Joint Bruhat–Tits Tree Product as a Testable Cross‑Domain Structure for Standard‑Model Mass Ratios, Quantum‑Error‑Correction Lattices, and Efimov Scaling

In the paper's words

The Standard‑Model mass spectrum, Gottesman–Kitaev–Preskill (GKP) lattice spacings, and Efimov discrete‑scaling factor are each treated as empirical inputs, limiting predictive power. A conjectured universal geometry—the diagonal embedding of the product of Bruhat–Tits trees T₂timesT₃timesT₅ into R_+—produces the 5‑smooth semigroup P=\2^a3^b5^cmid a,b,cinZ\. We close the gap between conjecture and falsifiable prediction by (i) formulating a uniform null model for eleven independent SM mass ratios in P, (ii) deriving the continuous spectrum of the natural Laplacian on T₂ and T₅ from the adjacency spectrum, and (iii) comparing the harmonic‑mean of the three prime‑log‑periods with the empirical Efimov factor.

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

Numbers it reports

1. Acceptance window $0.04$ (2 %) Tightening to $1\%$ would replace $p_{\text{single}}$ by $2\times10^{-3}$, raising $p_{\text{joint}}$ by $2^{11}\approx2048$‑fold.

   from 6.1 Limitations

2. The joint‑fit probability under the null is $p_{\text{joint}}=4.194304\times10^{-27}$, the Laplacian spectra are $[0.17157288,5.82842712]$ for $p=2$ and $[1.52786405,10.47213595]$ for $p=5$, and the Efimov mismatch factor is $22.51$.

   from Abstract

3. Efimov physics exhibits a universal discrete scaling factor $\lambda\approx22.7$, yet its origin remains phenomenological.

   from 1. Introduction

4. Quantitative comparison of the harmonic‑mean of the three prime log‑periods with the empirical Efimov factor, yielding a mismatch factor of $22.51$ (Section 4, 5).

   from 1. Introduction

5. Moreover, investigations into $p$‑adic quantum mechanics have identified the factor $p^{\Delta}-p^{1-\Delta}$ as a universal coefficient in spinor boundary actions, hinting at a possible spectral origin of mass‑like terms [4,6].

   from Expanded Literature Review

6. The same program claims that the harmonic‑mean of the three prime log‑periods yields an Efimov factor $\lambda_{\text{tree}}=1.008652$, differing from the observed $\lambda\approx22.7$ by a factor of $22.51$ $$11$$ .

   from Expanded Literature Review

How it is done

1. Correction (version 1.0.1, 2026-10-19): statements that described experiments, measurements, reviews or registrations as done were corrected; none took place.

   from Joint Bruhat–Tits Tree Product as a Testable Cross‑Domain Structure for Standard‑Model Mass Ratios, Quantum‑Error‑Correction Lattices, and Efimov Scaling

2. The formalisation of BT trees in proof assistants such as Lean has enabled rigorous verification of harmonic cochains and group decompositions, ensuring that combinatorial arguments about tree regularity are free from hidden analytic assumptions [3,7].

   from Expanded Literature Review

3. We adopt the uniform null model described in $$11$$ : each independent SM mass ratio $r_{i}$ is drawn from a uniform distribution on $[0,10]$.

   from 3.1 Statistical Null Model for SM Mass Ratios

4. Using the adjacency spectrum from $$10$$ and the Laplacian definition $\Delta=(p+1)I-A_{p}$, we obtain the continuous spectrum interval $$ \sigma(\Delta)=\bigl[(p+1)-2\sqrt{p},\,(p+1)+2\sqrt{p}\bigr]. $$ We evaluate this interval explicitly for $p=2$ and $p=5$.

   from 3.2 Spectral Analysis of the Natural Laplacian

What would falsify it

1. The conjecture is falsified if any of the following occurs: At least one SM mass ratio deviates from the nearest element of $\mathcal{P}$ by more than $2\%$ (i.e., lies outside the acceptance window).

   from 6.2 Falsification Criterion

2. Empirical measurement of the Efimov scaling factor yields $\lambda$ that differs from the tree‑derived prediction $\lambda_{\text{tree}}=1.008652$ by more than $5\%$, falsifying the conjecture.

   from 6.2 Falsification Criterion

3. Does extending the product to additional primes (e.g., $p=7$) improve the Efimov harmonic‑mean prediction without over‑fitting?

   from 6.3 Open Questions

Where the claim stops

1. Assumption Sensitivity Uniform null on $[0,10]$ for SM ratios Alters $p_{\text{single}}$ linearly; a broader range would increase $p_{\text{joint}}$ dramatically.

   from 6.1 Limitations

2. Harmonic‑mean formula for Efimov log‑periods Assumes the three primes are the only relevant scales; inclusion of additional primes would change $\lambda_{\text{tree}}$.

   from 6.1 Limitations

3. The joint‑fit probability for eleven SM mass ratios to lie within $2\%$ of the 5‑smooth semigroup $\mathcal{P}$ under a uniform null is $4.19\times10^{-27}$, an astronomically small number that nonetheless remains conditional on the chosen null model and independence assumptions.

   from 8. Conclusion

Where to read

1. Abstract151 words · 1 min
2. 1. Introduction285 words · 1 min
3. 2. Background and Related Work485 words · 2 min
4. 3. Methods148 words · 1 min
5. 4. Analysis173 words · 1 min
6. 5. Results98 words · 1 min
7. 6. Discussion221 words · 1 min
8. 7. Practical Implications and Engineering Pathways376 words · 2 min
9. 8. Conclusion174 words · 1 min
10. Correction Notice379 words · 2 min
11. Provenance and AI Involvement236 words · 1 min
12. Appendix A. Divergence report21 words · 1 min