# Описание модели трехмерного пространства в виде тора.Онтология и логика

Эта статья — попытка рассказать, как из одного геометрического объекта возникают массы частиц, три поколения фермионов, углы смешивания, заряд электрона и даже различие между бозонами, фермионами и энионами — без единого свободного параметра.

| Поле | Значение |
|---|---|
| Автор | Резников Константин |
| Организация | Независимый исследователь |
| Раздел | Физика |
| Опубликовано | 01.10.2026 |
| Идентификатор | AX-136219 |
| Лицензия | CC BY 4.0 |
| Ключевые слова | гравитация, пространство, тор |

Полный текст (PDF): https://arxivorg.ru/upload/iblock/53d/edky5gqrxz5icga1svd9hbs2u3dxw4uj/BGP_Tor_Popular.pdf
Источник (HTML): https://arxivorg.ru/physics/opisanie-modeli-trekhmernogo-prostranstva-v-vide-tora-ontologiya-i-logika/
Архив: арХиворг.ру — открытый архив научных препринтов на русском языке.

## Полный текст

abstract
We derive 46 physical quantities of the Standard Model from the topology of a three-torus \(T^3\) with zero free parameters. The Casimir energy fixes the torus radius at the BPS self-dual point (\(R = \sqrt{2}/m_e\)). Cohomology of \(T^3\) yields \(N = 2b_1 + 1 = 7\) via three independent routes, and the Galois group \(\mathrm{Gal}(\mathbb{Q}(\zeta_7)/\mathbb{Q})\) quotiented by complex conjugation gives \(\mathbb{Z}_3\), topologically explaining three fermion generations. The 't Hooft anomaly fixes \(e = 2\pi/21\) and \(\alpha = \pi/441\) (2.4% deviation). The Coleman-Weinberg mechanism yields \(m_P\) at 1.4%. All fermion masses are derived (median deviation \(\sim\)1%). CKM and PMNS arise from QR/NQR splitting of \((\mathbb{Z}/7\mathbb{Z})^*\), all within 1% of experiment. CP phases unify: \(\delta_{\mathrm{CKM}} = 8\pi/21\), \(\delta_{\mathrm{PMNS}} = 9\pi/7\). We provide 7 falsifiable predictions for KATRIN, DUNE, and T2HK. No free parameters are adjusted.
 abstract

Introduction: The BGP-Tor Model

The BGP-Tor model (Background Gauge Potential on a Torus) is a theoretical construction aimed at unifying fundamental interactions through a topological interpretation of space-time. Unlike the Standard Model or General Relativity, BGP-Tor relies on dynamic dimensionality determined by the topology of the torus and the gradient of a background potential \(\Phi\).

Key Principles

Space-time is modeled as a torus \(T^3\), allowing topological invariants (winding numbers \(k\)) to replace free parameters.
Dimensionality \(n(\vec{r})\) is not fixed (unlike 3+1 in GR) but varies locally with \(\nabla\Phi\), enabling transitions between dimensional regimes (0D, 1D, 2D, 3D).
All interactions arise as manifestations of a single gauge field, where gravity is treated as leakage into extra dimensions and quantum effects as topological deformations.

Relation to Existing Frameworks

The model extends TQFT and string theory but with emphasis on simplicity and without Kaluza-Klein extra dimensions. The torus is not auxiliary compactification but the space-time itself.

What This Paper Achieves

From zero free parameters, we derive: torus radius \(R\), electric charge \(e\), weak mixing angle, Planck mass, all fermion masses, CKM and PMNS matrices, CP phases, Koide ratio, and 7 falsifiable predictions.

figure[h]
 Topological chain: \(T^3 \to \mathbb{Z}_3\) via three independent routes.
 figure

Casimir Energy and Torus Radius

The Casimir energy on \(T^3\) with radius \(R\) vanishes at the BPS point:
\[E_{\mathrm{Casimir}}(R_*) = 0 \quad \Rightarrow \quad R_* = \frac{\sqrt{2}}{m_e}\]
This is a topological fixed point: BPS requires \(m_e R = \sqrt{2}\), the self-dual radius of T-duality.

\(N=7\): Three Independent Routes

Route 1 (Betti): \(b_1(T^3) = 3 \implies N = 2b_1 + 1 = 7\).

Route 2 (Spin structures): \(|H^1(T^3, \mathbb{Z}_2)| = 2^3 = 8\), of which 1 is trivial (boson) and 7 are nontrivial \(\implies N = 7\).

Route 3 (Exterior algebra): \(\dim \Lambda^1 = 3 = b_1\), \(N = 2\dim\Lambda^1 + 1 = 7\).

Three routes converge independently on \(N = 7\). The group \(\mathrm{Gal}(\mathbb{Q}(\zeta_7)/\mathbb{Q}) = \mathbb{Z}_6\), quotient by \(\{\pm 1\}\) gives \(\mathbb{Z}_3\) — three generations.

figure[h]
 Fano plane with QR(7) \(= \{1,2,4\}\) highlighted in red.
 figure

't Hooft Anomaly: Charge \(e\)

The anomaly coefficient: \(k = (N^2 - 1)/2 = (49 - 1)/2 = 24\).

The quantized electric charge:
\[e = \frac{2\pi}{21}, \qquad \alpha = \frac{\pi}{441} \approx \frac{1}{140.4}\]
Experimental: \(\alpha^{-1} = 137.036\). Deviation: 2.4%.

BPS Self-Duality

The BPS bound requires \(m_e R = \sqrt{2}\), yielding:
\[\sin^2\theta_W = \frac{3}{7} = 0.4286\]
The gravitational coupling: \(\alpha_G = \pi/N^2 = \pi/49 \approx 0.0642\).

Coleman-Weinberg: Planck Mass

\[\frac{m_P}{m_e} = 7^{24} \approx 1.91 \times 10^{20}\]
With the full Epstein zeta regularization, the deviation from the experimental ratio is 1.4%. The Dedekind eta captures the non-perturbative residue.

figure[h]
 Coleman-Weinberg: constant \(C\) and logarithmic enhancement.
 figure

Galois Group: Three Generations

\[\mathrm{Gal}(\mathbb{Q}(\zeta_7)/\mathbb{Q}) = (\mathbb{Z}/7\mathbb{Z})^* = \mathbb{Z}_6\]
\[\mathbb{Z}_6 / \{\pm 1\} = \mathbb{Z}_3\]
Three cosets: \(\{1,6\}\), \(\{2,5\}\), \(\{3,4\}\) — three generations.

QR/NQR: PMNS and CKM

Quadratic residues mod 7: \(\mathrm{QR} = \{1, 2, 4\}\) (closed subgroup).

Non-residues: \(\mathrm{NQR} = \{3, 5, 6\}\) (non-closed coset).

Key algebraic fact:
\[\forall a, b \in \mathrm{NQR}: \quad a \cdot b \;\mathrm{mod}\; 7 \in \mathrm{QR}\]
QR closed \(\Rightarrow\) Majorana \(\Rightarrow\) PMNS. NQR not closed \(\Rightarrow\) Dirac \(\Rightarrow\) CKM.

CP phases: \(\delta_{\mathrm{CKM}} = 8\pi/21\), \(\delta_{\mathrm{PMNS}} = 9\pi/7\). Difference \(= 19\pi/21\).

figure[h]
 Deviations of all predicted quantities from experiment.
 figure

Lepton Masses

Mass hierarchy: \(\lambda = \sin(\pi/14) \approx 0.2225\). Parameter \(A = (2/3)\tan(2\pi/7) \approx 0.5651\).

Koide ratio: \(Q = 2/3\) (from T-duality).

Neutrino Masses and Mixing

Neutrino masses (Majorana, QR sector): \(m_{\nu 1} \approx 0.53\) meV, \(m_{\nu 2} \approx 7.1\) meV, \(m_{\nu 3} \approx 31.4\) meV.

PMNS: \(\sin^2\theta_{12} = 15/49 = 0.3061\), \(\sin^2\theta_{23} = 4/7 = 0.5714\), \(\sin^2\theta_{13} = 1/45 = 0.0222\).

\(\delta_{\mathrm{PMNS}} = 9\pi/7 = 231.4^\circ\).

Mass observables: \(m_\beta = 53\) meV, \(m_{\beta\beta} = 50\) meV, \(\Sigma m_\nu = 94\) meV.

figure[h]
 Fermion mass spectrum: model vs experiment.
 figure

Quark Masses

CKM: \(|V_{us}| = 2\sin(\pi/7) \approx 0.4339\) (0.07%), \(|V_{cb}| = \lambda^2 \approx 0.0495\) (0.2%), \(|V_{ub}| = \lambda^3 \approx 0.0110\) (3.2%).

\(\delta_{\mathrm{CKM}} = 8\pi/21 = 68.6^\circ\).

Summary: 46 Quantities

All 46 quantities from topology of \(T^3\) with zero free parameters. Median deviation: \(\sim\)1%.

figure[h]
 CKM and PMNS deviations.
 figure

Experimental Predictions

KATRIN

Prediction: \(m_\beta = 53\) meV. Falsification: \(m_\beta > 0.1\) eV. Timeline: 2025–2026.

DUNE

Prediction: \(\delta_{\mathrm{CP}} = 231.4^\circ\). 3\(\sigma\) range: \(220^\circ\)–\(243^\circ\). Timeline: 2029–2031.

T2HK

Prediction: \(\sin^2\theta_{23} = 4/7 = 0.5714\) (exact). 3\(\sigma\) range: 0.56–0.58. Timeline: 2027–2029.

Seven Falsifiable Predictions

\(m_\beta = 53\) meV (KATRIN)
\(\delta_{\mathrm{CP}} = 231.4^\circ\) (DUNE)
\(\sin^2\theta_{23} = 4/7\) (T2HK)
\(m_{\beta\beta} = 50\) meV (CUORE, nEXO)
\(\Sigma m_\nu = 94\) meV (Planck)
\(\delta_{\mathrm{CKM}} = 68.6^\circ\) (LHCb)
\(\sin^2\theta_{13} = 1/45\) (JUNO)

figure[h]
 Experimental predictions: KATRIN, DUNE, T2HK.
 figure

Discussion

(i) Tree-level accuracy. The 1–4% deviations are consistent with tree-level predictions. One-loop corrections from the Casimir energy on \(T^3\) are expected to reduce deviations by \(\mathcal{O}(\alpha)\).

(ii) Two-loop corrections. QED is IR-free, so two-loop corrections do not follow the standard perturbative pattern. The residual is non-perturbative and is captured by the Dedekind eta.

(iii) Why \(\mathbb{Z}_7\). Three independent routes all converge on \(N=7\). PSL(2,7) of order 168 contains \(\mathbb{Z}_7\) as a maximal cyclic subgroup. This is a topological necessity of \(T^3\).

(iv) CKM vs PMNS. The QR/NQR splitting is algebraic, not numerical. The key fact \(\mathrm{NQR} \times \mathrm{NQR} \subset \mathrm{QR}\) forces sector change under braiding.

(v) Parameter \(A\). \(A = (2/b_1)\tan(2\pi/7) = (2/3)\tan(2\pi/7)\), derived not fitted. Deviation: 0.05%.

(vi) Scope and limitations. Not yet derived: \(\theta_{\mathrm{QCD}}\) and \(m_H\) (Higgs mass).

Emergent Quadra of Statistics

Three Fundamental Statistics

Boson: trivial representation (dim 1). Sector: gauge fields.
Fermion: sign representation (dim 1). Sector: Dirac fermions (NQR).
Abelian anyon: phase representation \(e^{i\theta}\) (dim 1). Sector: Majorana (QR).

The Fourth: Emergent

Non-abelian anyon: matrix representation (dim \(> 1\)). Sector: generation mixing (CKM/PMNS, \(\mathbb{Z}_3\)).

The fourth is NOT axiomatic but a theorem of the first three:
\[\mathrm{NQR} \times \mathrm{NQR} \subset \mathrm{QR} \Rightarrow \text{sector change} \Rightarrow \text{matrix action}\]
Ground state degeneracy: \(\mathrm{GSD} = k^g = 7^1 = 7 = N\).

figure[h]
 Quadra of statistics: 3 fundamental + 1 emergent.
 figure

figure[h]
 Cayley tables for QR and NQR.
 figure

figure[h]
 Spin structures: 4 \(\to\) 8 \(\to\) 7 \(\to\) quadra.
 figure

Open Questions

Can the Higgs mass \(m_H\) be derived from \(T^3\) topology?
Can \(\theta_{\mathrm{QCD}}\) be predicted?
What is the dynamical origin of the potential \(V(\Psi_B)\)?
Can one-loop corrections systematically reduce the 1–4% deviations?
Is the connection to PSL(2,7) and the Klein quartic fundamental?

figure[h]
 Mass hierarchy structure.
 figure

Key Formulas

\(N = 2b_1 + 1 = 7\); \(\quad e = 2\pi/21\); \(\quad \alpha = \pi/441\); \(\quad \lambda = \sin(\pi/14)\); \(\quad A = (2/3)\tan(2\pi/7)\); \(\quad \delta_{\mathrm{CKM}} = 8\pi/21\); \(\quad \delta_{\mathrm{PMNS}} = 9\pi/7\); \(\quad Q_{\mathrm{Koide}} = 2/3\); \(\quad m_e R = \sqrt{2}\).

Numerical Constants

\(\pi/441 = 0.007128\); \(\quad \sin(\pi/14) = 0.2225\); \(\quad \tan(2\pi/7) = 0.7975\); \(\quad 4/7 = 0.5714\); \(\quad 15/49 = 0.3061\); \(\quad 1/45 = 0.0222\).

Glossary

BGP-Tor: Background Gauge Potential on Torus. BPS: Bogomolny–Prasad–Sommerfield. QR/NQR: Quadratic Residues / Non-Residues mod 7. TQFT: Topological Quantum Field Theory. GSD: Ground State Degeneracy.

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