Does a CPT‑Symmetric $t=0$ Boundary Force the Majorana Condition? A Derivation of the Cross‑Sheet Conjugation Constraint

In plain language

The paper asked if a special start‑time rule forces particles to be their own twins. It built a math model of that rule that obeys the usual balance laws. The model shows the rule can be written as a simple local linear condition. With detect rate 0.8, background 4, width 4 keV and exposure 40 kg·yr, it gives a half‑life near 10^26 years. The result says the rule does not force the rare decay to happen. So experiments must still look for the decay to learn if particles are their own twins.

Restated for a non-expert reader; the original wording follows.

In the paper's words

Neutrinoless double‑beta decay experiments invest tonne‑scale resources to test whether neutrinos are Majorana particles, a question that currently lacks a dynamical explanation. A proposed dynamical origin is a CPT‑symmetric cosmological boundary at t=0 that might enforce the Majorana reality condition ψ=ψ^c. We formalize the t=0 CPT boundary as a constraint on the spinor bundle and derive the most general local, gauge‑invariant boundary condition compatible with CPT invariance.

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

Numbers it reports

1. The projected half‑life sensitivity $T_{1/2}^{0\nu}$ for a background‑free exposure $E$ (in kg·yr) scales as $$ T_{1/2}^{0\nu}\propto \frac{N\,\epsilon\,E}{\sqrt{B\,\Delta E}}, $$ where $\epsilon$ is detection efficiency (assumed $0.8$), $B=4$ counts/(tonne·yr) is the background rate, and $\Delta E=4\,$keV is the ROI width [2].

   from 4.5 Numerical cross‑checks: $^{76}$Ge isotope count and half‑life sensitivity

2. Half‑life sensitivity projection – Assuming a detection efficiency of $0.8$, background $B=4$ counts/(tonne·yr), ROI width $\Delta E=4$ keV, and a one‑year exposure of $40$ kg·yr, the projected neutrinoless double‑beta decay half‑life sensitivity is of order $10^{26}$ yr.

   from 5. Results

How it is done

1. By treating the boundary as a constraint on the spinor bundle, we derive the most general local, gauge‑invariant boundary condition compatible with CPT invariance and examine its implications for charged and neutral fields.

   from 1. Introduction

2. The methodology of deriving a concise analytic condition from underlying equations informs our approach to extracting a boundary‑induced constraint on spinors.

   from 2. Background and Related Work

3. We consider a four‑component Dirac spinor $\psi(x)$ defined on a globally hyperbolic spacetime with a spacelike hypersurface $\Sigma_{0}$ at $t=0$.

   from 3. Methods

4. (c) Smallest next experiment – A feasible near‑term test is to measure the ZBW topological invariant $\mathcal{O}_{\text{ZBW}}$ in a material where Majorana quasiparticles are engineered (e.g., topological superconductors).

   from 7. Practical Implications and Engineering Pathways

What would falsify it

1. Observation of a charged fermion exhibiting Majorana‑type lepton‑number‑violating processes (e.g., a charged lepton double‑beta decay) would directly falsify the claim that the $t=0$ CPT boundary cannot enforce a pointwise Majorana condition for charged fields.

   from Falsifiability

2. Conversely, a definitive null result in all charged‑lepton lepton‑number‑violation searches, combined with a positive neutrinoless double‑beta decay signal, would be consistent with the derived cross‑sheet constraint.

   from Falsifiability

3. Can the Zitterbewegung‑based observable $\langle j^\mu(x) j^\nu(0) \rangle$ proposed in [7, 8] be employed to test the cross‑sheet constraint experimentally?

   from Open Questions

Where the claim stops

1. Exact CPT invariance – The analysis assumes strict CPT symmetry at the cosmological boundary.

   from Limitations

2. Locality and linearity – The boundary condition is restricted to local, linear relations.

   from Limitations

3. Consequently, the boundary does not compel neutrinoless double‑beta decay, preserving the essential role of experimental searches.

   from 8. Conclusion

Where to read

1. Abstract158 words · 1 min
2. 1. Introduction256 words · 1 min
3. 2. Background and Related Work193 words · 1 min
4. 3. Methods172 words · 1 min
5. 4. Analysis526 words · 2 min
6. 5. Results137 words · 1 min
7. 6. Discussion204 words · 1 min
8. 7. Practical Implications and Engineering Pathways289 words · 1 min
9. 8. Conclusion96 words · 1 min
10. Appendix A. Divergence report19 words · 1 min
11. Appendix B. Claim attribution116 words · 1 min
12. Provenance and AI Involvement225 words · 1 min