Internal-Holonomy Assisted Quantum Tunneling in the Crystalline Axiverse: A Z48→Z24 Coupled-Channel Toy Model with N=23 Controls and Energy-Drift Hardening

Quantum tunneling is conventionally modeled as transmission through a classically forbidden region due to the nonzero tail of the wavefunction, leading to the familiar WKB scaling T ~ exp[-2∫κ(x)dx]. In this note we reformulate the same phenomenon through a minimal coupled-channel scattering mechanism that is naturally expressible in the language of the Crystalline Axiverse: a local “projected” sector (Z24) embedded in a larger internal space (Z48), where part of the state can traverse a barrier through a complementary internal subchannel and subsequently re-project back into the visible sector. The intent is not to replace standard quantum mechanics, but to provide an explicit, calculable toy mechanism showing how an internal governance constraint (a discrete Z24 grid) can be kinematically compatible with tunneling-like phenomena without invoking any exotic “vacuum softening”. We consider a two-component stationary Schrödinger problem Ψ(x) = (ψV, ψH)ᵀ with a piecewise-constant 2×2 Hermitian potential matrix. The visible channel V encounters a barrier core of height V0 and width a, while the hidden channel H may see a reduced core barrier VH,core. Mixing between channels is confined to two narrow “coupler” layers at the left and right edges of the core, with coupling strength g and a relative interferometric phase Δφ. In the Axiverse interpretation, Δφ is an internal holonomy (an interferometric valve) that controls constructive or destructive reconversion from the hidden virtual path back into the visible projection. We emphasize that in this stationary setting Δφ is a geometric/mixing phase (holonomy), not a dynamical time delay. Using an exact transfer-matrix propagator exp(ML) for each segment, we compute reflection and transmission amplitudes and enforce flux conservation. For a representative working point (dimensionless units with ħ²/2m = 1): E = 1.0, V0 = 6.0, a = 1.2, coupler width ℓ = 0.3, and g = 5.0, the single-channel baseline visible transmission is Tv0 = 0.010367. When both channels are open asymptotically, the coupled-channel mechanism yields strong enhancement of visible transmission (up to 11.83× for VH,core = 0.0 at the optimal Δφ), with transmitted flux splitting between V and H in a manner consistent with partial re-projection. As VH,core is increased the enhancement decreases smoothly but remains substantial (for VH,core = 3.0 the enhancement is ≈ 5.45× and the visible fraction rises above 0.51). The strongest result is obtained when the hidden channel is closed asymptotically (V_H,∞ > E), so that H carries no asymptotic flux and acts as a purely virtual intermediate state. In this “closed hidden sector” regime, the mechanism can enhance visible transmission by approximately 70.65× at the optimal holonomy phase (Δφ* = 2π), while still satisfying flux conservation over open channels. This demonstrates, in the most conservative sense, that a hidden internal subspace can mediate tunneling as a virtual path without modifying the visible barrier itself. To harden interpretation against numerology, we quantify the “cost” of discretizing Δφ on a fixed governance grid Δφ_n = 2πn/N. For VH,core = 2.0, the continuum optimum at E = 1.0 is φ* ≈ 4.43907 with Tv,max ≈ 0.073557. Restricting Δφ to the Z24 grid yields Tv ≈ 0.073555 at n = 17 (Δφ = 2π·17/24), giving an efficiency η24 = Tv(Z24)/Tv(continuum) ≈ 0.999976. A direct control with N = 23 is also highly efficient (η23 ≈ 0.999137), demonstrating that the mechanism does not “prove” the number 24 but is permissive of it: any sufficiently dense grid (N ≳ 20) incurs negligible penalty because the interference resonances are broad-band rather than high-Q. We report an energy-drift table showing how the optimal discrete bin index shifts with energy (E = 0.6–1.4), while efficiencies remain near unity for both N = 23 and N = 24. Finally, a holonomy-blur (phase diffusion) Monte Carlo shows that noise-averaged efficiencies remain high (≈0.90 even for σ = 0.8 rad) and that the mean advantage of N = 24 over N = 23 under blur is small (O(10^-4)), consistent with alignment effects rather than any arithmetic uniqueness. Overall, this Appendix-T toy model provides an existence proof of compatibility: if a fundamental Z24 governance constraint exists in the Crystalline Axiverse framework, it is kinematically allowed to coexist with coupled-channel tunneling mechanisms without fine-tuning. The model suggests a clean experimental mapping to superconducting circuits (e.g., an asymmetric SQUID analogue where an internal bypass arm and flux-controlled holonomy tune reconversion), while remaining fully consistent with standard coupled-channel scattering physics.