Mechanism‑First Pre‑Registered Testing of p‑adic Mass‑Ratio Hypotheses

In the paper's words

The Standard Model mass spectrum lacks a mechanistic explanation, and previous adelic and p‑adic numerology has produced only look‑elsewhere false positives. We close this gap by introducing a mechanism‑first testing framework that requires any proposed p‑adic or adelic mechanism to generate explicit infrared mass‑ratio predictions before statistical evaluation. The framework pre‑registers predictions, constructs surrogate spectra, and evaluates significance against a null hypothesis that the Standard Model provides no such relations.

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

Numbers it reports

1. Statistical threshold definition – Prior to any calculation we fix a family‑wise error rate $\alpha=0.05$ and a matching window $\delta=0.0035$ (3.5 %) for any ratio, matching conventions used in earlier adelic searches.

   from 3. Methods

2. Using concrete results from the literature we (i) derive the bound $n_{\max}=d$ for p‑adic $\gamma$‑equiangular lines, (ii) verify the surviving cross‑ratio hint $\text{CR}(e,\mu,\tau,H)=976/919$ with an absolute deviation of $5.76\times10^{-7}$, (iii) recover the exact phase‑transition condition $p=3$ for the three‑state p‑adic Potts model on a Cayley tree, (iv) compute the probability $P=0.1$ that a randomly chosen prime among the first ten yields a phase transition, (v) obtain the regulator $R=\pi/4\approx0.78539816$ for the quadratic field $\mathbb{Q}(\sqrt{2})$, and (vi) apply the protocol to an $SO(10)$ two‑loop RG prediction, finding no statistically significant hit.

   from Abstract

3. Derivation of the bound $n_{\max}=d$ for p‑adic $\gamma$‑equiangular lines (Section 4.2).

   from 1. Introduction

4. Numerical evaluation of the cross‑ratio hint $\text{CR}=976/919$ with deviation $5.76\times10^{-7}$ (Section 4.3).

   from 1. Introduction

5. QNFO cross‑ratio analysis – [8] reports a surviving hint: the cross‑ratio $\text{CR}(e,\mu,\tau,H)=976/919$.

   from 2. Background and Related Work

6. [8] reports the cross‑ratio $$ \text{CR}(e,\mu,\tau,H)=\frac{976}{919}. $$ Compute the rational value: $$\frac{976}{919}=1.062023939\ldots$$ The absolute deviation from unity is $$\Delta=\left|\frac{976}{919}-1\right| =\frac{976-919}{919} =\frac{57}{919} \approx6.202394\times10^{-2}.$$ The quoted deviation $5.76\times10^{-7}$ refers to the absolute deviation after propagating the experimental uncertainties of the four masses.

   from 4.3 Numerical Evaluation of the Surviving Cross‑Ratio Hint

How it is done

1. Using concrete results from the literature we (i) derive the bound $n_{\max}=d$ for p‑adic $\gamma$‑equiangular lines, (ii) verify the surviving cross‑ratio hint $\text{CR}(e,\mu,\tau,H)=976/919$ with an absolute deviation of $5.76\times10^{-7}$, (iii) recover the exact phase‑transition condition $p=3$ for the three‑state p‑adic Potts model on a Cayley tree, (iv) compute the probability $P=0.1$ that a randomly chosen prime among the first ten yields a phase transition, (v) obtain the regulator $R=\pi/4\approx0.78539816$ for the quadratic field $\mathbb{Q}(\sqrt{2})$, and (vi) apply the protocol to an $SO(10)$ two‑loop RG prediction, finding no statistically significant hit.

   from Abstract

2. Contributions Formal mechanism‑first pre‑registration protocol (Section 3).

   from 1. Introduction

3. The literature provides several concrete results that motivate a mechanism‑first approach. p‑adic Beilinson conjecture – [1] formulates a p‑adic analogue of Borel’s regulator relation for number fields, linking special values of $L$‑functions to $K$‑theory regulators.

   from 2. Background and Related Work

4. We construct a pre‑registered testing pipeline consisting of the following steps: Mechanism selection – Choose a concrete mechanism that yields explicit infrared (IR) mass‑ratio predictions.

   from 3. Methods

What would falsify it

1. The mechanism is falsified if any of the following occurs: The surrogate null distribution yields a $95^{\text{th}}$‑percentile hit count $k_{\alpha}\lt k_{\text{obs}}$.

   from Falsification Criterion

2. How to embed the equiangular‑line bound into a concrete particle‑physics model that predicts specific mass‑ratio vectors?

   from Open Questions

3. Using concrete results from the literature we (i) derive the bound $n_{\max}=d$ for p‑adic $\gamma$‑equiangular lines, (ii) verify the surviving cross‑ratio hint $\text{CR}(e,\mu,\tau,H)=976/919$ with an absolute deviation of $5.76\times10^{-7}$, (iii) recover the exact phase‑transition condition $p=3$ for the three‑state p‑adic Potts model on a Cayley tree, (iv) compute the probability $P=0.1$ that a randomly chosen prime among the first ten yields a phase transition, (v) obtain the regulator $R=\pi/4\approx0.78539816$ for the quadratic field $\mathbb{Q}(\sqrt{2})$, and (vi) apply the protocol to an $SO(10)$ two‑loop RG prediction, finding no statistically significant hit.

   from Abstract

Where the claim stops

1. Assumption Sensitivity The three‑state Potts model on a Cayley tree of order two accurately represents the prime‑dependent signature.

   from Limitations

2. Future work will extend the pipeline to incorporate lattice‑QCD inputs, higher‑loop RG flows, and experimental searches for the prime‑periodic noise signatures proposed in [11].

   from 8. Conclusion

3. Conversely, a failed test would prevent costly investment in models that cannot be distinguished from random coincidences.

   from 1. Introduction

Where to read

1. Abstract177 words · 1 min
2. 1. Introduction237 words · 1 min
3. 2. Background and Related Work464 words · 2 min
4. 3. Methods284 words · 1 min
5. 4. Analysis353 words · 2 min
6. 5. Results88 words · 1 min
7. 6. Discussion323 words · 1 min
8. 7. Practical Implications and Engineering Pathways325 words · 1 min
9. 8. Conclusion96 words · 1 min
10. Provenance and AI Involvement225 words · 1 min
11. Appendix A. Divergence report21 words · 1 min
12. Appendix B. Claim attribution549 words · 2 min