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Toroidal Topology of Reality: Nested φ-Tori as the Unification of Continuous and Discrete

Anton Pankratov(independent)·
torusnested toriKAM theoremgolden ratioπφspiral gapphotonquasiperiodic motiontokamakorbital mechanicscontinuous and discrete

Abstract

Аннотация

RU

Показано, что непрерывная фазовая динамика (π-вращение) и дискретные квантовые переходы (φ-скачки) являются проекциями одной геометрической структуры: квазипериодической траектории на вложенных φ-торах. Спиральный зазор (π−3)² — механизм связи непрерывного и дискретного. Отношение R/r=φ обеспечивает максимальную устойчивость по теореме КАМ. Фотон интерпретирован как квант зазора — мост между π-вращением и φ-переходом. Реальность — бесконечно вложенная тороидальная матрёшка.

Abstract

EN

It is shown that continuous phase dynamics (π-rotation) and discrete quantum transitions (φ-jumps) are projections of a single geometric structure: a quasiperiodic trajectory on nested φ-tori. The spiral gap (π−3)² is the coupling mechanism between continuous and discrete. R/r=φ ensures maximal stability by the KAM theorem. The photon is interpreted as a gap quantum — a bridge between internal rotation and inter-level jump. Reality is an infinitely nested toroidal matryoshka.

摘要

ZH

连续相位动力学（π旋转）和离散量子跃迁（φ跳跃）是嵌套φ-环上准周期轨迹这一单一几何结构的投影。螺旋间隙(π−3)²是连续与离散之间的耦合机制。R/r=φ根据KAM定理确保最大稳定性。光子被解释为间隙量子。现实是无限嵌套的环形套娃。

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Subjects & Identifiers

Subjects:
General Physics (physics.gen-ph) · torus · nested tori · KAM theorem · golden ratio · π · φ · spiral gap · photon · quasiperiodic motion · tokamak · orbital mechanics · continuous and discrete
Category:
Physics and Technology · Физика и технологии
Authors:
Anton Pankratov (independent researcher)
Submitted:
Last modified:
Languages:
Russian (primary), English
Permanent URL:
https://odtoe.org/en/articles/toroidal-topology
Journal:
Observer-Dependent Theory of Everything (ODTOE Corpus)
Comments:
For research collaboration or corrections, contact via /contact. Citations and academic engagement welcome.

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APA-like
Pankratov A. "Toroidal Topology of Reality: Nested φ-Tori as the Unification of Continuous and Discrete." Observer-Dependent Theory of Everything, odtoe.org, 2026. https://odtoe.org/en/articles/toroidal-topology
BibTeX[ click to expand ]
@article{pankratov2026toroidalTopology,
  author    = {Pankratov, Anton},
  title     = {Toroidal Topology of Reality: Nested φ-Tori as the Unification of Continuous and Discrete},
  journal   = {Observer-Dependent Theory of Everything},
  year      = {2026},
  month     = {Mar},
  url       = {https://odtoe.org/en/articles/toroidal-topology},
  publisher = {odtoe.org}
}
RIS (EndNote / Reference Manager)[ click to expand ]
TY  - JOUR
AU  - Pankratov, Anton
TI  - Toroidal Topology of Reality: Nested φ-Tori as the Unification of Continuous and Discrete
JO  - Observer-Dependent Theory of Everything
PY  - 2026
DA  - 2026-03-13
UR  - https://odtoe.org/en/articles/toroidal-topology
PB  - odtoe.org
ER  -
Toroidal Topology of Reality: Nested φ-Tori as the Unification of Continuous and Discrete

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