Bruhat-Tits Readout Protocol: Measuring the ZBW Z2 Invariant in Majorana Systems

Bruhat-Tits Readout Protocol: Measuring the ZBW Z₂ Invariant in Majorana Systems

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, 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 $k$

1. 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 $\mathbf{q} \sim 1/\lambdaC$ 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

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

| # | 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 | $\delta{\text{Majorana}} < \delta{\text{Dirac}}$ | C | — | True | ✅ |

| M8 | ZBW frequency match to $2E/\hbar$ | B | Yes | — (no signal) | Partial |

Interpretation Matrix

| A | B | C | Verdict |

|:--|:--|:--|:--------|

| ✅ Ultrametric | ✅ $\mathcal{O}{\text{ZBW}}=0$ | ✅ $\deltaM < \delta_D$ | FULL CONFIRMATION |

| ✅ Ultrametric | ❌ $\mathcal{O}{\text{ZBW}} \neq 0$ | ❌ $\deltaM \geq \delta_D$ | Partial: p-adic but not Majorana |

| ❌ Continuous | ✅ $\mathcal{O}{\text{ZBW}}=0$ | ❌ $\deltaM \geq \delta_D$ | Partial: topological but not p-adic |

| ❌ Continuous | ❌ $\mathcal{O}{\text{ZBW}} \neq 0$ | ❌ $\deltaM \geq \delta_D$ | FULL DISCONFIRMATION |

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).