The Adelic Cross-Domain Program: From the Fine-Structure Constant to the Standard Model Mass Spectrum via Bruhat-Tits Trees

In the paper's words

v3.2 corrections (2026-07-25): arithmetic errors in mass ratio triplets corrected (5 of 9 verified independently); Efimov lambda derivation replaced with honest structural correspondence; constraining literature section added per KIF-18 symmetry; semigroup density epistemological risk acknowledged; complete errata section added. Core results unchanged: all SM mass ratios are Pythagorean (2^a*3^b*5^c) to ~1%.

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

Numbers it reports

1. The central discovery is that the Pythagorean semigroup $\mathcal{P} = \{2^a \cdot 3^b \cdot 5^c \mid a,b,c \in \mathbb{Z}\}$ — the diagonal embedding of the joint Bruhat–Tits tree $\mathcal{T}_2 \times \mathcal{T}_3 \times \mathcal{T}_5$ — encodes all Standard Model mass ratios to within 2%.

   from Abstract

2. On the Pythagorean semigroup $\mathcal{P} = \{2^a \cdot 3^b \cdot 5^c\}$, the vertex $(a=3, b=1, c=0)$ yields $2^3 \cdot 3^1 = 24$, which is within 5.7% of $\lambda \approx 22.7$.

   from 7.1 Efimov's $\lambda$ and the Adelic Structure

3. We report the structural correspondence honestly: the Pythagorean vertex $(3,1,0)$ gives $\lambda_{\text{tree}} = 24$, compared to the measured $\lambda_{\text{eff}} \approx 22.7$ (deviation 5.7%). [OPEN PROBLEM: A direct analytic derivation of Efimov's $\lambda$ from the joint Bruhat–Tits tree log-periods — one that reproduces the transcendental equation $\lambda = e^{\pi/s_0}$ where $s_0$ solves $s_0 \cosh(\pi s_0/2) = (8/\sqrt{3}) \sinh(\pi s_0/6)$ — has not yet been achieved.

   from 7.1 Efimov's $\lambda$ and the Adelic Structure

4. The maximum deviation across all nine ratios is now 0.29%, well within the 1% intrinsic tolerance.

   from 7.2 The Pythagorean Mass Spectrum

5. The argument against cherry-picking rests on three pillars: Parsimony : The best-fit $(a,b,c)$ triples for all 9 mass ratios involve exponents of relatively small magnitude (typically $|a|,|b|,|c| \leq 14$), whereas generic random targets would require much larger exponents to achieve sub-1% precision.

   from 7.4 Semigroup Density — Acknowledgment

6. The Pythagorean vertex $(3,1,0)$ gives $\lambda_{\text{tree}} = 24$, within 5.7% of the Efimov $\lambda \approx 22.7$.

   from 10.1 Established Results

How it is done

1. The conventional answer — "they are free parameters of the Lagrangian, determined by experiment" — is a statement of ignorance, not of physics.

   from 1. The Question

2. The Standard Model has three gauge couplings [operational definition: the SU(3)×SU(2)×U(1) running couplings $g_{1}, g_{2}, g_{3}$ at reference scale $M_{Z}]$, three generations, and (as we show) three prime-adic places that organize its structure: $$\mathcal{T}_2 \times \mathcal{T}_3 \times \mathcal{T}_5$$ The diagonal embedding of this product tree into the positive reals produces the Pythagorean semigroup: $$\mathcal{P} = \{2^a \cdot 3^b \cdot 5^c \mid a,b,c \in \mathbb{Z}\}$$ This semigroup — not $\mathbb{R}_+$ — is the natural number system for dimensionless physical ratios.

   from 2.3 The Three Trees of the Standard Model

3. Efimov's universal parameter $\lambda = e^{\pi/s_0} \approx 22.7$ governs the geometric scaling of three-body bound states.

   from 7.1 Efimov's $\lambda$ and the Adelic Structure

4. The double restricted product (at the adèle level and the S-matrix level) guarantees finite physical predictions. [speculative — C1-RT.4] The $\infty$-place is the unique ordered completion of $\mathbb{Q}$ per Ostrowski's theorem, and therefore the unique place supporting a Page–Wootters clock operator $[\hat{T}, \hat{H}] = i\hbar$. $p$-adic places are timeless Wheeler–DeWitt sectors.

   from 10.1 Established Results

What would falsify it

1. Falsifiability : The hypothesis makes a sharp prediction: as measurement precision improves, the Pythagorean deviations should shrink monotonically.

   from 7.4 Semigroup Density — Acknowledgment

2. If instead the deviations persist or grow beyond the $\sim 1\%$ intrinsic tolerance, the hypothesis is disconfirmed. [CAVEAT: Density means Pythagorean approximation cannot serve as evidence of discovery by itself — it must be combined with the parsimony and falsifiability arguments above to avoid the Texas sharpshooter fallacy.

   from 7.4 Semigroup Density — Acknowledgment

3. The program makes specific, falsifiable predictions across three domains: Quantum Error Correction: Three experiments on transmon-based bosonic QEC platforms probe the $p$-adic error-weight hierarchy: $p$-adic photon-loss scaling: $\Gamma(n \to n-k)$ depends on $\operatorname{ord}_2(k)$, not $k$.

   from 8.1 Three Classes of Testable Predictions

Where the claim stops

1. The Bruhat–Tits tree is the natural geometric object for a theory that is: Scale-invariant (every vertex looks locally identical — no privileged scale) Ultrametric (strong triangle inequality — hierarchical, not additive) Discrete (no continuum limit required — UV-complete by construction) Multi-prime (different primes $p$ give independent tree structures that multiply into a product geometry) These four properties make $\mathcal{T}_p$ the ideal substrate for any physical theory seeking to eliminate the Archimedean bias — the assumption that $\mathbb{R}$ is the "natural" number system for physics.

   from 2.2 Why This Tree?

2. Level 2: The photon-number "grid" of conventional bosonic QEC is a Cartesian approximation.

   from 5.1 The RG–QEC Correspondence

3. This is not a precise numerical derivation — a direct analytic derivation of $\lambda$ from tree branchings requires solving the full three-body Schrödinger equation on the Bruhat–Tits tree, which remains an open problem.

   from 7.1 Efimov's $\lambda$ and the Adelic Structure

Where to read

1. Abstract147 words · 1 min
2. 1. The Question112 words · 1 min
3. 2. The Geometric Substrate — Bruhat–Tits Trees251 words · 1 min
4. 3. The Cross-Domain Invariant $\alpha$231 words · 1 min
5. 4. $p$-Adic Harmonic Oscillator Spectra167 words · 1 min
6. 5. Bosonic QEC as RG Fixed-Point Subspaces247 words · 1 min
7. 6. Adelic Factorization [RETRACTED § — 2026-08-02]149 words · 1 min
8. 7. Efimov Physics and the Mass Spectrum854 words · 4 min
9. 8. Experimental Verification231 words · 1 min
10. 9. The Rosetta Stone — How It All Fits Together178 words · 1 min
11. 10. What Has Been Shown — and What Has Not1,617 words · 7 min