Paying the Tick: Quantized Leakage from a Minimal Z₄₈ Generator

**Abstract.** We present an explicit, minimal two‑channel mechanism that turns an order‑48 bank tick into a leaf‑level leakage parameter in the Bank/IRIS framework. Starting from the affine Z₄₈ holonomy that produces the primitive phase ω₂₄ = e^{2πi/24}, we model one fine tick as a swap‑plus‑phase operator U acting on a hidden Z₂ fiber (a parity doublet). The unitary driver closes exactly after 48 ticks (U^{48} = I) while swapping the hidden fiber at each step. Assuming leaf observables are fiber‑blind (p‑blind), we compute the per‑tick projection leakage as the transition probability from the visible even channel |+⟩ to the hidden odd channel |−⟩, yielding p = sin²(π/24). The odds‑ratio parameter appearing in the time‑kinetic kernel is then fixed as ε = p/(1−p) = tan²(π/24). We provide a compact dictionary γ = cos(2θ), p = (1−γ)/2, ε = (1−γ)/(1+γ), showing that sin² and tan² forms are reparametrizations of the same channel rather than competing claims. Finally, we state the correct consistency condition behind “clean restart” over 48 ticks: stroboscopic closure of the unitary driver together with repinning/reprojection at cycle boundaries, rather than reversal of coarse‑grained dissipation. This resolves a key modeling ambiguity and supplies a shared bridge behind multiple numerical hints, while keeping any mapping to specific Standard Model structures explicitly model‑dependent.