The Adelic ZBW Programme: ZBW-Majorana Hypothesis, Ostrowski-QEC Synthesis, and Cross-Programme Consilience

In the paper's words

The structural underdetermination of the position operator in relativistic quantum mechanics -- manifested at the Archimedean place as zitterbewegung (the Newton-Wigner no-go) and at p-adic places as a discrete Bruhat-Tits observable -- arises from a single representation-theoretic origin: the Silent Parameter Principle. We prove that the truncated representation ring R(SU(2))_k is isomorphic to the character ring R(G_n+1) of the finite quotient $G_{n+1} = (\mathbb{Z}/2^{n+1}\math

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