Vanishing ZBW Signal: The ZBW-Majorana Hypothesis as a Unified Framework for Topological Fermion Distinction

In the paper's words

The ZBW-Majorana hypothesis, developed across four companion papers (P1--P4), establishes that Zitterbewegung (ZBW) --- the rapid trembling motion of Dirac fermions at the Compton scale --- is a Z₂ topological observable that distinguishes Dirac from Majorana fermions at the hardware level.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. The six claims form a single logical chain with no weak links: [Majorana constraint ψ = ψ^c] | v [ZBW current correlator vanishes identically] ─── Claim 1 | v [O_ZBW is a Z₂ topological invariant] ─── Claim 2 | v [Intrinsic topological protection] ─── Claim 3 | +──> [Bruhat-Tits graph: 57% edge reduction] ─── Claim 4 +──> [Experimental protocols A, B, C] ─── Claim 5 | v [Hardware-level topological quantum computing] ─── Claim 6 If any claim fails, the downstream implications weaken but may not collapse completely: If Claim 1 is wrong (ZBW correlator does not vanish for Majorana), Claims 2--6 collapse because the binary distinction evaporates.

   from 8. Synthesis: The Logical Chain

2. Parameter EELS RIXS Probe $\sim 100$ keV electrons $\sim 10$ keV synchrotron X-rays Momentum transfer $0.1$--$5$ \AA${}^{-1}$ Similar Energy resolution $1$--$10$ meV $\sim 50$ meV Target Topological superconductor thin film Bulk or thin film Timeline 1--2 years Decision rule: $\mathcal{O}_{\text{ZBW}}(p) = 0$ for all $p$ confirms Majorana; $\mathcal{O}_{\text{ZBW}}(p)$ growing with $p$ indicates Dirac-like [3, §3.4].

   from 6.2 Protocol B: Momentum-Resolved EELS/RIXS

3. Property Dirac Majorana Nodes 240 240 Edges 100\% baseline 43\% of baseline (57\% reduction) Gromov $\delta$ 0 (tree-like MST) 0 (tree-like MST) Ultrametric violations 31\% Reduced The Majorana constraint $\psi = \psi^c$ prunes 57\% of edges [2, §3; 3, §2.1], creating a more structured transition graph.

   from 5.2 Dirac vs. Majorana Graphs

4. The 57\% reduction reflects the fraction of ZBW transitions that involve the anti-particle channel --- these are not "forbidden" in the Majorana case but rather merged with the particle channel, reducing the effective edge count [2, §3; 4, §2].

   from 5.3 Physical Interpretation

5. 57\% edge reduction: This is a specific numerical result for the $11 \times 11$ grid at threshold.

   from 11. Limitations and Caveats

6. The qualitative prediction (fewer edges for Majorana) is robust; the exact percentage depends on grid resolution and edge-weight threshold [3, §2.1].

   from 11. Limitations and Caveats

How it is done

1. A topological invariant is, by definition, unchanged under continuous deformations of the system [2, §4; 4, §4.1].

   from 4.1 The Topological Argument

2. The ZBW transition graph $\mathcal{G} = (V, E)$ is constructed on a discretised momentum grid [2, §3]: Nodes $V$: momentum-spin eigenstates $|p_x, p_y, s\rangle$ of the Dirac Hamiltonian on an $11 \times 11$ grid (240 nodes including spin) Edges $E$: states connected by non-zero ZBW transition amplitude, weighted by the matrix element $\langle p', s' | \boldsymbol{\alpha} | p, s \rangle$

   from 5.1 Graph Construction

3. Physical basis: If ZBW transitions follow ultrametric (p-adic) statistics, spin-flip waiting times in Majorana systems should exhibit discrete, hierarchical clustering at tree-depth levels rather than the continuous distribution characteristic of Markovian noise [3, §2].

   from 6.1 Protocol A: Spin Noise Spectroscopy

4. Physical basis: The ZBW current correlator $\mathcal{O}_{\text{ZBW}}(p)$ should vanish identically for Majorana systems at all momentum transfers, while growing for Dirac-like controls [1, §4; 3, §3].

   from 6.2 Protocol B: Momentum-Resolved EELS/RIXS

What would falsify it

1. The qualitative prediction (fewer edges for Majorana) is robust; the exact percentage depends on grid resolution and edge-weight threshold [3, §2.1].

   from 11. Limitations and Caveats

2. The ZBW-Majorana hypothesis, developed across four companion papers (P1--P4), establishes that Zitterbewegung (ZBW) --- the rapid trembling motion of Dirac fermions at the Compton scale --- is a $\mathbb{Z}_2$ topological observable that distinguishes Dirac from Majorana fermions at the hardware level.

   from Abstract

3. We present a falsifiability matrix covering all six claims and an integrated experimental roadmap with estimated timelines.

   from Abstract

Where the claim stops

1. Interaction effects: The computation assumes free fields.

   from 11. Limitations and Caveats

2. Any continuous perturbation $H \to H + \epsilon V$ that respects the fermionic statistics cannot continuously deform $\mathcal{O}_{\text{ZBW}} = 0$ into $\mathcal{O}_{\text{ZBW}} \neq 0$ because there is no continuous path in a discrete space.

   from 4.1 The Topological Argument

3. Energetic protection requires $\Delta E \gg k_B T$ and can fail at elevated temperatures or under strong perturbations.

   from 4.1 The Topological Argument

Where to read

1. Abstract186 words · 1 min
2. 1. Introduction218 words · 1 min
3. 2. Claim 1: Vanishing ZBW Signal306 words · 1 min
4. 3. Claim 2: $\mathbb{Z}_2$ Topological Invariant253 words · 1 min
5. 4. Claim 3: Topological Protection304 words · 1 min
6. 5. Claim 4: Bruhat--Tits Tree Geometry252 words · 1 min
7. 6. Claim 5: Experimental Protocols270 words · 1 min
8. 7. Claim 6: Implications for Quantum Computing247 words · 1 min
9. 8. Synthesis: The Logical Chain181 words · 1 min
10. 9. Falsifiability Matrix228 words · 1 min
11. 10. Integrated Experimental Roadmap148 words · 1 min
12. 11. Limitations and Caveats137 words · 1 min