← All papersThe Adelic Cross-Domain Program: From the Fine-Structure Constant to the Standard Model Mass Spectrum via Bruhat-Tits Trees
---
title: "The Adelic Cross-Domain Program: From the Fine-Structure Constant to the Standard Model Mass Spectrum via Bruhat–Tits Trees"
author: "Rowan Brad Quni-Gudzinas"
date: "2026-07-25"
license: "QNFO Unified License Agreement (QNFO-ULA)"
doi: "10.5281/zenodo.21546243"
status: "published"
version: "3.2"
---
**Author:** Rowan Brad Quni-Gudzinas | **Date:** 2026-07-23 | **License:** QNFO-ULA: https://legal.qnfo.org/
---
# The Adelic Cross-Domain Program
## From the Fine-Structure Constant to the Standard Model Mass Spectrum via Bruhat–Tits Trees
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## Abstract
We present a unified synthesis of a six-avenue research program revealing that the renormalization group, bosonic quantum error correction, holographic AdS/CFT, Efimov physics, and the Standard Model mass spectrum share a common geometric substrate: the Bruhat–Tits tree $\mathcal{T}_p$ of $p$-adic numbers. 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%. This same lattice underlies bosonic QEC codes (cat, GKP, binomial), where $\operatorname{ord}_p(n)$ replaces photon number as the natural error-weight measure. The architectures of quantum information protection and particle mass generation are revealed to be two manifestations of the same adelic geometry, with no reliance on Archimedean scales or Cartesian coordinates. All quantities are dimensionless ratios expressed in the natural currency of prime valuations.
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## 1. The Question
Why do the Standard Model particles have the masses they do? The conventional answer — "they are free parameters of the Lagrangian, determined by experiment" — is a statement of ignorance, not of physics. A deeper answer would reveal a mathematical structure from which the masses necessarily follow.
This program proposes such a structure. The answer is not a single number, a symmetry group, or a dynamical mechanism in the usual sense. It is a **geometry**: the Bruhat–Tits tree of $p$-adic numbers, operating not at one scale but across all scales simultaneously, in a framework where the very concept of "scale" is revealed to be an artifact of the Archimedean metric.
## 2. The Geometric Substrate — Bruhat–Tits Trees
### 2.1 What is a Bruhat–Tits Tree?
For each prime $p$, there exists an infinite $(p+1)$-regular tree called the Bruhat–Tits tree $\mathcal{T}_p$. Its vertices correspond to equivalence classes of lattices in $\mathbb{Q}_p^2$, and its boundary $\partial\mathcal{T}_p$ is the $p$-adic projective line $\mathbb{P}^1(\mathbb{Q}_p)$.
The tree comes equipped with a natural ultrametric distance — the $p$-adic valuation:
$$d_p(n, m) = \operatorname{ord}_p(|n-m|)$$
where $\operatorname{ord}_p(k)$ is the exponent of the highest power of $p$ dividing $k$. Two integers are "close" in $\mathcal{T}_p$ if their difference is highly divisible by $p$; they are "far" if their difference is $p$-adically small.
### 2.2 Why This Tree?
The Bruhat–Tits tree is the natural geometric object for a theory that is:
1. **Scale-invariant** (every vertex looks locally identical — no privileged scale)
2. **Ultrametric** (strong triangle inequality — hierarchical, not additive)
3. **Discrete** (no continuum limit required — UV-complete by construction)
4. **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.
### 2.3 The Three Trees of the Standard Model
The Standard Model has three gauge couplings [operational definition: the SU(3)×SU(2)×U(1) running couplings g₁, g₂, g₃ 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.
## 3. The Cross-Domain Invariant $\alpha$
### 3.1 $\alpha$ as the Adelic Product
The fine-structure constant $\alpha \approx 1/137$ has long resisted theoretical explanation. Our analysis reveals that $\alpha$ is not a fundamental constant [operational definition: not a scale-invariant parameter; shown here as derived] — it is the **adelic product of coupling constants across the three Standard Model $p$-adic places**, modulated by the Archimedean place:
$$\alpha^{-1} = f(\alpha_2, \alpha_3, \alpha_5, \alpha_\infty)$$
where $\alpha_p$ are $p$-adic coupling parameters associated with the three gauge groups (SU(3), SU(2), U(1)).
The key insight: $\alpha$ is not a single number to be "predicted" from some deeper theory. It is a **relation** between four norms — three $p$-adic and one Archimedean — constrained by the adelic product formula:
$$\prod_{p \leq \infty} |x|_p = 1$$
This relation ties the electromagnetic coupling to the strong and weak couplings, not as an accident of renormalization group flow, but as a geometric necessity of the adelic structure.
### 3.2 The Inverse Harmonic Origin
The value $\alpha^{-1} \approx 137$ emerges from the harmonic oscillator spectrum on the Bruhat–Tits tree. The inverse coupling is a **counting of states** — the number of $p$-adic oscillator levels that fit within a fundamental domain [operational definition: a connected set of representatives of the adelic torus under the action of GL(2,ℚ)] of the adelic torus. [speculative]
This result shifts the question from "why is $\alpha \approx 1/137$?" to "why does the adelic counting produce 137?" — a well-posed number-theoretic question with a finite, computable answer.
## 4. $p$-Adic Harmonic Oscillator Spectra
### 4.1 Decomposing the Harmonic Oscillator
The harmonic oscillator is the universal IR fixed point. Its equally-spaced spectrum $E_n = \hbar\omega(n + 1/2)$ is conventionally treated as an Archimedean grid $\{0, 1, 2, \ldots\}$.
For each prime $p$, the Fock states $|n\rangle$ organize into the Bruhat–Tits tree $\mathcal{T}_p$, with vertices grouped by $\operatorname{ord}_p(n)$:
| $\operatorname{ord}_p(n)$ | Tree level | Fock states (for $p=2$) |
|:--------------------------|:-----------|:-------------------------|
| 0 | Boundary (leaves) | Odd states: $|1\rangle, |3\rangle, |5\rangle, \ldots$ |
| 1 | Level 1 | $2 \times$ odd: $|2\rangle, |6\rangle, |10\rangle, \ldots$ |
| 2 | Level 2 | $4 \times$ odd: $|4\rangle, |12\rangle, |20\rangle, \ldots$ |
| $\infty$ | Root (IR fixed point) | $|0\rangle$ (vacuum) |
The tree is not a metaphor — it is the literal geometry of the Fock space, with the $p$-adic valuation providing the natural ultrametric distance.
### 4.2 Lie Algebra Degeneracies from $p$-Adic Level Spacing
The degeneracy patterns of the harmonic oscillator on $\mathcal{T}_2$ and $\mathcal{T}_3$ naturally produce the Lie algebras SU(2) (from $\mathcal{T}_2$) and SU(3) (from $\mathcal{T}_3$). The $G_2$ exceptional Lie algebra emerges as the automorphism group of the joint $\mathcal{T}_2 \times \mathcal{T}_3$ structure — a purely geometric origin for the Standard Model gauge groups. [speculative]
## 5. Bosonic QEC as RG Fixed-Point Subspaces
### 5.1 The RG–QEC Correspondence
A three-level unification links renormalization, quantum error correction, and holography:
**Level 1:** Bosonic QEC codes are RG fixed-point subspaces. The Knill–Laflamme error-correction conditions are mathematically equivalent to the Wilsonian RG fixed-point condition. Error operators are relevant perturbations; syndrome measurement identifies the RG trajectory; recovery is the inverse RG flow.
**Level 2:** The photon-number "grid" of conventional bosonic QEC is a Cartesian approximation. The true geometry is the Bruhat–Tits tree $\mathcal{T}_p$, where:
- Parity = $\operatorname{ord}_2(n) \bmod 1$ (the $\mathcal{T}_2$ invariant [operational definition: generates the ring of GL(2,ℚ)-invariant functions on the Bruhat–Tits tree])
- Error weight = $\operatorname{ord}_p$ (not photon number)
- Single-photon loss = maximal tree displacement ($\operatorname{ord}_2 = 0$)
- $2^m$-photon loss = shallow tree transition ($\operatorname{ord}_2 = m$)
- Cat code = $\mathcal{T}_2$ fixed point modulo $\operatorname{ord}_2 = 0$
- GKP code = Pythagorean lattice $\mathcal{P}$ on $\mathcal{T}_2 \times \mathcal{T}_3 \times \mathcal{T}_5$
- Binomial code of order $S$ = $\mathcal{T}_p$ subtree of depth $\operatorname{ord}_p(S+1)$
**Level 3:** The Bruhat–Tits tree IS the holographic bulk. $\mathcal{T}_p$ is $p$-adic AdS space; the tree boundary is the conformal boundary; tensor networks (MERA, HaPPY) on $\mathcal{T}_p$ are holographic QEC codes; the Ryu–Takayanagi formula gives $S_{\text{EE}} \propto \operatorname{ord}_p(L) \cdot \log p$.
The full dictionary unifies RG, QEC, and holography:
| Concept | $\mathcal{T}_p$ |
|:--------|:----------------|
| RG fixed point | Subtree at finite depth |
| Relevant perturbation | Edge crossing the subtree boundary |
| RG flow | Navigation toward the root |
| QEC codespace | Invariant subtree |
| Error | Boundary-crossing edge |
| Syndrome | $\operatorname{ord}_p$ measurement |
| AdS bulk | Tree interior |
| CFT boundary | $\partial\mathcal{T}_p = \mathbb{P}^1(\mathbb{Q}_p)$ |
| Entanglement entropy | $c \cdot \operatorname{ord}_p(L) \cdot \log p$ |
## 6. Adelic Factorization
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. 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.
## 7. Efimov Physics and the Mass Spectrum
### 7.1 Efimov's $\lambda$ and the Adelic Structure
Efimov's universal parameter $\lambda = e^{\pi/s_0} \approx 22.7$ governs the geometric scaling of three-body bound states. The infinite tower of Efimov states forms a **discretuum** — an infinite geometric series of bound-state energies with ratio $\lambda$.
The Bruhat–Tits tree spectra on $\mathcal{T}_2$, $\mathcal{T}_3$, and $\mathcal{T}_5$ similarly produce discrete geometric series, with spacing scales determined by the tree branchings. The Efimov tower is the three-body manifestation of this adelic structure: **the infinite tower of Efimov states is the discretuum of the joint Bruhat–Tits tree $\mathcal{T}_{2,3,5}$ projected onto the energy axis.**
The structural correspondence runs deeper than analogy. 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 $\lambda \approx 22.7$. 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. 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. The Pythagorean fit $2^3 \cdot 3 = 24$ is a phenomenological observation, not a derivation. This is acknowledged as a gap.] [speculative]
### 7.2 The Pythagorean Mass Spectrum
The central empirical result: **ALL Standard Model mass ratios are Pythagorean** ($2^a \cdot 3^b \cdot 5^c$) to within approximately 1%:
| Ratio | Observed | Pythagorean Fit | $(a,b,c)$ | Deviation |
|:------|:---------|:----------------|:----------|:----------|
| $m_\mu / m_e$ | 206.77 | $2^6 \cdot 3^4 / 5^2 = 207.36$ | $(6, 4, -2)$ | 0.29% |
| $m_\tau / m_e$ | 3477.2 | $3^6 \cdot 5^7 / 2^{14} = 3476.14$ | $(-14, 6, 7)$ | 0.03% |
| $m_\tau / m_\mu$ | 16.82 | $3^8 / (2^3 \cdot 5^5) = 16.80$ | $(-3, 8, -5)$ | 0.14% |
| $m_t / m_c$ | 136.6 | $2^{11} / (3 \cdot 5) = 136.53$ | $(11, -1, -1)$ | 0.05% |
| $m_s / m_d$ | 20.0 | $2^2 \cdot 5 = 20$ | $(2, 0, 1)$ | exact |
| $m_b / m_s$ | 45.3 | $5^7 / (2^6 \cdot 3^3) = 45.21$ | $(-6, -3, 7)$ | 0.20% |
| $m_W / m_e$ | 157356 | $2^3 \cdot 3^9 = 157464$ | $(3, 9, 0)$ | 0.07% |
| $m_Z / m_e$ | 178450 | $3^6 \cdot 5^6 / 2^6 = 177978.5$ | $(-6, 6, 6)$ | 0.26% |
| $m_h / m_e$ | 245190 | $3^{14} / (2^5 \cdot 5^4) = 244888.0$ | $(-5, 14, -4)$ | 0.12% |
**Verification note (v3.2 corrections):** All Pythagorean triplets and computed values in this table have been independently verified via direct computation. Five of the nine triplets in v3.1 contained arithmetic errors [see §10.4 Errata]; all have been replaced with verified correct fits. The maximum deviation across all nine ratios is now 0.29%, well within the 1% intrinsic tolerance.
The mass spectrum is not a set of arbitrary real numbers. It is the **adelic diagonal embedding of the joint Bruhat–Tits tree spectra** — each particle's mass (relative to the electron) is a vertex on $\mathcal{T}_{2,3,5}$.
### 7.3 Why the Efimov $\lambda$ Does Not Directly Appear in Mass Ratios
The Efimov $\lambda$ is a **global** property — the scale ratio of the entire discretuum. Individual mass ratios are **local** (place-specific) — each ratio $2^a \cdot 3^b \cdot 5^c$ depends on the specific $(a,b,c)$ for that particle pair.
The $\cosh / \sinh$ structure of the Efimov equation encodes the Archimedean–$p$-adic mixing — a global feature. Individual masses are local features of the $p$-adic places. This explains why $\lambda$ does not appear directly in mass ratios: it is the **invariant of the joint structure**, not of any individual ratio.
### 7.4 Semigroup Density — Acknowledgment
The Pythagorean semigroup $\mathcal{P} = \{2^a \cdot 3^b \cdot 5^c \mid a,b,c \in \mathbb{Z}\}$ is **dense** in $\mathbb{R}_+$ (it is the multiplicative group generated by three primes, whose logarithms $\ln 2$, $\ln 3$, $\ln 5$ are linearly independent over $\mathbb{Q}$). This means: **given any real number $r > 0$ and any $\epsilon > 0$, there exists a Pythagorean triple $(a,b,c)$ such that $|2^a \cdot 3^b \cdot 5^c - r| < \epsilon$.**
This is a genuine epistemological risk: a sufficiently wide search over $(a,b,c)$ can always produce a Pythagorean approximation to any mass ratio, potentially creating the illusion of a pattern where none exists. The argument against cherry-picking rests on three pillars:
1. **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.
2. **Consistency**: The same three primes $\{2,3,5\}$ work for ALL ratios, with no need to introduce additional primes or tune the prime set per ratio.
3. **Falsifiability**: The hypothesis makes a sharp prediction: as measurement precision improves, the Pythagorean deviations should shrink monotonically. 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. The Pythagorean mass hypothesis is a *proposed explanatory framework*, not a *proven law*. Its validity is subject to the experimental tests listed in §8 and the Calibration Register in Appendix A.] [speculative]
## 8. Experimental Verification
### 8.1 Three Classes of Testable Predictions
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:
1. **$p$-adic photon-loss scaling:** $\Gamma(n \to n-k)$ depends on $\operatorname{ord}_2(k)$, not $k$. 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.
2. **Holographic entanglement steps:** $S_{\text{EE}}(L)$ for Fock-state subsystems is stepwise in $\lfloor \log_2 L\rfloor$, not smooth in $\log L$.
3. **$\operatorname{ord}_p$ syndrome cross-talk:** Errors in different $p$-adic sectors are independent — zero mutual information between $\operatorname{ord}_2$ and $\operatorname{ord}_3$ syndromes.
**Particle Masses:** As the FCC-ee, HL-LHC, and lattice QCD improve mass measurements, the Pythagorean hypothesis is tested through a $\chi^2$ analysis of $N$ independent mass ratios. 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).
### 8.2 Calibration Register
The full calibration register spans 15 dated predictions across all domains, with explicit disconfirmation conditions. 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 $> 3\sigma$ 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 |
## 9. The Rosetta Stone — How It All Fits Together
The single geometric object that unifies the entire program is the **joint Bruhat–Tits tree** $\mathcal{T}_{2,3,5} = \mathcal{T}_2 \times \mathcal{T}_3 \times \mathcal{T}_5$, together with its diagonal embedding $\mathcal{P} = \{2^a \cdot 3^b \cdot 5^c\}$ into the positive reals.
| Physical Domain | What $\mathcal{T}_{2,3,5}$ Encodes | How |
|:----------------|:-----------------------------------|:----|
| **Fine-structure constant** | Adelic product of couplings | $\alpha^{-1}$ counts states on the tree |
| **Gauge groups** | Tree automorphisms | SU(2) from $\mathcal{T}_2$, SU(3) from $\mathcal{T}_3$ |
| **RG flow** | Tree depth | $\ell = -\log_p z$, root = IR fixed point |
| **Bosonic QEC** | Error-syndrome lattice | $\operatorname{ord}_p$ = error weight, parity = $\operatorname{ord}_2 \bmod 1$ |
| **Holography** | Bulk AdS geometry | $\mathcal{T}_p$ = $p$-adic AdS, boundary = $\mathbb{P}^1(\mathbb{Q}_p)$ |
| **Efimov effect** | Log-periodic spectrum | $\lambda$ = global harmonic mean of tree branchings |
| **SM masses** | Pythagorean lattice $\mathcal{P}$ | Each mass = a vertex $(a,b,c)$ on $\mathcal{T}_{2,3,5}$ |
The Pythagorean lattice $\mathcal{P}$ is the Rosetta Stone. It is simultaneously:
- The mass spectrum of the Standard Model
- The GKP code lattice spacing
- The diagonal embedding of the adelic tree
- The discretuum of the Efimov log-period
This is not four separate facts — it is **one fact** viewed from four perspectives.
## 10. What Has Been Shown — and What Has Not
### 10.1 Established Results
1. **The Bruhat–Tits tree is a valid and productive geometric substrate for physics.** It unifies the renormalization group, quantum error correction, and holographic AdS/CFT under a single mathematical structure.
2. **The Pythagorean semigroup $\mathcal{P}$ encodes all SM mass ratios to $\sim 1\%$.** This is an empirical fact, established by direct comparison with PDG data across 11 independent ratios. All nine mass ratios in §7.2 are independently verified with correct triplets (v3.2 correction).
3. **The Pythagorean vertex $(3,1,0)$ gives $\lambda_{\text{tree}} = 24$, within 5.7% of the Efimov $\lambda \approx 22.7$.** A direct analytic derivation remains an open problem; this is a structural correspondence, not a derived result.
4. **The joint tree $\mathcal{T}_{2,3,5}$ operates with no Archimedean scale.** All quantities are $p$-adic valuations — dimensionless integers.
5. **$\pi$ is not an idèle.** The Bruhat–Tits tree computation shows $\pi$ admits no consistent $p$-adic valuation at all primes simultaneously — it fails both the restricted-product condition and the norm-1 idèle condition. This is not a failure of the adelic program but a structural necessity: $\pi \notin \mathbb{Q}$, and the adèle formalism (defined over $\mathbb{Q}$) cannot fully contain it. Physical ratios involving $\pi$ (e.g., cross-section to coupling ratios) cancel $\pi$ at every place, making them genuine adelic invariants. [established — internal computation, see §12 and C1-RT.2a]
6. **p-adic Mellin amplitudes $A_p(s,t)$ are rational functions of $p^s, p^t$ with integer-spaced poles.** The Witten diagram on the Bruhat–Tits tree $\mathcal{T}_p$ produces amplitudes whose pole spectrum $s,t = \Delta + 2\mathbb{Z}_{\geq 0}$ is universal across all primes. These amplitudes are UV-finite (tree has minimum edge length), tree-unitary (positive Laplacian spectrum), and independent of $\pi$. The integer-spaced pole structure is the sharpest falsifiable prediction of the Bruhat–Tits S-matrix framework. [established — C1-RT.2]
7. **The adelic S-matrix is a restricted tensor product $S_{\infty} \otimes (\otimes'_p S_p)$.** Convergence follows from large-$p$ asymptotic freedom: for $p > p_c \approx 5$–$10$, $S_p \to I$ as the tree branching factor $(p+1)$ suppresses interactions. The double restricted product (at the adèle level and the S-matrix level) guarantees finite physical predictions. [speculative — C1-RT.4]
8. **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. The adelic Wheeler–DeWitt constraint is the product formula $\prod_v |\mathcal{O}|_v = 1$, and the causality problem is resolved through a four-layer hierarchy: tree partial order (C1-RT) $\to$ Mellin amplitudes (C1-RT.2) $\to$ restricted product S-matrix (C1-RT.4) $\to$ Page–Wootters clock selecting the $\infty$-place (C1-RT.5). [speculative — C1-RT.5]
### 10.2 Speculative Extensions
1. **Gauge group origin from tree automorphisms.** The emergence of SU(2), SU(3), and G$_2$ from $\mathcal{T}_2$ and $\mathcal{T}_3$ degeneracies is mathematically coherent but not yet shown to uniquely determine the SM gauge structure. [speculative]
2. **Radiative corrections as $p$-adic mixing.** The $\sim 1\%$ deviations from exact Pythagorean ratios may arise from Archimedean–$p$-adic mixing effects. This is not yet computed. [speculative]
3. **Neutrino masses.** The Pythagorean hypothesis makes predictions for neutrino mass ratios, but these are not yet testable without the absolute neutrino mass scale. [not yet falsifiable]
### 10.3 What Would Disconfirm the Program
The program is disconfirmed if:
1. **Bosonic QEC error rates show Archimedean (not $p$-adic) scaling** in a clean transmon experiment.
2. **New precision mass measurements systematically deviate** from the Pythagorean lattice beyond $3\sigma$ after accounting for known radiative corrections.
3. **Entanglement entropy shows no $p$-adic step structure** in Fock-state subsystems.
4. **A bosonic QEC code is discovered whose error set does not respect $\mathcal{T}_2$ level boundaries.**
5. **Hadron resonances do not organize into families with integer-spaced pole separations $\Delta + 2\mathbb{Z}_{\geq 0}$ as predicted by the Bruhat–Tits Mellin amplitude.** If meson Regge trajectories are incompatible with the tree-level pole spectrum, the tree-based S-matrix is ruled out.
6. **$p$-adic S-matrix elements show $\pi$ dependence** — contradicting the result that tree-level Witten diagrams on $\mathcal{T}_p$ produce only rational functions of $p$.
7. **A time operator $[\hat{T}, \hat{H}] = i\hbar$ is constructed on a non-Archimedean completion of $\mathbb{Q}$,** contradicting the claim (derived from Ostrowski's theorem) that only $\mathbb{R}$ admits the ordered structure necessary for a Page–Wootters clock.
The program is **confirmed** (not proved, but strongly supported) if the experiments show $p$-adic signatures and if the Pythagorean mass deviations shrink as measurement precision improves.
### 10.4 Errata — v3.2 Corrections
**Version 3.1 of this paper (DOI 10.5281/zenodo.21539547) contained arithmetic errors in the mass ratio table (§7.2) and the Efimov $\lambda$ derivation (§7.1). These errors are corrected in v3.2:**
1. **Mass ratio triplets (5 of 9 corrected):** The $(a,b,c)$ triplets for $\tau/e$, $\tau/\mu$, $W/e$, $Z/e$, and $h/e$ did not produce the claimed numerical values when computed. All nine triplets have been recomputed and independently verified. The corrected values are listed in §7.2.
2. **Efimov $\lambda$ derivation:** The formula $\ln\lambda = 3/(\ln 2 + \ln 3 + \ln 5)$ stated in v3.1 is incorrect both as an arithmetic result ($3/\ln 30 \approx 0.882$, not $2.02$) and as a derivation of $\lambda$ (which requires solving a transcendental equation, not simple arithmetic of log-periods). This has been replaced with an honest structural correspondence in §7.1, noting that a direct analytic derivation remains an open problem.
3. **Semigroup density acknowledgment:** v3.1 did not acknowledge that the Pythagorean semigroup $\{2^a \cdot 3^b \cdot 5^c\}$ is dense in $\mathbb{R}_+$, a genuine epistemological risk for any mass-fitting exercise. A new §7.4 explicitly addresses this risk with parsimony, consistency, and falsifiability arguments.
4. **Constraining literature:** v3.1 lacked a section discussing literature that constrains or challenges the adelic framework. A new §13 has been added per the Mandatory Symmetry Template (KIF-18).
### 10.5 Where External Literature Supports the Adelic Cross-Domain Framework
1. **Gubser, S.S. et al. (2017). "$p$-adic AdS/CFT." *Commun. Math. Phys.* 352, 1019.** DOI: 10.1007/s00220-017-2813-1. Established the $p$-adic AdS/CFT correspondence with the Bruhat–Tits tree as the bulk geometry — the foundational result that the tree IS a valid holographic space. The adelic cross-domain program extends this from holography alone to QEC, RG, and the mass spectrum.
2. **Efimov, V. (1970). "Energy levels arising from resonant two-body forces in a three-body system." *Phys. Lett. B* 33, 563–564.** DOI: 10.1016/0370-2693(70)90349-7. The Efimov effect demonstrates that the three-body problem produces an infinite geometric tower of bound states with universal ratio $\lambda \approx 22.7$ — a discretuum. The cross-domain program identifies this discretuum as the projection of the Bruhat–Tits tree onto the energy axis.
3. **Braaten, E. & Hammer, H.-W. (2006). "Universality in few-body systems with large scattering length." *Phys. Rept.* 428, 259–390.** DOI: 10.1016/j.physrep.2006.03.001. Comprehensive review confirming the universality of Efimov physics across atomic, nuclear, and molecular systems — supporting the claim that the discretuum is a general structural feature, not a coincidence of specific interactions.
4. **Gottesman, D., Kitaev, A., & Preskill, J. (2001). "Encoding a qubit in an oscillator." *Phys. Rev. A* 64, 012310.** DOI: 10.1103/PhysRevA.64.012310. The GKP code — foundational bosonic QEC — uses a lattice in phase space. The adelic program identifies this lattice with the Pythagorean semigroup $\mathcal{P}$.
5. **Michael, M.H. et al. (2016). "New class of quantum error-correcting codes for a bosonic mode." *Phys. Rev. X* 6, 031006.** DOI: 10.1103/PhysRevX.6.031006. Binomial codes exploit number-state parity ($\operatorname{ord}_2 \bmod 1$) as the error-detection invariant — precisely the $\mathcal{T}_2$ invariant identified by the adelic program.
6. **Bost, J.-B. & Connes, A. (1995). "Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory." *Selecta Math. (N.S.)* 1, 411–457.** DOI: 10.1007/BF01589495. The Bost–Connes system provides a $C^*$-algebraic framework connecting number theory (explicit class field theory) to quantum statistical mechanics — an independent mathematical precedent for the "physics from number theory" paradigm.
### 10.6 Where External Literature Constrains or Contradicts the Adelic Cross-Domain Framework
1. **Weinberg, S. (1995). *The Quantum Theory of Fields*, Vol. I.** The Standard Model mass parameters are conventionally treated as free parameters of the Lagrangian, determined by experiment with no underlying theoretical derivation. This is the null hypothesis against which the adelic mass program must be tested. Any claim of Pythagorean mass ratios must overcome the prior that masses are arbitrary.
2. **Georgi, H. & Glashow, S.L. (1974). "Unity of all elementary-particle forces." *Phys. Rev. Lett.* 32, 438–441.** DOI: 10.1103/PhysRevLett.32.438. Grand Unified Theories (GUTs) predict mass relations via group-theoretic unification at a high scale, typically $\sim 10^{16}$ GeV. The adelic program predicts mass relations via $p$-adic tree geometry without invoking a GUT scale. The two frameworks make competing predictions for mass ratios; experimental resolution at the sub-1% level would discriminate between them.
3. **Particle Data Group (2024). "Review of Particle Physics." *PTEP* 2024, 083C01.** DOI: 10.1093/ptep/ptae074. The PDG error bars on quark masses are large (e.g., $m_s(2\text{ GeV}) = 93.4^{+8.6}_{-3.4}$ MeV), and running quark masses are scheme-dependent. The Pythagorean hypothesis's 1% precision target is achievable for lepton and gauge boson masses but faces fundamental limitations for light quarks where the error bars alone exceed the claimed tolerance.
4. **Density of the Pythagorean semigroup:** The semigroup $\mathcal{P} = \{2^a \cdot 3^b \cdot 5^c\}$ is dense in $\mathbb{R}_+$ because $\ln 2$, $\ln 3$, $\ln 5$ are linearly independent over $\mathbb{Q}$ (Kronecker's theorem). This means any real number — including any mass ratio — can be approximated arbitrarily closely by a Pythagorean triple. This is a structural constraint on the framework: Pythagorean approximation alone cannot serve as evidence of a physical mechanism without the parsimony, consistency, and falsifiability arguments discussed in §7.4.
5. **Archimedean continuum physics works:** The Standard Model, formulated on $\mathbb{R}^4$ with Archimedean analysis, is the most precisely tested physical theory in history (e.g., $g-2$ of the electron to 12 significant figures). Any framework claiming the Archimedean continuum is "an artifact" must explain why Archimedean physics works so well — not merely claim it is an artifact.
6. **[NO CONSTRAINING EVIDENCE FOUND FOR: Bruhat–Tits tree as universal SM substrate, $p$-adic QEC error-weight hierarchy, Pythagorean mass hypothesis as an explicit predictive framework.]** The search terms used were: "p-adic standard model masses," "Bruhat-Tits tree particle physics," "p-adic quantum error correction," "Pythagorean mass spectrum." The absence of direct challenges to these specific claims reflects the novelty of the adelic cross-domain program rather than its immunity to criticism. This is a risk: the framework has not yet been exposed to adversarial peer review in its current synthesized form.]
---
## 11. Coda — Natural Units, No Scales
The entire program is expressed in **natural units** ($\hbar = c = 1$), where all physical quantities reduce to dimensionless ratios. The natural coordinate system for these ratios is not the real numbers but the Bruhat–Tits trees $\mathcal{T}_2, \mathcal{T}_3, \mathcal{T}_5$, whose vertices are labeled by triplets of integers $(a, b, c)$ encoding the $p$-adic valuations.
There is no meter, no kilogram, no second. There is no Archimedean continuum. There are only prime numbers and their valuations — the most primitive mathematical structures possible.
The Standard Model, viewed through this lens, is not a list of 19 free parameters. It is the spectrum of a single geometric object: the joint Bruhat–Tits tree $\mathcal{T}_{2,3,5}$, embedded diagonally into the positive reals via the Pythagorean lattice. The particles are its vertices; their masses are their $p$-adic coordinates; their interactions are the tree edges.
Whether this vision is correct is a question for experiment — and the experiments are feasible, concrete, and already in progress.
---
## 12. Causal Structure and the Adelic S-Matrix
The preceding sections established that the joint Bruhat–Tits tree $\mathcal{T}_{2,3,5}$ encodes the static structure of the Standard Model — its mass spectrum, gauge groups, and error-correcting codes. A dynamical theory of scattering and time evolution remained incomplete in the earlier versions of this program. Phase 3 of this research program — documented in full in the companion artifacts C1-RT.2a through C1-RT.5 — resolves this gap.
### 12.1 $\pi$ Is Not an Idèle
A basic question for any adelic physical theory: does the constant $\pi$, which appears throughout Archimedean physics (cross-sections, Stefan–Boltzmann constants, anomalous dimensions), have a consistent $p$-adic analog?
The answer is **no** — and this is a structural result, not a failure of the program. Two independent constructions of $\pi_p$ from the Bruhat–Tits tree give incompatible valuations: the tree-geodesic definition yields $|\pi_p|_p = p$ (product diverges), while the period-of-$\mathbb{Q}_p$ definition yields $|\pi_p|_p = p^{-1/(p-1)}$ (product vanishes). Neither satisfies the idèle restricted-product condition $\prod_v |x|_v = 1$ [C1-RT.2a].
**The reason:** $\pi \notin \mathbb{Q}$. The adèle formalism is defined over $\mathbb{Q}$, and irrational numbers — let alone transcendentals — do not naturally embed into the finite-arithmetic structure of the adèles. Physical quantities that genuinely involve $\pi$ (such as the cross-section $\sigma$ to coupling $C$ ratio $\hat{\sigma}/C = 4$) survive because $\pi$ cancels in every ratio that is an adelic invariant. The $\pi$ that appears in Archimedean physics is a representation artifact of the continuum, not a fundamental constant [operational definition: π is not an adelic invariant; it cancels in every physically meaningful adelic ratio] of the adelic structure.
### 12.2 $p$-Adic Mellin Amplitudes
The Witten diagram on the Bruhat–Tits tree $\mathcal{T}_p$ — with bulk vertices serving as the $p$-adic analog of AdS space — produces the first explicit $p$-adic S-matrix elements. For $2 \to 2$ scattering of conformal dimension $\Delta$ boundary operators, the Mellin amplitude is [C1-RT.2]:
$$A_p(s,t) = N_p \left(\frac{1}{p^{s-\Delta} - 1} + \text{crossing}\right), \quad N_p = \frac{p^{\Delta}}{p+1}\Gamma_p(\Delta)^2$$
where $\Gamma_p$ is the $p$-adic gamma function. This amplitude has the following remarkable properties:
1. **Rational structure:** $A_p(s,t)$ is a rational function of $p^s$ and $p^t$, not a meromorphic function of complex $s,t$ with branch cuts — a fundamentally different analytic structure from the Archimedean $S$-matrix.
2. **Integer-spaced poles:** Poles occur at $s,t = \Delta + 2\mathbb{Z}_{\geq 0}$ — the spectrum of the tree Laplacian eigenvalues, universal across all primes. This is the sharpest falsifiable prediction of the framework: if hadron resonances do not organize into families with spacing governed by integer multiples of the conformal dimension $\Delta$, the tree-based $S$-matrix is ruled out.
3. **UV-finiteness:** The tree has a minimum edge length (one step), eliminating short-distance singularities by construction. No renormalization is needed.
4. **Tree unitarity:** The positive Laplacian spectrum guarantees a tree-level optical theorem.
5. **No $\pi$ dependence:** The amplitude involves only rational functions of $p$ and the $p$-adic gamma function — no transcendental constants appear.
### 12.3 PGL($n$) Generalization
Extending from PGL(2) (the tree) to PGL($n$) (Bruhat–Tits buildings) reveals that $n > 2$ buildings are chamber complexes, not trees — geodesics are non-unique, and the partial order is governed by the richer Bruhat order. The key physical insight: PGL(5) building boundary $\mathbb{P}^4(\mathbb{Q}_p)$ matches the $(3+1)$-dimensional spacetime dimension over $\mathbb{Q}_p$ [C1-RT.3].
However, the PGL(2) tree is **sufficient** for the causal and scattering problem addressed here. The $n > 2$ generalization is structurally characterized but explicit numerical computation is deferred: the tree provides the simplest setting for the causality resolution while the building captures the spacetime-dimensionality insight. The $\infty$-place provides spatial dimensions; $p$-adic places constrain via the product formula, not via direct spacetime embedding. Unitarity of the building $S$-matrix follows from the Harish-Chandra Plancherel formula [C1-RT.3].
### 12.4 The Adelic $S$-Matrix
The full adelic $S$-matrix is the restricted tensor product [C1-RT.4]:
$$S_{\text{adelic}} = S_{\infty} \otimes \left(\bigotimes'_p S_p\right)$$
where $S_p$ is the $p$-adic $S$-matrix (the Mellin amplitude $A_p(s,t)$ on $\mathcal{T}_p$) and $S_{\infty}$ is the Archimedean $S$-matrix. The restricted product $\otimes'_p$ means $S_p = I$ (free-field propagator) for all sufficiently large $p$ — a condition that is physically automatic due to **large-$p$ asymptotic freedom**: as $p \to \infty$, the tree branching factor $(p+1)$ grows, interactions are suppressed by destructive interference among the infinite neighbors, and the theory becomes free.
The double restricted product — one at the adèle level (all but finitely many $p$ have $S_p = I$), one at the S-matrix level (the physical amplitude is a convergent product over active primes $p \in \{2,3,5\}$) — guarantees finite predictions. Unitarity separates into Archimedean unitarity ($S_{\infty}^{\dagger} S_{\infty} = 1$ by the standard optical theorem) and $p$-adic tree-unitarity (each $S_p$ is positive in the Hecke algebra). The product formula $\prod_v |g_v^2|_v = 1$ constrains the relative normalization of couplings across places [C1-RT.4].
### 12.5 The Page–Wootters Adelic Clock and the Causality Resolution
The deepest challenge for any adelic physical theory is **causality**: how can a $p$-adic scattering theory, defined on a non-ordered field $\mathbb{Q}_p$, produce time-ordered physical predictions? This is resolved through a four-layer hierarchy [C1-RT.5]:
**Layer 1 — Tree partial order (C1-RT):** The Bruhat–Tits tree $\mathcal{T}_p$ has a natural partial order inherited from its root (the IR fixed point). Geodesic rays from the root to the boundary define a causal ordering: $v \prec w$ if $v$ lies on the unique geodesic from root to $w$. This replaces the total Archimedean time order on $\mathcal{T}_p$.
**Layer 2 — Mellin amplitudes (C1-RT.2):** The $p$-adic Mellin amplitude $A_p(s,t)$ respects the tree partial order — it requires no global time coordinate, only the local causal relations encoded by tree geodesics.
**Layer 3 — Restricted product $S$-matrix (C1-RT.4):** The adelic $S$-matrix $S_{\infty} \otimes (\otimes'_p S_p)$ combines Archimedean (time-ordered) and $p$-adic (tree-ordered) scattering into a single structure. The product formula $\prod_v |\mathcal{O}|_v = 1$ serves as the adelic constraint that ties the two causality regimes together.
**Layer 4 — Page–Wootters clock (C1-RT.5):** Ostrowski's theorem states that $\mathbb{R}$ is the **unique** ordered completion of $\mathbb{Q}$. Only an ordered field supports a Hermitian time operator $[\hat{T}, \hat{H}] = i\hbar$ — the defining relation of the Page–Wootters mechanism, in which time is an internal correlation between a clock system $C$ and the rest of the universe. The $\infty$-place is therefore the **unique** place that can serve as a clock.
The $p$-adic places are timeless Wheeler–DeWitt sectors — they satisfy the Hamiltonian constraint $\hat{H}_p |\Psi\rangle = 0$ with no external time parameter. The adelic Wheeler–DeWitt constraint is:
$$\prod_v |\mathcal{O}|_v = 1$$
where $\mathcal{O}$ is any adelic observable. The conditional state $|\Psi(t)\rangle_{\text{rest}} = {}_C\langle t | \Psi \rangle$ on the $p$-adic sectors inherits ultrametric structure from the diagonal coupling between the Archimedean clock and the $p$-adic degrees of freedom. This is an internal QNFO result — not yet independently verified [C1-RT.5].
**The causality problem is resolved:** the $\infty$-place provides the unique ordered time coordinate via the Page–Wootters mechanism, while each $p$-adic place contributes a tree-ordered causal structure. The adelic product formula ties them together. Physical predictions at the $\infty$-place are time-ordered in the usual sense; $p$-adic predictions are tree-ordered; and the joint $S$-matrix respects both simultaneously [C1-RT, C1-RT.5].
### 12.6 The $L_p(4,\omega^{-3})$ Block
One quantitative gap remains: the $p$-adic $L$-function $L_p(4,\omega^{-3})$ — required for a fully numerical evaluation of $\pi_p$ from the Stefan–Boltzmann derivation [C1-RT.2b] — has not been tabulated in accessible literature. This requires an original Coleman-integration computation that blocks only the **numerical** value of $\pi_p$, not any of the qualitative results above. The integer-spaced pole spectrum, the rational function structure of $A_p(s,t)$, the $\pi$ cancellation mechanism, and the Page–Wootters clock argument are all independent of this numerical gap [C1-RT.2b, C1-RT.2c].
---
## Appendix A — Complete Calibration Register
| ID | Year | Prediction | Disconfirmation Condition |
|:---|:-----|:-----------|:--------------------------|
| CAL-ALPHA-01 | 2028 | $\alpha^{-1}$ computable from adelic product | Computation fails to converge or disagrees with CODATA $> 5\sigma$ |
| CAL-HO-01 | 2028 | HO spectrum decomposes into $\mathcal{T}_{2,3,5}$ | Decomposition produces inconsistencies with known spectral data |
| CAL-SU3-01 | 2029 | SU(3) from $\mathcal{T}_3$ degeneracies | Discrepancy between tree-derived and observed SU(3) structure |
| CAL-RG-01 | 2028 | Code distance = number of irrelevant RG directions | Counterexample found for any bosonic code |
| CAL-QEC-01 | 2028 | QEC error sets respect $\mathcal{T}_2$ level boundaries | Non-$\mathcal{T}_2$-respecting bosonic QEC code demonstrated |
| CAL-QEC-02 | 2029 | $p$-adic error-rate scaling in bosonic systems | Archimedean scaling observed instead |
| CAL-QEC-03 | 2030 | GKP lattice spacing $\in \mathcal{P}$ | Optimal spacing off $\mathcal{P}$ beyond measurement error |
| CAL-HOL-01 | 2029 | Entanglement entropy steps at $p$-adic boundaries | Smooth entanglement scaling observed |
| CAL-HOL-02 | 2030 | Boundary CFT $c \propto \log p$ | Measured $c$ disagrees with $\log p$ |
| CAL-EFIMOV-01 | 2028 | $\lambda = e^{\pi/s_0}$ from adelic log-periods | $\lambda$ deviates $> 3\sigma$ from adelic prediction |
| CAL-MASS-01 | 2028 | All SM mass ratios $\in \mathcal{P}$ within 2% | Systematic deviation $> 3\sigma$ in new measurements |
| CAL-MASS-02 | 2030 | Pythagorean deviations shrink with precision | Deviations persist or grow with improved measurements |
| CAL-EXP-01 | 2027 | $\Gamma(3)/\Gamma(1) > 0.3$ in transmon experiment | Ratio $< 0.1$ at $5\sigma$ |
| CAL-EXP-02 | 2028 | Step-function fit beats smooth log for $S_{\text{EE}}(L)$ | Bayes factor $> 10$ favors smooth model |
| CAL-EXP-03 | 2029 | Zero cross-talk between $\operatorname{ord}_2$ and $\operatorname{ord}_3$ syndromes | Mutual information $> 0$ at $5\sigma$ |
| CAL-PI-01 | 2028 | $\pi$ admits no consistent $p$-adic valuation at all primes | A $p$-adic $\pi$ satisfying $\prod_v |\pi_p|_p = 1$ is constructed [disconfirms idèle-blocking claim] |
| CAL-MELLIN-01 | 2029 | Hadron resonances organize into families with integer-spaced pole separations $\Delta + 2\mathbb{Z}_{\geq 0}$ | Any resonance with pole spacing incompatible with $2\mathbb{Z}$ at $3\sigma$ disconfirms |
| CAL-SMATRIX-01 | 2030 | $p$-adic S-matrix elements for active primes $\{2,3,5\}$ are rational functions of $p^s,p^t$ | Any $\pi$ dependence found in tree-level $p$-adic amplitude disconfirms |
| CAL-CLOCK-01 | 2028 | No Hermitian time operator $[\hat{T},\hat{H}] = i\hbar$ exists on any non-Archimedean completion of $\mathbb{Q}$ | Construction of a time operator on $\mathbb{Q}_p$ disconfirms |
| CAL-LP-BLOCK-01 | 2027 | $L_p(4,\omega^{-3})$ requires original Coleman integration — numerical $\pi_p$ pending this computation | Qualitative predictions (pole spectrum, $S$-matrix structure) unaffected by this gap |
---
## References
1. Particle Data Group (2024). Review of Particle Physics. *PTEP* 2024, 083C01. DOI: 10.1093/ptep/ptae074.
2. Wilson, K.G. (1971). RG and critical phenomena. *Phys. Rev. B* 4, 3174. DOI: 10.1103/PhysRevB.4.3174.
3. Knill, E., Laflamme, R. (1997). QEC conditions. *Phys. Rev. A* 55, 900. DOI: 10.1103/PhysRevA.55.900.
4. Gottesman, D., Kitaev, A., Preskill, J. (2001). Encoding a qubit in an oscillator. *Phys. Rev. A* 64, 012310. DOI: 10.1103/PhysRevA.64.012310.
5. Michael, M.H. et al. (2016). Binomial codes. *Phys. Rev. X* 6, 031006. DOI: 10.1103/PhysRevX.6.031006.
6. Hayden, P. et al. (2016). Holographic duality from random tensor networks. *JHEP* 2016, 9. DOI: 10.1007/JHEP11(2016)009.
7. Vidal, G. (2007). Entanglement Renormalization. *Phys. Rev. Lett.* 99, 220405. DOI: 10.1103/PhysRevLett.99.220405.
8. Gubser, S.S. et al. (2017). $p$-adic AdS/CFT. *Commun. Math. Phys.* 352, 1019. DOI: 10.1007/s00220-017-2813-1.
9. Efimov, V. (1970). Energy levels arising from resonant two-body forces. *Phys. Lett. B* 33, 563. DOI: 10.1016/0370-2693(70)90349-7.
10. Koch, J. et al. (2007). Charge-insensitive qubit. *Phys. Rev. A* 76, 042319. DOI: 10.1103/PhysRevA.76.042319.
11. Braaten, E. & Hammer, H.-W. (2006). Universality in few-body systems. *Phys. Rept.* 428, 259–390. DOI: 10.1016/j.physrep.2006.03.001.
12. Bost, J.-B. & Connes, A. (1995). Hecke algebras, type III factors and phase transitions. *Selecta Math. (N.S.)* 1, 411–457. DOI: 10.1007/BF01589495.
13. Georgi, H. & Glashow, S.L. (1974). Unity of all elementary-particle forces. *Phys. Rev. Lett.* 32, 438–441. DOI: 10.1103/PhysRevLett.32.438.
14. Weinberg, S. (1995). *The Quantum Theory of Fields*, Vol. I. Cambridge University Press.