Testable, falsifiable predictions. Concrete proposals for first experiments.
Prediction: In a hierarchical system, for any three states $A, B, C$, the two largest pairwise distances are equal (all triangles are isosceles).
For a thermally activated system, $r \propto e^{-\Delta E / k_B T}$, so larger distance → lower rate. With $n = 100$ measurements per pair, construct a 95% CI for $\Delta$. If it includes zero, ultrametricity is supported. If $\Delta$ is significantly positive, the system is not ultrametric.
Prediction: $\log P(\text{error}) \approx \log C - (\Delta E_D / \varepsilon)^\beta$.
Method: Vary temperature $T$ (varying $\varepsilon = k_B T$) at fixed encoding depth $D$. Measure error rate. Plot $\log P$ vs. $1/T$. The low-temperature slope gives $-\Delta E_D/k_B$ (for $\beta = 1$). Repeat for different $D$ to verify $\Delta E_D = E_0 \cdot b^{-\alpha D}$.
Expected: Linear regime (exponential suppression) at low $T$; saturation at high $T$ where barrier is irrelevant; bend at $\varepsilon \approx \Delta E_D$.
Prediction: $N$ sub-threshold perturbations → total error bounded by max individual perturbation. Compare:
Signature: Flat error rate vs. growing error magnitude. In the hierarchical system, you either have an error or you don't — no "partial" error.
Molecule: 1,2,3,4-tetrafluorobenzene ($\text{C}_6\text{H}_2\text{F}_4$) — four $^{19}$F nuclei with natural coupling hierarchy:
| Coupling | Type | $J$ (Hz) |
|---|---|---|
| $J_{12}$ | Ortho | 20 |
| $J_{13}$ | Meta | 8 |
| $J_{14}$ | Para | 2 |
Encoding: Logical bit in F1–F2 pair state. Weaker couplings form deep hierarchy. Apply controlled RF noise, vary noise power, measure logical error rate. Equipment: Standard 400–600 MHz NMR spectrometer with $^{19}$F capability.
Platform: Linear chain of $^{171}\text{Yb}^+$ or $^{40}\text{Ca}^+$ ions. Motional modes have a natural frequency hierarchy — center-of-mass mode (highest frequency) down to buckling modes (lowest). Encode logical bit in phonon number of the highest mode.
Apply controlled electric field noise. Measure logical error rate vs. noise power. Advantages: Single-quantum precision, near-unit fidelity, tunable mode hierarchy, long coherence times.