric realizability, (II) local homogeneity without any globally transitive realization (hidden asymmetry), and (III) forced local heterogeneity. For oriented local windows, we establish a products-versus-powers criterion in terms of admissible group elements and pointwise stabilizers of window templates. We derive structural necessary conditions for hidden asymmetry and prove that it cannot occur when the prescribed alphabet-recoding group is abelian. The analysis is extended to local window types identified under reflection through a two-orientation, group-labelled closed-path criterion. We also provide a regular-equivariance criterion for finite Cayley graphs, examples involving cycle colourings and other uniform local relations, and reproducible computational checks. The results are stated within explicitly defined finite mathematical models. Their independent novelty relative to finite-group extensions of shifts of finite type and reversal systems remains subject to further literature assessment. No physical primary defect, spacetime geometry, or vacuum dynamics is derived in this work.