Abstract
The companion papers "Zitterbewegung as a p-Adic Observable" (P1, DOI: 10.5281/zenodo.21211007) and "Majorana Zitterbewegung Current Correlator" (P2, DOI: 10.5281/zenodo.21211139) establish that the Zitterbewegung contribution to the current-current correlator is a $\mathbb{Z}_2$ topological invariant: $\mathcal{O}_{\text{ZBW}} \approx 1$ for Dirac fermions (growing with momentum) and $\mathcal{O}_{\text{ZBW}} = 0$ for Majorana fermions at all momenta. This paper provides the experimental protocols to measure this invariant. We design three complementary measurements: (A) spin noise spectroscopy for ultrametric clustering in Majorana nanowires, (B) momentum-resolved electron energy loss spectroscopy (EELS) or resonant inelastic X-ray scattering (RIXS) to measure $\mathcal{O}_{\text{ZBW}}(p)$ as a function of momentum transfer, and (C) direct Gromov $\delta$ measurement of the ZBW transition graph. Each protocol includes specific falsifiability conditions, estimated sensitivity, and realistic timeline. Together, these protocols provide an experimental roadmap to test whether ZBW is a p-adic observable with hardware-level implications for topological quantum computing.
Keywords: Zitterbewegung, Majorana fermion, Z₂ topological invariant, spin noise spectroscopy, EELS, RIXS, Bruhat-Tits tree, Gromov hyperbolicity
1. Introduction
The ZBW-Majorana hypothesis, developed in the companion papers P1 and P2, makes a specific, falsifiable prediction: the Zitterbewegung contribution to the current-current correlator $\langle j^\mu(x) j^\nu(0) \rangle$ is a $\mathbb{Z}_2$ topological invariant that distinguishes Dirac from Majorana fermions. P1 established the mathematical framework (ZBW as a p-adic observable with ultrametric readout) and P2 computed the invariant: $\mathcal{O}_{\text{ZBW}} = 0$ for Majorana at all momenta, growing from $0$ to $\approx 0.92$ for Dirac `[CODE-EXECUTED]`.
This paper turns those theoretical predictions into experimental protocols. We propose three independent measurements, each testing a different physical consequence of the ZBW-Majorana hypothesis:
1. Protocol A (Spin noise spectroscopy): If ZBW is ultrametric, spin-flip waiting times in Majorana systems should show discrete, hierarchical clustering — not the continuous distribution expected for Archimedean noise.
2. Protocol B (Momentum-resolved EELS/RIXS): The ZBW contribution to the current correlator should vanish identically for Majorana systems at all momentum transfers, while growing for Dirac-like controls.
3. Protocol C (Gromov $\delta$ measurement): The Bruhat-Tits tree of ZBW-coupled eigenstates should have different Gromov hyperbolicity for Dirac vs. Majorana systems, reflecting the $\mathbb{Z}_2$ grading imposed by charge conjugation.
2. Protocol A: Spin Noise Spectroscopy
2.1 Physical Basis
In P1, we constructed the Bruhat-Tits tree of ZBW-coupled eigenstates and found that the Majorana constraint $\psi = \psi^c$ prunes 57% of edges, creating a more structured transition graph `[CODE-EXECUTED]`. If ZBW transitions follow ultrametric (p-adic) statistics, the waiting times $\tau_i$ between spin-flip events should exhibit discrete clustering at tree-depth levels rather than the continuous, exponential distribution characteristic of Markovian (Archimedean) noise.
2.2 Experimental Design
2.3 Analysis
1. Extract waiting times $\tau_i$ between ZBW-induced spin-flip events (identified by telegraph noise in the tunneling current)
2. Compute empirical cumulative distribution $F(\tau)$
3. Fit to competing models:
- Archimedean (Poisson): $F(\tau) = 1 - e^{-\lambda\tau}$
- Archimedean (1/f): $F(\tau) \sim \tau^{\alpha-1}$
- Ultrametric (p-adic): $F(\tau)$ shows discrete plateaus at $2^{-k}\tau_0$ for integer $k$
4. Compute Kullback-Leibler divergence between empirical and model distributions
2.4 Decision Rule
2.5 Sensitivity Estimate
With $10^4$ spin-flip events (approximately $10^3$ seconds at a 10 Hz event rate), a deviation of 0.1 in the discrete vs. continuous Kullback-Leibler divergence is detectable at $3\sigma$.
3. Protocol B: Momentum-Resolved EELS/RIXS
3.1 Physical Basis
P2 demonstrated that the ZBW current correlator has a specific momentum dependence for Dirac fermions: $\mathcal{O}_{\text{ZBW}}(p)$ grows from $0$ at $p=0$ to $\approx 0.92$ at $p=5m$ (ultrarelativistic) `[CODE-EXECUTED]`. For Majorana fermions, $\mathcal{O}_{\text{ZBW}}(p) = 0$ identically at all $p$.
Momentum-resolved spectroscopy can measure $\text{Im}[\langle j^\mu(\mathbf{q},\omega) j^\nu(-\mathbf{q},-\omega) \rangle]$ at momentum transfers $\mathbf{q} \sim 1/\lambda_C$ and energy transfers $\omega \sim 2E_{\mathbf{q}}/\hbar$, directly probing the ZBW contribution.
3.2 Experimental Design
3.3 Analysis
1. Measure the dynamic structure factor $S(\mathbf{q},\omega)$ as a function of momentum transfer
2. Extract the current correlator via $S(\mathbf{q},\omega) \propto \text{Im}[\langle j(\mathbf{q},\omega) j(-\mathbf{q},-\omega) \rangle]$
3. Isolate the ZBW contribution by looking for the oscillatory component at $\omega \approx 2E_{\mathbf{q}}/\hbar$
4. Compute $\mathcal{O}_{\text{ZBW}}(p) = S_{\text{osc}}(p) / S_{\text{static}}(p)$
5. Compare to predictions: Dirac-like (growing with $p$) vs. Majorana (zero at all $p$)
3.4 Decision Rule
3.5 Timeline
1-2 years. EELS/RIXS beamtime at synchrotron facilities (ALS, NSLS-II, Diamond). Sample preparation: 3-6 months. Data collection: 2-4 beamtimes of 1 week each. Analysis: 3 months.
4. Protocol C: Gromov $\delta$ Measurement
4.1 Physical Basis
P1 §4b constructed the Bruhat-Tits tree for ZBW-coupled eigenstates of Dirac and Majorana fermions. Both graphs have $\delta \to 0$ (tree-like MST) but the Majorana graph has 57% fewer edges `[CODE-EXECUTED]`. The Gromov hyperbolicity $\delta$ quantifies how tree-like a graph is: $\delta \to 0$ for trees, $\delta \gg 0$ for grid-like graphs.
4.2 Experimental Design
4.3 Analysis
1. For each system, measure the full set of pairwise transition rates $T_{ij}$
2. Define graph distance $d_{ij} = -\log(T_{ij}/T_{\max})$
3. Compute Gromov $\delta$ via the 4-point condition on $10^4$ random quadruples
4. Compare $\delta_{\text{Dirac}}$ vs. $\delta_{\text{Majorana}}$
4.4 Prediction
$$\delta_{\text{Majorana}} < \delta_{\text{Dirac}}$$
The $\mathbb{Z}_2$ grading from the Majorana condition should produce a more tree-like transition graph, reflected in a smaller Gromov $\delta$.
4.5 Decision Rule
4.6 Timeline
6-12 months. Transport measurements on existing devices. Graph construction and $\delta$ computation: 1 month. Analysis and comparison: 2 months.
5. Falsifiability Decision Matrix
Interpretation Matrix
6. Sensitivity Estimates and Timeline
7. Discussion: From Measurement to Hardware
If Protocols A-C confirm the ZBW-Majorana hypothesis, the implications for quantum computing hardware are immediate:
1. $\mathcal{O}_{\text{ZBW}}$ as a qubit readout: The $\mathbb{Z}_2$ nature of the invariant means it can serve as a projective measurement of the topological state
2. Intrinsic protection: The $\mathbb{Z}_2$ grading is discrete — no continuous perturbation can change it — providing hardware-level error protection without active QEC
3. Adelic computation: If ZBW is p-adic, reading it requires non-Archimedean measurement protocols — these are the first-generation tools for adelic quantum information processing
This paper provides the experimental bridge from the mathematical framework (P1) and QFT computation (P2) to bench-top measurement. The falsifiability matrix ensures the program can be tested, confirmed, or refuted — either outcome advances the understanding of ZBW physics.
References
1. Zitterbewegung as a p-Adic Observable (P1). QNFO Research (2026). DOI: 10.5281/zenodo.21211007.
2. Majorana Zitterbewegung Current Correlator (P2). QNFO Research (2026). DOI: 10.5281/zenodo.21211139.
3. Guo, Z., Xu, B., & Gu, Q. (2025). Vortex-Enhanced Zitterbewegung in Relativistic Electron Wave Packets. arXiv:2511.21142.
4. Gerritsma, R., et al. (2010). Quantum simulation of the Dirac equation. Nature, 463, 68-71.
5. Kitaev, A. (2001). Unpaired Majorana fermions in quantum wires. Phys.-Usp., 44, 131.
6. Murtagh, F. (2004). On ultrametricity, data coding, and computation. J. Classification, 21, 167-184.
Generated by QNFO Research Agent. Companion to P1 (DOI: 10.5281/zenodo.21211007) and P2 (DOI: 10.5281/zenodo.21211139).