Order complexes — spherical topology
The poset of weak compositions has an order complex homeomorphic to a \((d-1)\)-sphere, with \(S(n,d)\) facets. The generation tower therefore triangulates a sphere.
This explains the trivial automorphism group (spheres have highly constrained symmetries) and the absence of loops in decay chains via vanishing higher Betti numbers.
RSK correspondence and fusion rules
\(S(n,d)\) counts semistandard Young tableaux of shape \((n-1)\) with entries \(\le d+1\). The Robinson–Schensted–Knuth algorithm gives a deterministic combinatorial fusion rule for combining modes.
The \(5+6=11\) collision is revealed as the natural Pieri rule acting on tableaux — geometrically expected rather than accidental.
Root systems of type \(A_d\)
Each mode corresponds to a weight in the \(A_d\) root lattice. The hockey-stick recursion is the Kostant partition function. The Weyl group \(S_{d+1}\) explains the observed symmetry across sectors.
The full Lie-theoretic toolkit (Weyl character formula, Kostant multiplicity) now becomes available for degeneracies and generating functions.
The binomial Hopf algebra
The divided-power Hopf algebra with basis \(x_{n,d}\) has structure constants given by the hockey-stick numbers. Its antipode yields the algebraic inverse of the mass law via signed Möbius inversion.
This gives a closed-form, search-free inverse and explains the lack of preferred direction in the generation graph via cocommutativity.
Implications for IDWT
These deeper floors show that the rigid 15-mode tower is not a numerical accident but the natural realization of structures that appear across polytopes, topology, tableaux, Lie theory, and Hopf algebras.
We now have powerful new levers: RSK for fusion rules, Hopf antipodes for exact inverses, and root-system methods for symmetries — all purely combinatorial.
See also Inevitable Structure, Further Shelves, and The Combinatorial Skeleton.