Tick-SU(2)-OCT: Discrete Tick Geometry, Acoustic Scale and the α-Sector

This work formulates a discrete tick model in which the fundamental dynamics is represented by a local update operator of distinguishability on a graph. The basic cell has six oriented ports corresponding to an octahedral packing inside a smooth three-dimensional ball. This packing defines the dimensionless coefficient κ_OCT = 3/π and the area gap ε_A = 1 − κ_OCT². The geometric sector is described in terms of SU(2) holonomies on graph edges and flux variables on plaquettes. In the large-scale limit, this structure is interpreted as a discrete Palatini/BF-like framework from which an effective metric and a ΛCDM background are expected to emerge after geometric constraints and coarse-graining. The main testable hypothesis concerns the early acoustic scale: the octahedral microgeometry may renormalize the scalar-sector spatial stiffness as λ_Y = Y_tick/Y_smooth = 3/π while keeping the leading norm Z_tick/Z_smooth ≈ 1. This gives an acoustic projection r_d → r_d√(3/π). The paper also discusses a related 24-block electromagnetic α-sector as a separate hypothesis requiring an independent derivation of U(1), charge normalization and running α.