Internal-Holonomy Assisted Quantum Tunneling in the Crystalline Axiverse: A Z48→Z24 Coupled-Channel Toy Model with N=23 Controls and Energy-Drift Hardening
Diogenes
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. ### Abstract v.2 We revisit and harden a two-channel, piecewise-constant quantum tunneling toy model in which an internal holonomy phase acts as an interferometric “valve” controlling visible-channel transmission. The system is formulated as coupled-channel scattering for a two-component wavefunction Ψ=(ψ_V,ψ_H)^T under a Hermitian 2×2 potential that is diagonal in the core barrier (width a) and mixed only within thin coupling layers (width ℓ) on either side. The physically relevant control parameter is the relative holonomy Δφ=φ_R−φ_L between the left and right couplers, which governs constructive/destructive V→H→V interference at the exit. We implement an exact segment propagator Y(x+L)=exp(ML)Y(x), solve boundary matching for scattering amplitudes, and certify numerical stability by explicit flux conservation (machine-precision closure in sampled grids). At the canonical working point (E,V0,a,ℓ,g)=(1,6,1.2,0.3,5) in units ħ²/2m=1, the single-channel baseline (g=0) yields T_V0=0.01036713563683. With both channels open asymptotically, optimizing over Δφ reproduces the open-channel enhancement reported in the original note: for V_H,core=0 the visible transmission reaches T_V⋆≈0.122668 at Δφ⋆≈3.322667, corresponding to an 11.832× boost over T_V0, while the transmitted flux partitions between channels with a phase-dependent visible fraction T_V/(T_V+T_H). In the closed-channel regime (V_H,∞>E), the hidden channel serves as a virtual intermediate state and produces a much larger enhancement, with T_V,max≈0.732437 (≈70.65× over baseline) near Δφ=2π. Robustness under holonomy diffusion is assessed using a deterministic wrapped-Gaussian expectation over precomputed T_V(Δφ), reproducing the blur table (e.g. ≈69.38× at σ=0.4 and ≈65.53× at σ=0.8) without Monte Carlo variance. A central question is whether a discrete “governance” rule on Δφ can be imposed without reintroducing fine tuning. We define the governance efficiency η_N(E) as the best discrete-bin transmission over Δφ∈{2πn/N} relative to the continuum optimum. At (E,V_H,core)=(1,2), we reproduce near-lossless governance for Z24 (η_24=0.999976) and Z23 (η_23=0.999137). Hardening against energy drift over E∈{0.6,0.8,1.0,1.2,1.4} reveals a binding point at E=0.8 where η_24 drops to 0.997641, motivating a data-driven drift gate η_min≳0.997. An extended N-scan over N∈[6,48] exhibits a sharp “cliff” (e.g. Z12 gives η_12(E=1)=0.988401), while Z24 and Z48 remain near-unity at the canonical point. **Proposition (Minimum Viable Divisor-of-48 Governance).** Let the admissible governance grids be restricted by the Z48 architecture to divisors of 48 (N|48). Let the drift-hardened gate require η_min≡min_{E∈{0.6,0.8,1.0,1.2,1.4}} η_N(E) ≥ 0.997. Then the smallest admissible N that passes the gate is N=24. (Numerically, the first N in [6,48] satisfying the drift gate is N=18, but it is excluded by the Z48 divisor filter; among N|48, the minimum viable choice is N=24.) These results establish a compact, auditable benchmark: the enhancement mechanism is reproducible, the discretization cost is quantified by explicit efficiency gates, and the stability of the solver is verified by flux closure. The model is intended as a constrained toy laboratory for “crystalline” phase governance and does not by itself supply a microphysical derivation of the Z48→Z24 restriction; rather, it demonstrates how such a restriction, when imposed, can be tested and hardened against energy drift and phase diffusion in a transparent gate-based framework.
•Jan 6, 2026•0 citations