Tunneling has two principles, joined at the Wick rotation t ↔ iτ, the two axes of the complex exponential e^z: the reversible golden-ratio spectrum (order, in space) and the irreversible dynamics (flow, in time).

Left: Order (φ · spatial · reversible). The Kohmoto–Kadanoff–Tang trace map (1983) computes the Cantor band-membership of a Fibonacci golden-ratio superlattice in O(log N): a gen-35 ≈ 9.2-million-cell lattice in milliseconds. Here φ is genuinely load-bearing: the golden ratio is the recurrence (the silver-ratio word breaks the invariant). The allowed-fraction shown is a grid-sampled estimate of the shrinking Cantor measure.

Bottom: the real-world payoff (φ · optics). The exact same Kohmoto trace map drives a fabricable device: a quarter-wave TiO₂/SiO₂ dielectric multilayer stacked in the Fibonacci word: a photonic quasicrystal mirror. Its transmission spectrum T(λ) carries the φ signature as fractal (Cantor) band gaps (gold), where a periodic Bragg stack has only simple, evenly-spaced gaps. The optical layer matrices are unimodular, so their half-traces obey the identical golden trace map with a conserved Kohmoto invariant: φ is load-bearing here by the same mechanism, now in classical optics you could build on a bench.

Right: Flow (irreversible dynamics · time). The same two wells, now in time: H = −(Δ/2)X drives coherent tunneling between them, while a bath (rate γ_φ) dephases it. With γ_φ=0 the population ⟨Z⟩ oscillates and recurs forever, reversible, no decay. Turn the bath on and the oscillation irreversibly damps to localization: the dissipator (the OPERATOR) is the engine. The constant e washes (it's notation); the rising entropy is a readout, not the cause.

The two are orthogonal registers joined only at the Wick rotation: φ orders the spatial spectrum, the operator drives the irreversible flow. They compose; there is no φ×e fusion. We do not claim φ governs the dynamics, nor that e governs the spectrum, nor any speed-up over diagonalization.