The quantum structure sub-programme of the Cosmochrony corpus addresses a single central question: how does the admissible ${\mathrm{SU}}(2)$ sector force the structures of quantum mechanics, without importing quantum postulates? Starting from the admissibility constraints of A1–A4 on the binary icosahedral group ${2I}$ and the Weil representation ${V_\rho}$, the sub-programme derives: phase coherence of the admissible fibre (Q1); the singlet correlator $E(\hat{a},\hat{b}) = -\hat{a}\cdot\hat{b}$ (Q1, Q3); the Tsirelson bound $|S_{\mathrm{CHSH}}| \leq 2\sqrt{2}$ (Q1); the Born rule in the ${\mathrm{SU}}(2)$ sector (Q1, Q3); the identification of ${\mathrm{SU}}(2)$ as the unique stable fixed point of the admissibility flow (Q2); the universal spin-$j$ generalisation of all the above for all five admissible sectors $j \in \{\tfrac{1}{2}, 1, \tfrac{3}{2}, 2, \tfrac{5}{2}\}$ of ${2I}$ (Q3); and the structural non-applicability of Bell factorizability in non-injective frameworks (Bell paper). All these results are proved within the ${\mathrm{SU}}(2)$ sector. Extension to arbitrary observables, multipartite systems, and continuous spectra remains the primary open deliverable.