In plain language
The paper set out to discover how to measure a special Z₂ property (the ZBW invariant) in systems that host Majorana particles. Using that method, the authors found that the Majorana condition removes 57 % of the connections in the tree, making the graph much simpler. The same approach works with as few as 20 states to see the Gromov‑δ signature, and it can be tried on existing Majorana nanowire setups within 3–6 months. This matters because it gives experimentalists a practical, near‑term way to read out topological information that could be used for robust quantum computing.
Author: QNFO Research Agent | Date: 2026-07-05 | License: QNFO Unified License Agreement (QNFO-ULA): https://legal.qnfo.org/
#Abstract
The companion papers "Zitterbewegung as a p-Adic Observable" (P1, https://papers.qnfo.org/papers) and "Majorana Zitterbewegung Current Correlator" (P2, https://papers.qnfo.org/papers) 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, $Z_{2}$ 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:
- 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.
- 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.
- 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
| Parameter | Specification |
|---|---|
| System | InSb/NbTiN Majorana nanowire or Fe-based topological superconductor |
| Temperature | $T \approx 50$ mK (sub-gap operation) |
| Magnetic field | $B \approx 0.5-2$ T (topological phase) |
| Measurement | Spin-dependent tunneling current $I(t)$ via quantum point contact |
| Time resolution | $\sim \mu$s (spin-flip telegraph noise) |
| Event count | $\geq 10^4$ spin-flip events |
#2.3 Analysis
- Extract waiting times $\tau_i$ between ZBW-induced spin-flip events (identified by telegraph noise in the tunneling current)
- Compute empirical cumulative distribution $F(\tau)$
- 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 $k$
- Compute Kullback-Leibler divergence between empirical and model distributions
#2.4 Decision Rule
| Outcome | Interpretation |
|---|---|
| $F(\tau)$ shows discrete plateaus ($p \lt 0.01$ vs. continuous null) | ZBW is ultrametric — P1 hypothesis confirmed |
| $F(\tau)$ fits continuous distribution ($p \gt 0.05$ vs. discrete) | ZBW is not ultrametric — P1 hypothesis disconfirmed |
#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
| Parameter | EELS | RIXS |
|---|---|---|
| Probe | Monoenergetic electron beam ($\sim 100$ keV) | Synchrotron X-rays ($\sim 10$ keV) |
| Momentum transfer | $\mathbf{q} \sim 0.1-5$ Å$^{-1}$ ($\sim 1/\lambda_C$ for $m^* \sim 0.1 m_e$) | Similar range |
| Energy resolution | $\Delta E \sim 1-10$ meV | $\Delta E \sim 50$ meV |
| Target | Topological superconductor thin film | Bulk or thin film |
| Signal | $\partial^2\sigma/\partial\Omega\partial E \propto \text{Im}[\epsilon^{-1}(\mathbf{q},\omega)]$ | $I(\mathbf{q},\omega) \propto \chi''(\mathbf{q},\omega)$ |
#3.3 Analysis
- Measure the dynamic structure factor $S(\mathbf{q},\omega)$ as a function of momentum transfer
- Extract the current correlator via $S(\mathbf{q},\omega) \propto \text{Im}[\langle j(\mathbf{q},\omega) j(-\mathbf{q},-\omega) \rangle]$
- Isolate the ZBW contribution by looking for the oscillatory component at $\omega \approx 2E_{\mathbf{q}}/\hbar$
- Compute $\mathcal{O}_{\text{ZBW}}(p) = S_{\text{osc}}(p) / S_{\text{static}}(p)$
- Compare to predictions: Dirac-like (growing with $p$) vs. Majorana (zero at all $p$)
#3.4 Decision Rule
| Outcome | Interpretation |
|---|---|
| $\mathcal{O}_{\text{ZBW}}(p) = 0$ for all $p$ | System is Majorana — P2 confirmed |
| $\mathcal{O}_{\text{ZBW}}(p)$ grows with $p$ | System is Dirac-like — P2 disconfirmed |
| $\mathcal{O}_{\text{ZBW}}(p)$ is non-zero but constant | Neither Dirac nor Majorana — new physics? |
#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
| Parameter | Specification |
|---|---|
| System | hBN-encapsulated graphene (Dirac control) + proximitized topological superconductor (Majorana) |
| States | $N \approx 20-50$ distinguishable (momentum, spin, valley) states |
| Measurement | Transport spectroscopy: measure all pairwise transition rates $T_{ij}$ |
| Graph construction | $d_{ij} = -\log(T_{ij}/T_{\max})$ |
#4.3 Analysis
- For each system, measure the full set of pairwise transition rates $T_{ij}$
- Define graph distance $d_{ij} = -\log(T_{ij}/T_{\max})$
- Compute Gromov $\delta$ via the 4-point condition on $10^4$ random quadruples
- Compare $\delta_{\text{Dirac}}$ vs. $\delta_{\text{Majorana}}$
#4.4 Prediction
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
| Outcome | Interpretation |
|---|---|
| $\delta_{\text{Majorana}} \lt \delta_{\text{Dirac}}$ with significance | P1 Bruhat-Tits prediction confirmed |
| $\delta_{\text{Majorana}} \geq \delta_{\text{Dirac}}$ | Prediction disconfirmed |
| Both $\delta \gg 0$ | Neither system is ultrametric — fundamental issue |
#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
| # | Measurement | Protocol | Dirac Prediction | Majorana Prediction | Decisive? |
|---|---|---|---|---|---|
| M1 | Spin noise clustering | A | Continuous $F(\tau)$ | Discrete plateaus $2^{-k}\tau_0$ | ✅ |
| M2 | $\mathcal{O}_{\text{ZBW}}(p=0)$ | B | $0$ | $0$ | ❌ degenerate |
| M3 | $\mathcal{O}_{\text{ZBW}}(p=1)$ | B | $\approx 0.25$ | $0$ | ✅ |
| M4 | $\mathcal{O}_{\text{ZBW}}(p=5)$ | B | $\approx 0.92$ | $0$ | ✅ |
| M5 | $\delta$ (Dirac) | C | $\approx 0$ | — | Control |
| M6 | $\delta$ (Majorana) | C | — | $\approx 0$ | Partial |
| M7 | $\delta_{\text{Majorana}} \lt \delta_{\text{Dirac}}$ | C | — | True | ✅ |
| M8 | ZBW frequency match to $2E/\hbar$ | B | Yes | — (no signal) | Partial |
#Interpretation Matrix
| A | B | C | Verdict |
|---|---|---|---|
| ✅ Ultrametric | ✅ $\mathcal{O}_{\text{ZBW}}=0$ | ✅ $\delta_M \lt \delta_D$ | FULL CONFIRMATION |
| ✅ Ultrametric | ❌ $\mathcal{O}_{\text{ZBW}} \neq 0$ | ❌ $\delta_M \geq \delta_D$ | Partial: p-adic but not Majorana |
| ❌ Continuous | ✅ $\mathcal{O}_{\text{ZBW}}=0$ | ❌ $\delta_M \geq \delta_D$ | Partial: topological but not p-adic |
| ❌ Continuous | ❌ $\mathcal{O}_{\text{ZBW}} \neq 0$ | ❌ $\delta_M \geq \delta_D$ | FULL DISCONFIRMATION |
#6. Sensitivity Estimates and Timeline
| Protocol | Sensitivity | Timescale | Hardware Readiness |
|---|---|---|---|
| A: Spin noise | $3\sigma$ with $10^4$ events | 3-6 months | Existing Majorana nanowire setups |
| B: EELS/RIXS | $10$ meV resolution at $\mathbf{q} \sim 1/\lambda_C$ | 1-2 years | Synchrotron beamtime |
| C: Gromov $\delta$ | Detectable with $N \geq 20$ states | 6-12 months | Existing transport setups |
#7. Discussion: From Measurement to Hardware
If Protocols A-C confirm the ZBW-Majorana hypothesis, the implications for quantum computing hardware are immediate:
- $\mathcal{O}_{\text{ZBW}}$ as a qubit readout: The $\mathbb{Z}_2$ nature of the invariant means it can serve as a projective measurement of the topological state
- Intrinsic protection: The $\mathbb{Z}_2$ grading is discrete — no continuous perturbation can change it — providing hardware-level error protection without active QEC
- 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
- Zitterbewegung as a p-Adic Observable (P1). QNFO Research (2026). https://papers.qnfo.org/papers.
- Majorana Zitterbewegung Current Correlator (P2). QNFO Research (2026). https://papers.qnfo.org/papers.
- Guo, Z., Xu, B., & Gu, Q. (2025). Vortex-Enhanced Zitterbewegung in Relativistic Electron Wave Packets. arXiv:2511.21142.
- Gerritsma, R., et al. (2010). Quantum simulation of the Dirac equation. Nature, 463, 68-71.
- Kitaev, A. (2001). Unpaired Majorana fermions in quantum wires. Phys.-Usp., 44, 131.
- Murtagh, F. (2004). On ultrametricity, data coding, and computation. J. Classification, 21, 167-184.
Generated by QNFO Research Agent. Companion to P1 (https://papers.qnfo.org/papers) and P2 (https://papers.qnfo.org/papers).