Zitterbewegung as a $p$-Adic Ultrametric Observable: A $\mathbb{Z}_2$ Invariant on the Bruhat–Tits Tree, Its Readout, and Its Non-Additivity Signature

In the paper's words

Zitterbewegung (ZBW) — the Compton-frequency trembling of Dirac wave packets — is conventionally modeled as an Archimedean interference oscillation between positive- and negative-energy branches. We develop a conjectural alternative in which the ZBW readout is ultrametric: the observable lives on a Bruhat–Tits tree over Q₂, and the physically accessible content is not a continuous oscillation but a discrete Z₂-valued invariant carried by the current correlator. [...]

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

Numbers it reports

1. Define the scale map $$\nu: \mathbb{R}_{+} \to \mathbb{Z}, \qquad \nu(x) = \left\lfloor \log_2 x \right\rfloor,$$ assigning to a physical scale $x$ (a frequency, a length) the level of $\mathcal{T}_2$ at which it is read out.

   from 3.2 $p$-adic spectral decomposition

2. The scale map $\nu(x) = \lfloor \log_2 x \rfloor$ is a convention; a different base or a level offset shifts $\chi$, so the specific value $\chi = 1$ is convention-dependent even if the discreteness of the signature is not.

   from 6. Discussion

3. This is the tool we need for the conservation claim in Section 3.3: if the ZBW readout dynamics is a stationary Markov process on a tree isometrically embeddable in $\mathbb{Q}_2$, the reduction theorem guarantees that the evolution can be analyzed with $p$-adic machinery, and isometric embeddings preserve the ultrametric distances on which our invariant is defined.

   from 2. Background and Related Work

4. Hence $$\log_2 \lambda_C = -41.2359, \qquad \nu(\lambda_C) = \lfloor -41.2359 \rfloor = -42.$$ (Note: $\lfloor -41.2359 \rfloor = -42$, not $-41$; the floor of a negative non-integer rounds down.) Step 5 (the $\mathbb{Z}_2$ invariant). $$\chi = \nu(\omega_C) \bmod 2 = 69 \bmod 2 = 1.$$ Result 1.

   from 4.2 Compton scales and tree levels

5. The fractional part of $\log_2 \omega_C$ is $0.3951$ octaves; the ratio of $\omega_C$ to the nearest lower power of two is $$2^{0.3951} = e^{0.3951 \times 0.69315} = e^{0.27387} = 1.3150.$$ Result 3. $\omega_C = 2^{69} \times 1.3150$: the Compton frequency lies a factor $1.3150$ above the level-69 vertex, i.e., $0.3951$ octaves into the level — this fractional position is the $p$-adic "phase" analogue, but it enters the observable only through the floor (the parity), which is why the predicted signature is discrete.

   from 4.3 The non-additivity observable

6. (ii) A demonstration that the floor function $\nu$ is not physically operative — i.e., that the fractional octave $0.3951$ enters the observable, not just its parity — would collapse the predicted discrete signature into a conventional phase measurement.

   from 6. Discussion

How it is done

1. A complementary crossing-count protocol gives the same discrete structure through the $2$-adic norm $|N|_2$.

   from Abstract

2. Our contributions: (1) a $p$-adic analogue of the Dirac spectral decomposition (Sections 3.1–3.2); (2) a proof that the $\mathbb{Z}_2$ invariant is well-defined and conserved under isometric ultrametric evolution (Section 3.3); (3) explicit numerical evaluation of the Archimedean and $p$-adic readouts for the electron, with all arithmetic shown (Section 4); (4) a concrete distinguishing observable — the failure of multiplicativity of the tree readout — and three detection channels (Sections 4–5).

   from 1. Introduction

3. The supplied summary of [1] gives no further detail on the sequential construction, and we use it only at the level stated. $p$-adic stochastic dynamics.

   from 2. Background and Related Work

4. The Bruhat–Tits tree $\mathcal{T}_2$ is an infinite $(2+1)$-regular tree; we use its vertex set as the readout space.

   from 3.2 $p$-adic spectral decomposition

What would falsify it

1. We develop a conjectural alternative in which the ZBW readout is ultrametric: the observable lives on a Bruhat–Tits tree over $\mathbb{Q}_2$, and the physically accessible content is not a continuous oscillation but a discrete $\mathbb{Z}_2$-valued invariant carried by the current correlator.

   from Abstract

2. If the position operator is structurally underdetermined, the "oscillation" readout is one convention among several; we propose that an ultrametric readout — in which the ZBW observable is a hierarchical connectivity pattern on a Bruhat–Tits tree rather than a displacement on the real line — is an equally admissible convention with a sharply different experimental signature.

   from 1. Introduction

3. The Compton-frequency beat between positive- and negative-energy branches admits an adelic formulation in which the ZBW observable is a $\mathbb{Z}_2$-valued topological invariant carried by the current correlator, read out as ultrametric (Bruhat–Tits tree) connectivity, and conserved under the ultrametric evolution.

   from 1. Introduction

Where the claim stops

1. The conservation proof (Section 3.3) holds only for the isometric stationary-Markov ansatz of [2]; we do not prove parity preservation for a general adelic Dirac dynamics, and any dynamics that moves the readout across the root would break the proof.

   from 6. Discussion

2. The non-additivity observable $\Delta$ compares a tree distance to a logarithm — dimensionally both are pure numbers here, but the physical operability of $\Delta$ as a measured quantity is assumed, not demonstrated.

   from 6. Discussion

3. If a Majorana platform shows a nonzero ZBW current-correlator signal, the $\mathbb{Z}_2$ distinction of [11] fails, removing the physical content of $\chi$.

   from 6. Discussion

Where to read

1. Abstract239 words · 1 min
2. 1. Introduction457 words · 2 min
3. 2. Background and Related Work1,055 words · 5 min
4. 3. Methods787 words · 3 min
5. 4. Analysis863 words · 4 min
6. 5. Results247 words · 1 min
7. 6. Discussion388 words · 2 min
8. 7. Conclusion135 words · 1 min
9. Appendix A. Divergence report191 words · 1 min
10. Appendix B. Claim attribution188 words · 1 min