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

In the paper's words

The Bruhat–Tits tree T_p of Q_p is the infinite (p+1)-regular tree that serves as the bulk of p-adic AdS/CFT. We test the conjecture that a natural Laplacian on T_p — or the prime-indexed arithmetic of the tree — encodes Standard Model mass ratios. We compute, with all arithmetic shown: (i) the exact adjacency spectrum σ(A_p)=[-2√p, 2√p] and a proof that the ℓ² point spectrum of the adelic-conditioned Laplacian is empty for every prime p; (ii) the finite-truncation spectrum for p=2, N=3, giving Laplacian eigenvalues μ₁=1, μ₂=3, μ₃=5 and ratio R=5, which we classify as a truncation artifact rather than a physical output; (iii) the Koide ratio Q=0.6666664 for charged leptons, within 4×10^-7 relative of 2/3; and (iv) bounded semigroup tests of the T₂,3,5=\2^a3^b5^c\ calibration hypothesis, which fail at the 3\%–8\% level with internal multiplicative inconsistency.

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

Numbers it reports

1. The appeal: the adelic product formula $\prod_v |x|_v = 1$ ties the tree at every prime $p$ into a single rigid structure, so a spectral statement on $\mathcal{T}_p$ is not freely adjustable; if a mass ratio appeared as an eigenvalue ratio, its origin would be arithmetic, in the spirit of the adelic cross-domain programme [10], [12].

   from 1. Introduction

2. Third, we compute the Koide ratio for charged leptons and test the $\mathbb{T}_{2,3,5}$ semigroup calibration hypothesis against lepton and quark mass ratios with a bounded search.

   from 1. Introduction

3. Muon-to-electron ratio, from the stated inputs: $$ \frac{m_\mu}{m_e} = \frac{105.6583755}{0.51099895} = 206.7683, $$ (long division: $0.51099895 \times 206 = 105.26578$; remainder $0.39259$; $0.39259/0.51099895 = 0.76828$).

   from 4.6 Standard Model comparators

4. Tau-to-muon ratio: $$ \frac{m_\tau}{m_\mu} = \frac{1776.86}{105.6583755} = 16.8170 $$ ($105.6583755 \times 16.8 = 1775.06$; remainder $1.7966$; $1.7966/105.6584 = 0.01701$).

   from 4.6 Standard Model comparators

5. Tau-to-electron ratio: $1776.86/0.51099895 = 3477.19$.

   from 4.6 Standard Model comparators

6. Against the universal tree ratio $2$: no fundamental mass ratio equals $2$; the closest charged-lepton ratio is $m_\tau/m_\mu = 16.8170$, off by a factor $16.8170/2 = 8.41$.

   from 4.6 Standard Model comparators

How it is done

1. We test the conjecture that a natural Laplacian on $\mathcal{T}_p$ — or the prime-indexed arithmetic of the tree — encodes Standard Model mass ratios.

   from Abstract

2. A mature literature has transplanted relativistic and field-theoretic structures onto this tree: BTZ-type black-hole geometries and Wilson lines [2], boundary spinor theories obtained by integrating out the bulk [4], geodesic bulk diagrams computing conformal blocks [5], effective field theories on tree subspaces flowing to boundary conformal field theories [6], and formal verification of harmonic cochains in the Lean theorem prover [3].

   from 1. Introduction

3. Heyman [3] machine-formalises the Bruhat–Tits tree in the Lean proof assistant and verifies a result on harmonic cochains — functions on the tree satisfying a discrete Laplace equation — establishing that the combinatorial facts we use (degree $p+1$, no cycles, boundary $\mathbb{Q}_p$) are theorem-level, which raises the reproducibility bar for any spectral claim built on them.

   from 2. Background and Related Work

4. Let $\mathcal{T}_p$ be the Bruhat–Tits tree of $\mathbb{Q}_p$: the infinite $(p+1)$-regular tree, every vertex of degree $q=p+1$, with no cycles, and boundary $\partial\mathcal{T}_p=\mathbb{Q}_p\cup\{\infty\}$ (facts established formally in [3]).

   from 3.1 The tree and the operator

What would falsify it

1. We conclude that the leading-order spectral encoding is falsified, specify what a surviving version would require, and provide a reproducible computational benchmark.

   from Abstract

2. Fourth, we compare all resulting ratios against Standard Model mass ratios and argue that the leading-order encoding is falsified, while identifying the escape routes — non-radial boundary conditions, quotient trees, and dynamical (running) operators — that a surviving version would need.

   from 1. Introduction

3. Heyman [6] computes effective actions on two subspaces of $\mathcal{T}_p$ and shows both limits agree with the same $p$-adic boundary CFT — evidence that boundary data are insensitive to bulk details, which cuts against any claim that a specific bulk spectrum fixes observable numbers.

   from 2. Background and Related Work

Where the claim stops

1. At the leading level — a natural, adelic-compatible Laplacian on the full tree $\mathcal{T}_p$ — the encoding conjecture fails structurally, not numerically: Claim 4.1 shows the $\ell^2$ point spectrum is empty for every prime $p$, so there is no discrete eigenvalue hierarchy to compare with particle masses at all.

   from 6. Discussion

2. At the calibration level, the bounded semigroup tests fail at $3\%$–$8\%$ with an internal $2.4\%$ multiplicative inconsistency, which is the strongest single piece of evidence against $\mathbb{T}_{2,3,5}$ as the mass spectrum's organizing structure: a genuine generating relation would compose exactly.

   from 6. Discussion

3. If the semigroup tests are repeated with a larger exponent window and the lepton chain composes exactly (e.g. a single $t$ with $t = t_1 t_2$ to high precision), the internal-inconsistency objection fails.

   from 6. Discussion

Where to read

1. Abstract171 words · 1 min
2. 1. Introduction398 words · 2 min
3. 2. Background and Related Work618 words · 3 min
4. 3. Methods494 words · 2 min
5. 4. Analysis1,128 words · 5 min
6. 5. Results421 words · 2 min
7. 6. Discussion836 words · 4 min
8. 7. Conclusion191 words · 1 min