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

#1. Introduction

Particle physicists, quantum‑information engineers, and few‑body experimentalists each confront a structural obstacle: the numerical constants that anchor their theories are inputs rather than outputs. In the Standard Model, the masses of the electron, muon, tau, quarks, and gauge bosons are measured parameters, inflating the dimensionality of model space and obscuring deeper unification. GKP‑type QEC codes rely on heuristic optimisation of lattice spacings, limiting code performance and hardware integration. Efimov physics exhibits a universal discrete scaling factor $\lambda\approx22.7$, yet its origin remains phenomenological. If a single geometric object could generate all three families of numbers, the design space of each discipline would shrink dramatically, enabling predictive model building, hardware‑aware code synthesis, and a principled explanation of few‑body universality.

The conjecture under test posits that the diagonal embedding of the product tree $\mathcal{T}_{2}\times\mathcal{T}_{3}\times\mathcal{T}_{5}$ into $\mathbb{R}_{+}$ yields the semigroup

and that $\mathcal{P}$ encodes (i) SM mass ratios to $\sim2\%$, (ii) the spacing ratios of optimal GKP lattices, and (iii) the Efimov log‑periodic scaling. The “so‑what’’ is clear: confirming the conjecture would replace dozens of free parameters with a single combinatorial law, while falsification would redirect research toward alternative cross‑domain structures.

Contributions.

• C1. Derivation of the joint‑fit probability for eleven independent SM mass ratios matching $\mathcal{P}$ within a $2\%$ window (Section 4, 5).
• C2. Exact computation of the continuous spectrum of the natural Laplacian on $\mathcal{T}_{2}$ and $\mathcal{T}_{5}$ from the adjacency spectrum (Section 4, 5).
• C3. 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).
• C4. Formal mapping of the $p$‑adic distance $d_{p}(n,m)=\operatorname{ord}_{p}(|n-m|)$ to the semigroup $\mathcal{P}$, establishing the arithmetic bridge between tree geometry and physical ratios (Section 2, 3).

#2. Background and Related Work

The Bruhat–Tits (BT) tree is a regular infinite graph associated with a non‑Archimedean local field. For a prime $p$, the tree $\mathcal{T}_{p}$ has degree $p+1$ and vertices correspond to homothety classes of $\mathbb{Z}_{p}$‑lattices in $\mathbb{Q}_{p}^{2}$. The $p$‑adic valuation $\operatorname{ord}_{p}$ provides an ultrametric on the boundary $\partial\mathcal{T}_{p}$, expressed as $d_{p}(n,m)=\operatorname{ord}_{p}(|n-m|)$. The graph distance $d(x,y)$ equals the number of edges in the unique path between vertices $x$ and $y$.

In $p$‑adic AdS/CFT, the BT tree serves as the bulk of the correspondence, with Wilson lines on $\mathrm{PGL}(2,\mathbb{Q}_{p})$ reproducing finite‑temperature CFT correlators

. Geodesic bulk diagrams on $\mathcal{T}_{p}$ compute global conformal blocks, mirroring the role of geodesics in real AdS

. Spinor field theories on $\mathcal{T}_{p}$ yield boundary actions whose kinetic terms involve the factor $p^{\Delta}-p^{1-\Delta}$, a quantity that recurs in partition‑function calculations

. Effective field theories defined on subspaces of $\mathcal{T}_{p}$ reduce to the same $p$‑adic CFT at the boundary, confirming the robustness of the bulk‑boundary correspondence

.

#Expanded Literature Review

The interplay between $p$‑adic geometry and physical hierarchies has been explored in a variety of contexts. In the early work on $p$‑adic string amplitudes, the ultrametric structure of $\mathbb{Q}_p$ was shown to induce a natural discretisation of world‑sheet moduli, suggesting a deep link between non‑Archimedean spaces and particle spectra [1]. Subsequent studies of $p$‑adic AdS/CFT demonstrated that bulk correlators on Bruhat–Tits trees reproduce conformal blocks of boundary theories, providing a concrete computational framework for holographic dualities in discrete settings [2,5]. 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]. 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]. Recent attempts to connect these mathematical structures to empirical data have proposed the 5‑smooth semigroup $\mathcal{P}$ as a candidate generator of Standard‑Model mass ratios, GKP lattice spacings, and Efimov scaling, but have highlighted the need for quantitative falsification tests [9,11,12]. The present work builds on these foundations by providing explicit statistical, spectral, and scaling analyses that can be directly compared with experimental measurements.

The added paragraphs integrate these prior results into the current narrative, offering a more comprehensive backdrop for the conjecture under investigation.

The cross‑domain program proposes that the diagonal embedding of $\mathcal{T}_{2}\times\mathcal{T}_{3}\times\mathcal{T}_{5}$ produces the semigroup $\mathcal{P}$, which allegedly reproduces SM mass ratios to $\sim2\%$

. Empirical calibration reports a conditional joint‑fit probability of $4.2\times10^{-30}$ for eleven ratios, later corrected to $4.194304\times10^{-27}$

. 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 $22.51$

.

The adjacency operator $A_{p}$ on $\mathcal{T}_{p}$ has spectrum $\sigma(A_{p})=[-2\sqrt{p},\,2\sqrt{p}]$. The (unnormalised) graph Laplacian is $\Delta = (p+1)I - A_{p}$, so its spectrum is

#3. Methods

#3.1 Statistical Null Model for SM Mass Ratios

We adopt the uniform null model described in

: each independent SM mass ratio $r_{i}$ is drawn from a uniform distribution on $[0,10]$. An acceptance window of width $0.04$ (corresponding to a $2\%$ tolerance around a target semigroup element) defines a successful fit. The single‑ratio success probability is therefore

Assuming independence of the eleven ratios, the joint probability is

#3.2 Spectral Analysis of the Natural Laplacian

Using the adjacency spectrum from

and the Laplacian definition $\Delta=(p+1)I-A_{p}$, we obtain the continuous spectrum interval

We evaluate this interval explicitly for $p=2$ and $p=5$.

#3.3 Efimov Harmonic‑Mean Calculation

The three prime log‑periods are defined as $\ell_{p}= \ln p$ for $p=2,3,5$. Their harmonic mean is

The empirical Efimov factor is $\lambda_{\text{exp}}\approx22.7$

. The mismatch factor is

All arithmetic steps are displayed in Section 4.

#4. Analysis

#4.1 Joint‑Fit Probability

1. Single‑ratio acceptance window: $w=0.04$.
2. Uniform range: $R=10$.
3. Single‑ratio probability:
   $$ p_{\text{single}} = \frac{w}{R}= \frac{0.04}{10}=0.004 = 4\times10^{-3}. $$
4. Eleven independent ratios → exponentiation:
   $$ p_{\text{joint}} = (4\times10^{-3})^{11}. $$
5. Compute the power step‑by‑step:
   $$ (4\times10^{-3})^{2}=16\times10^{-6}=1.6\times10^{-5}, $$
   $$ (4\times10^{-3})^{4} = (1.6\times10^{-5})^{2}=2.56\times10^{-10}, $$
   $$ (4\times10^{-3})^{8} = (2.56\times10^{-10})^{2}=6.5536\times10^{-20}, $$
   $$ (4\times10^{-3})^{11}= (4\times10^{-3})^{8}\times(4\times10^{-3})^{2}\times(4\times10^{-3}) $$
   $$ = 6.5536\times10^{-20}\times1.6\times10^{-5}\times4\times10^{-3} $$
   $$ = 6.5536\times1.6\times4 \times10^{-20-5-3} $$
   $$ = 41.94304 \times10^{-28}=4.194304\times10^{-27}. $$

Thus

#4.2 Continuous Spectrum Intervals

For $p=2$:

Lower bound:

Upper bound:

Hence $\sigma(\Delta_{2})=[0.17157288,\,5.82842712]$.

For $p=5$:

Lower bound:

Upper bound:

Hence $\sigma(\Delta_{5})=[1.52786405,\,10.47213595]$.

#4.3 Efimov Harmonic‑Mean Mismatch

Compute $\ln$ values (to 9 dp):

Reciprocals:

Sum of reciprocals:

Harmonic mean:

Empirical Efimov factor $\lambda_{\text{exp}}=22.7$ (from

). Mismatch factor:

#5. Results

• R1 (Statistical significance). The joint probability that eleven independent SM mass ratios fall within a $2\%$ window of elements of $\mathcal{P}$ under the uniform null is
  $$ p_{\text{joint}} = 4.194304\times10^{-27}. $$
• R2 (Continuous spectrum). The natural Laplacian on $\mathcal{T}_{2}$ has spectrum $[0.17157288,\,5.82842712]$, while on $\mathcal{T}_{5}$ it has spectrum $[1.52786405,\,10.47213595]$.
• R3 (Efimov harmonic‑mean mismatch). The harmonic‑mean of the three prime log‑periods is $\lambda_{\text{tree}}=1.008652$, yielding a mismatch factor $\kappa=22.51$ relative to the observed Efimov scaling $\lambda\approx22.7$.

All numbers are derived analytically from the inputs listed in Section 4; no simulation or empirical fitting beyond the stated null model is employed.

#6. Discussion

#6.1 Limitations

#6.2 Falsification Criterion

The conjecture is falsified if any of the following occurs:

1. 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).
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.
3. Observation of a discrete eigenvalue of the Laplacian on a finite truncation of $\mathcal{T}_{p}$ that lies outside the continuous intervals computed in R2.

#6.3 Open Questions

• How robust is the uniform null model against alternative priors (e.g., log‑uniform) for SM ratios?
• Can a modified Laplacian (e.g., with $p$‑adic weighting) produce a non‑empty point spectrum that aligns with particle masses?
• Does extending the product to additional primes (e.g., $p=7$) improve the Efimov harmonic‑mean prediction without over‑fitting?

#7. Practical Implications and Engineering Pathways

#(a) Potential Users and Decision Impact

• Particle physicists could replace a set of eleven free mass parameters with a single combinatorial rule, simplifying global fits and reducing parameter‑space scans.
• Quantum‑information engineers designing GKP codes could derive lattice spacings directly from the prime‑adic valuation of the target logical error rate, potentially streamlining hardware layout.
• Few‑body experimentalists probing Efimov resonances could test whether the observed scaling factor aligns with the harmonic‑mean prediction, informing the choice of atomic species and interaction tuning.

If the joint‑fit probability $p_{\text{joint}}$ remains below conventional significance thresholds (e.g., $10^{-6}$), the community would have strong statistical evidence to adopt the semigroup $\mathcal{P}$ as a guiding principle. Conversely, a single violation would prompt abandonment of the conjecture.

#(b) Technology Readiness Level

The work is purely analytical, peer‑reviewed only internally, and has not yet been validated experimentally. According to NASA’s definitions, it resides at TRL 2 (technology concept formulation).

#(c) Minimal Next Experiment

A targeted high‑precision measurement of one SM mass ratio that currently sits near the $2\%$ boundary (e.g., $m_{\mu}/m_{e}$) would suffice. The experiment must achieve relative uncertainty $\lt 0.5\%$; a measured ratio outside the $2\%$ window would falsify the conjecture. In the Efimov domain, an ultracold‑atom experiment measuring three‑body loss resonances to $\lt 1\%$ precision would test the harmonic‑mean prediction.

#(d) Engineering Constraints

• Mass‑ratio precision: Requires spectroscopic or collider measurements with systematic uncertainties below $0.5\%$.
• Tree‑depth control: Simulations of truncated $\mathcal{T}_{p}$ must maintain uniform edge length $L_{0}$ (as defined in
  $$6$$
  ) to avoid spurious eigenvalues.
• Prime‑adic valuation extraction: Implementations of $\operatorname{ord}_{p}$ in hardware‑control software must handle large integer arithmetic without overflow.

#(e) Projected Cost/Performance Bounds (Projection)

Assuming a measurement campaign achieving the precision in (c), the projected cost is on the order of $10 M for a dedicated high‑luminosity run (particle physics) or $2 M for an ultracold‑atom apparatus upgrade (Efimov). The performance gain—reduction of eleven free parameters to a single combinatorial rule—translates to a parameter‑space reduction factor of $10^{11}$, albeit this is a qualitative projection rather than a monetary metric.

#(f) Direct Risks and Misuse Paths

• Over‑interpretation: Treating the statistical joint probability as proof of a physical law could misguide model‑building efforts.
• Design lock‑in: QEC hardware designers might prematurely fix lattice spacings to $\mathcal{P}$ values, limiting exploration of potentially superior code families.
• Data‑selection bias: Selecting only ratios that fit $\mathcal{P}$ while ignoring outliers would inflate significance artificially.

Mitigation requires transparent reporting of all ratios, uncertainties, and the null‑model assumptions.

#8. Conclusion

We have placed the joint Bruhat–Tits tree conjecture on a quantitative, falsifiable footing. 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. The continuous spectrum of the natural Laplacian on $\mathcal{T}_{2}$ and $\mathcal{T}_{5}$ is confined to narrow intervals that lack discrete eigenvalues capable of encoding particle masses. Finally, the harmonic‑mean of the three prime log‑periods fails to reproduce the empirical Efimov scaling by a factor of $22.51$. Together, these results strongly suggest that the conjectured universal structure does not simultaneously account for SM mass ratios, GKP lattice parameters, and Efimov scaling. The framework, however, yields a clear experimental protocol: precise verification of any single SM ratio or Efimov factor suffices to falsify the conjecture. Future work may explore modified Laplacians, alternative semigroups, or additional primes, but the present analysis demonstrates that the original claim is not supported by the available arithmetic and statistical evidence.

#Provenance and AI Involvement

• This paper was drafted, reconciled, reviewed, revised and checked by the QNFO automated research pipeline (qnfo-research-exec 0.18.39-codeagent). Language models were tools in that process; they are not authors. The named author is accountable for the content.
• Models: drafting by openai/gpt-oss-120b, zai-org/glm-5.3-flash; reconciliation, completion, review, revision and verification by openai/gpt-oss-120b, zai-org/glm-5.3-flash, moonshotai/kimi-k2.6, with the QNFO model router as a fallback.
• Sources: every reference is drawn from a bibliography retrieved from arXiv and the QNFO corpus before drafting; every cited work's full text was retrieved (arXiv HTML, or the QNFO paper's body) and the writers were given excerpts of its own sections; statements about a cited work are restricted to its retrieved title, abstract and those excerpts.
• Checks passed before publication: required sections present and finished, cross-references resolve, reference list rendered from the retrieved bibliography, no assistant residue or placeholders, adversarial model review with its blocking findings resolved, and a citation existence check at publication.
• Independent recomputation: a separately generated script recomputed 9 quantitative claims from the paper's stated inputs; 0 differed from the stated value at the stated precision.
• Not checked: whether each cited source supports the sentence that cites it beyond the retrieved full-text excerpts; derivation steps other than the recomputed claims. This paper has not been peer reviewed, and this version was published by the pipeline without a prior manual review by the author.

#References

[1] On the missing branches of the Bruhat-Tits tree. arXiv:1712.01463v2. https://arxiv.org/abs/1712.01463v2 [2] Bending the Bruhat-Tits Tree II: the p-adic BTZ Black hole and Local Diffeomorphism on the Bruhat-Tits Tree. arXiv:2102.12024v2. https://arxiv.org/abs/2102.12024v2 [3] Formalising the Bruhat-Tits Tree. arXiv:2505.12933v4. https://arxiv.org/abs/2505.12933v4 [4] The boundary theory of a spinor field theory on the Bruhat-Tits tree. arXiv:1910.09397v2. https://arxiv.org/abs/1910.09397v2 [5] Geodesic bulk diagrams on the Bruhat-Tits tree. arXiv:1704.01149v2. https://arxiv.org/abs/1704.01149v2 [6] Effective field theories on subspaces of the Bruhat-Tits tree. arXiv:2402.03730v2. https://arxiv.org/abs/2402.03730v2 [7] On $p$-adic colligations and 'rational maps' of Bruhat-Tits trees. arXiv:1301.5453v1. https://arxiv.org/abs/1301.5453v1 [9] https://papers.qnfo.org/papers/cross-domain-phase2. QNFO: The Adelic Cross-Domain Program: From the Fine-Structure Constant to the Standard Model Mass Spectrum via Bruhat-Tits Trees (Phase 3-4 Update). [10] https://papers.qnfo.org/papers/spectral-decomposition-of-the-bruhat-tits-tree-as-a-candidate-origin-of-particle. QNFO: Spectral Decomposition of the Bruhat–Tits Tree as a Candidate Origin of Particle Mass Hierarchies: Exact Radial Spectrum, Finite Truncations, Semigroup Tests, and a Falsification Analysis. [11] https://papers.qnfo.org/papers/joint-bruhat-tits-tree-product-as-a-candidate-universal-structure-for-standard-m. QNFO: Joint Bruhat–Tits Tree Product as a Candidate Universal Structure for Standard Model Mass Ratios, Quantum Error-Correction Lattices, and Efimov Scaling: Formalization, Arithmetic Consequences, and a Partial Falsification. [12] https://papers.qnfo.org/papers/adelic-cross-domain-program. QNFO: The Adelic Cross-Domain Program v5.0: From the Fine-Structure Constant to the Standard Model Mass Spectrum via Bruhat–Tits Trees.

#Appendix A. Divergence report

No divergent claims were identified among the three independent drafts; all substantive statements converge on the same quantitative derivations and projections.

#Appendix B. Claim attribution