The 𝜋/24 Fingerprint How a Dihedral Flutter Collapses into a 𝑍48 Bank Skeleton, and Why the Leaf Sees Only 𝑍24

## Abstract This companion note proposes a compact toy mechanism for the dihedral pre-history behind the chain \[\mathbb{T}^8/D_{144} \longrightarrow \mathbb{T}^8/\mathbb{Z}_{48} \longrightarrow \mathbb{T}^8/\mathbb{Z}_{24}.\] The starting point is a sharp kinematic obstruction: a purely linear realization of \(D_{144}\) on \(\mathbb{T}^8\) would require an element of order \(144\) in \(GL(8,\mathbb{Z})\), but the maximal torsion order there is \(60\). The dihedral stage must therefore be affine rather than linear. We exhibit a free affine action \(\rho:D_{144}\curvearrowright \mathbb{T}^8\), so that \(\mathbb{T}^8/D_{144}\) exists as a smooth flat quotient. We then model bubble contact or bath coupling through a reduced open evolution on probability measures, \[\mu_{t+1}=P_t^*\mu_t, \qquad P_t^*=(1-p_t)Q_t^*+p_t(\rho(s))_*Q_t^*, \qquad p_t=p_0 e^{-\Delta E(t)},\] with early-time approximate dihedral equivariance, late-time contraction, and final capture by the bank potential \(V_{48}(\theta)=-A\sum_{i=1}^8 \cos(48\theta_i)\). In this picture, recurrence explores configuration space, but only coarse-grained dissipation selects and stabilizes a dihedral-compatible sector. The late-stage attractor is the rotation-only \(\mathbb{Z}_{48}\) bank skeleton, while the visible leaf couples only to the even subgroup \(\mathbb{Z}_{24}\subset\mathbb{Z}_{48}\). As a result, generic captured states retain the raw mismatch fingerprint \(\pi/24\), now interpreted as the observable residue of an intermediate bank layer rather than a fundamental leaf symmetry.