A Toy Origin Mechanism for a Crystalline Axiverse / {$\T^8/\Z_{144}\rightarrow \T^8/\Z_{48}\rightarrow \T^8/\Z_{24}$ via Planck-Scale Wandering, Bank Capture, and Even-Subgroup Projection}}

This note presents a toy “origin storyboard” for a divisibility-driven symmetry chainT^8/Z144 → T^8/Z48 → T^8/Z24, motivated by a crystalline/holonomy picture in which a high-order phaseorganization seeds a bank-level holonomy and a leaf-visible subgroup. The intent is explicitlyhypothesis-building rather than certainty: we do not claim a physical cosmogenesis derivation, a UVcompletion, or that our simplified dynamics is the literal behavior of Planck-scale physics. We arepathfinding and providing machine-checkable artifacts. We work with eight angular coordinates θ ∈ [0,2π)^8 as a proxy for an internal R^8 phase fiber reducedto an 8-torus T^8. For n ∈ ℕ we define the Z_n phase grid θ_i = 2π k_i/n and a folded distance-to-gridobservable d_n(θ_i). A “high-symmetry window” is defined by D_n(θ) = max_i d_n(θ_i) < ε for a smalltolerance ε. To capture the intuition that a Planck-sized causal cell may explore a compact phaseconfiguration space, we model “wandering” using two deliberately crude stochastic proxies: (i) IIDuniform draws on T^8 and (ii) a correlated random walk on T^8. For ε=0.003 rad and n=144, an IIDestimate gives a per-step hit probability p ≈ (nε/π)^8 and an expected waiting time E ≈ 7.8×10^6steps; this scaling is observed in finite trial runs. Random walks exhibit longer waiting times due tocorrelations and mixing, but multi-walker runs produce hits at practical budgets. Starting from a Z144-window hit, we model bank formation as deterministic capture into the Z48sublattice via gradient descent in V48(θ) = −A Σ_i cos(48 θ_i), whose minima are exactly the Z48 gridpoints. In the recorded run the state is driven to Z48 at machine precision (max distance to the grid~10^−15 rad). Since Z48 is a sublattice of Z144, the captured state also becomes Z144-compatible atmachine precision. A key point is that leaf-visible Z24 does not need to be a dynamical attractor. A generic Z48-capturedstate may occupy odd Z48 indices, and therefore lie one coarse tick away from the Z24 sublattice. Thisproduces a characteristic mismatch Δθ = 2π/48 = π/24 ≈ 0.1308997 rad in the raw distance-to-Z24 afterZ48 capture. In our run this mismatch appears exactly within floating precision. We therefore treatZ24 visibility as a selection rule: an explicit even-subgroup projection k48 → k48_even that zeros theparity vector (k48 mod 2) and restores Z24 compatibility at machine precision (~10^−16 rad). Thiscleanly separates a dynamical process (bank capture) from an algebraic constraint (leaf visibility),while preserving a sharp quantitative signature (the 2π/48 mismatch) of the intermediate Z48 layer. We provide a provenance-locked verification pack (environment, parameters, JSON/MD summaries, logs, andSHA256 fingerprints) and a storyboard figure with a vertical line marking the projection activationstep. The mechanism should be read as a minimal consistency scaffold, not as a certainty claim, and itis intended to motivate more rigorous dynamical formulations.