Modelling of the properties of high-spin isotopomers, as polyhedra- on-lattice-points which yield various symbolic-computational <InlineEquation ID="IEq3"> <EquationSource Format="TEX"> ${\mathcal{S}_{12} }$ </EquationSource> </InlineEquation>-encodings of nuclear permutation (upto some specific SU(m) branching level), is important in deriving the spin-ensemble weightings of clusters, or cage-molecules. The mathematical determinacies of these, obtained here for higher m-valued <InlineEquation ID="IEq4"> <EquationSource Format="TEX"> $SU\left( m \right) \times \mathcal{S}_{12} \downarrow \mathcal{I}$ </EquationSource> </InlineEquation> group embeddings, are compared with that of an established group embedding, in order to collate the spin physics of [<Superscript>11</Superscript>BH]<Stack> <Subscript>12</Subscript> <Superscript>2−</Superscript> </Stack> <InlineEquation ID="IEq5"> <EquationSource Format="TEX"> $\left( {SU\left( {2\left( {m \leqslant 4} \right)} \right) \times \mathcal{S}_{12} \downarrow \mathcal{I}} \right)$ </EquationSource> </InlineEquation> with that for [<Superscript>10</Superscript>BH]<Stack> <Subscript>12</Subscript> <Superscript>2−</Superscript> </Stack> (SU(m ≤ 7) × ..)-analogue. The most symmetrical form of <InlineEquation ID="IEq6"> <EquationSource Format="TEX"> $\left[ {\left( {^{10} BH} \right) \left( {^{11} BH} \right)} \right]_6^{2 - } \left( {\left( {\mathcal{S}_6 \otimes \mathcal{S}_6 } \right) \downarrow \left( {\mathcal{S}_3 \otimes \mathcal{S}_3 } \right)} \right)$ </EquationSource> </InlineEquation> anion provides a pertinent example of the <InlineEquation ID="IEq7"> <EquationSource Format="TEX"> $SU\left( {m > n} \right) \times \mathcal{S}_n \downarrow \mathcal{G}$ </EquationSource> </InlineEquation> physics discussed in [10]. Retention of determinacy in the two <InlineEquation ID="IEq8"> <EquationSource Format="TEX"> $\mathcal{S}_{12} \downarrow \mathcal{I}$ </EquationSource> </InlineEquation> cases is correlated to the completeness of the 1:1 bijective maps for natural embeddings of automorphic dual group NMR spin symmetries. The Kostka transformational coefficients of a suitable model (<InlineEquation ID="IEq9"> <EquationSource Format="TEX"> $\mathcal{S}_n$ </EquationSource> </InlineEquation> module, Schur fn.) play a important role. Our findings demonstrate that determinacy persists (to <InlineEquation ID="IEq10"> <EquationSource Format="TEX"> $SU\left( {m \sim {n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\mathcal{S}_n$ </EquationSource> </InlineEquation> branching levels) more readily for embeddings derived from (automorphic) finite groups dominated by odd-permutational class algebras, such as the above <InlineEquation ID="IEq11"> <EquationSource Format="TEX"> $\mathcal{S}_{12} \downarrow \mathcal{I}$ </EquationSource> </InlineEquation>, or the <InlineEquation ID="IEq12"> <EquationSource Format="TEX"> $SU\left( {m \leqslant 3} \right) \times \mathcal{S}_6 \downarrow \mathcal{D}_3$ </EquationSource> </InlineEquation> case discussed in [16a,15,3d], compared to other examples — (e.g. as respectively, in press, and in [17b]): <InlineEquation ID="IEq13"> <EquationSource Format="TEX"> $SU\left( m \right) \times \mathcal{S}_8 \downarrow \mathcal{D}_4$ </EquationSource> </InlineEquation>, <InlineEquation ID="IEq14"> <EquationSource Format="TEX"> $SU\left( m \right) \times \mathcal{S}_{10} \downarrow \mathcal{D}_5$ </EquationSource> </InlineEquation>. Generality of the symbolic algorithmic difference approach is stressed throughout and the corresponding dodecahedral <InlineEquation ID="IEq15"> <EquationSource Format="TEX"> $SU\left( m \right) \times \mathcal{S}_{20} \downarrow \mathcal{I}$ </EquationSource> </InlineEquation> maps are outlined briefly — for the wider applicability of SF-difference mappings, or of comparable <InlineEquation ID="IEq16"> <EquationSource Format="TEX"> $\mathcal{S}_n$ </EquationSource> </InlineEquation>-symbolic methods, (e.g.) via [7]. Copyright Società Italiana di Fisica, Springer-Verlag 1999
