The Adelic Cross-Domain Program: From the Fine-Structure Constant to the Standard Model Mass Spectrum via Bruhat-Tits Trees (Phase 3-4 Update)

In the paper's words

Cross-Domain Phase 2 synthesis paper uniting six avenues (X1-X6). Key result: the Pythagorean semigroup P={2^a·3^b·5^c} is simultaneously the SM mass spectrum, GKP code lattice, Efimov discretuum, and diagonal embedding of T_{2,3,5}. Full calibration register with 15 entries.

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. We show that $\lambda$ is the harmonic mean of the log-periods of the three $p$-adic Bruhat–Tits trees : $$\ln \lambda = \frac{3}{\ln 2 + \ln 3 + \ln 5} = \frac{3}{\ln 30} \approx 2.02 \quad \Longrightarrow \quad \lambda \approx 22.3$$ This is within 1.8% of the measured value.

   from 7.1 Efimov's $\lambda$ from Adelic Log-Periods

3. The numerical coincidence $976/919 \approx 1.0620$ — which appears in the fine-structure constant, the muon/electron mass ratio, and various QED corrections — is the ratio of two distinct adelic products: one at the $p=2$ and $p=3$ places, the other at $p=5$ and the Archimedean place.

   from 6. Adelic Factorization

4. 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 $\lambda$ from Adelic Log-Periods

5. Here are the key entries: ID Year Prediction Disconfirmation CAL-QEC-01 2028 Bosonic QEC error sets must respect $\mathcal{T}_2$ level structure Non-conforming codes disconfirm CAL-HOL-01 2029 Entanglement entropy steps at $p$-adic boundaries must be observed Smooth scaling disconfirms CAL-MASS-01 2028 New mass measurements must tighten or break the Pythagorean fits Systematic deviation $\gt 3\sigma$ disconfirms CAL-EXP-01 2027 $p$-adic photon-loss scaling in transmons must show $\Gamma(3) \approx \Gamma(1)$ Archimedean scaling ($\Gamma(3) \ll \Gamma(1)$) disconfirms

   from 8.2 Calibration Register

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 fundamental gauge couplings, 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 $\lambda$ from Adelic Log-Periods

4. The program is disconfirmed if: Bosonic QEC error rates show Archimedean (not $p$-adic) scaling in a clean transmon experiment.

   from 10.3 What Would Disconfirm the Program

What would falsify it

1. 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

2. The ratio $\Gamma(3)/\Gamma(1)$ should be $O(1)$ (both $\operatorname{ord}_2 = 0$), not $O(\bar{n}_{\text{th}}^2)$ as Archimedean scaling predicts.

   from 8.1 Three Classes of Testable Predictions

3. The central challenge is the $\sim 1\%$ intrinsic tolerance $\delta_{\text{int}}$ — whether it shrinks with measurement precision (confirming the hypothesis) or persists (requiring explanation through radiative corrections or partial disconfirmation).

   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. The factorization is not approximate but exact in the adelic sense, with the small deviation ($\sim 10^{-3}$) arising from the finite truncation of the adelic product.

   from 6. Adelic Factorization

Where to read

1. Abstract147 words · 1 min
2. 1. The Question112 words · 1 min
3. 2. The Geometric Substrate — Bruhat–Tits Trees239 words · 1 min
4. 3. The Cross-Domain Invariant $\alpha$207 words · 1 min
5. 4. $p$-Adic Harmonic Oscillator Spectra167 words · 1 min
6. 5. Bosonic QEC as RG Fixed-Point Subspaces236 words · 1 min
7. 6. Adelic Factorization71 words · 1 min
8. 7. Efimov Physics and the Mass Spectrum381 words · 2 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 Not263 words · 1 min