Published February 3, 2026 | Version v1
Preprint Open

Hybrid Linear–Cyclic Topological Structures for Digital Sequence Encoding and Technosignature Analysis

Authors/Creators

  • 1. Academy of Ministry of Inteiror

Description


This work develops a formal topological analysis of a planar geometric configuration commonly known as the “Pi Crop Circle” (2008), treated here purely as an abstract hybrid structure. The configuration is modeled as a composition of a distinguished center, a one-dimensional spiral trajectory, and three nested cyclic components. Parametrization in the manifold $S^{1} \times \mathbb{R}^{+}$ separates the linear and cyclic contributions and reveals a coherent hybrid organization.

A minimal set of structural invariants is identified—center, continuous spiral, discrete radial segmentation, and embedded cycles—each stable under homeomorphisms and jointly determining the topological type of the configuration. The resulting discrete--continuous architecture exhibits low redundancy and high internal order, characteristic of hybrid linear--cyclic systems.

Such structures are relevant to geometric encoding theory and to methodological approaches in technosignature analysis, where emphasis is placed on structural coherence rather than provenance. The identified invariants correspond to widely recurring geometric motifs—center, cycle, spiral, and radial progression—appearing across independent mathematical and cosmological traditions. These parallels are noted solely at the level of abstract form.

The study makes no claims regarding authorship or intent. Its contribution lies in demonstrating that the configuration constitutes a well-defined example of hybrid linear--cyclic topology with potential applications in encoding theory, discrete--continuous systems, and the analysis of structured patterns in applied contexts.

Notes

⭐ Highlights

  • A hybrid linear–cyclic topological space is formalized as the planar set
                                 X = {p₀} ∪ γ([0,1]) ∪ ⋃ᵢ Cᵢ ⊂ ℝ²,
    consisting of a distinguished center, a continuous spiral trajectory, and a finite collection of embedded cyclic components.

  • Digital sequences are embedded into X through a reversible geometric mapping that integrates linear ordering with rotational symmetry, producing a structurally enriched representation.

  • The resulting topological invariants expose correlational, periodic, and structural features relevant to analytical frameworks in technosignature research, where hybrid geometric patterns may encode non‑random informational content.

  • The construction provides a geometric basis for encoding schemes that rely on hybrid linear–cyclic topological signatures, offering a structurally robust method for representing digital information in a continuous geometric medium.

Series information

This publication forms part of the broader research cycle Linear Infinity → Cyclic Space → Expansive Chaos, a long‑term program investigating how linear asymptotic structures, cyclic geometries, reconstruction manifolds, and emergent organizational regimes can be described within a common geometric–informational framework.

The series currently consists of the following interconnected contributions:

  • Part I. — Hybrid Linear–Cyclic Topological Structures for Digital Sequence Encoding and Technosignature Analysis | DOI: 10.5281/zenodo.18473473
  • Part II. — Informational Geometry of the Positive Half-Line. A World Without Negative Numbers | DOI: 10.5281/zenodo.18474513
  • Part III. —  A Symmetric Classification of Prime Numbers. Correlational, Identity, and Inversion Symmetry | DOI: 10.5281/zenodo.18520138

The present work should be regarded as a companion volume to Part III:

Research Notes Companion — Symmetry‑Based Classification of Prime Number Families: A FAIR Dataset | DOI:10.5281/zenodo.18328852

  • Part IV. — Global Affine Time and Metric Uniqueness: A Geometric Characterization of Linear and Cyclic Temporal Structure | DOI: 10.5281/zenodo.18505857
  • Part V. — Symmetric Signatures of Global Configurations: Topological Rigidity and the Epoch of Convergence in a Case Study of Orion–Giza Correspondence. A Unified Framework for 2D Similarity Invariants, 3D Orientation Geometry, and 4D Precessional Dynamics | DOI:10.5281/zenodo.18651258
  • Part VI. — Linear Time, Cyclic Geometry. Spectral Phase Structure of Cosmic Emitters | DOI:10.5281/zenodo.18698140
  • Part VII. — From Topological Models to Telescope Data: Empirical Tests of Hybrid Linear–Cyclic Dynamics in High-Energy Astrophysical Systems | DOI:10.5281/zenodo.20545297

The present work should be regarded as a companion volume to Part VII:

Research Notes Companion — Research Notes on Reconstruction Manifolds Derived from FAIR High‑Energy Astrophysical Data: A Collection of Observations, Tests, Negative Results, and Working Hypotheses Derived from Reconstructed Descriptor Spaces of High‑Energy Astrophysical Systems | DOI:10.5281/zenodo.20680593

Unlike the primary FAIR data publication, which focuses on the construction of reconstruction descriptors and their cross‑mission properties, the companion notes document a broader set of exploratory investigations. These include outlier persistence, descriptor‑space geometry, cross‑mission correlations, manifold structure, curvature analysis, direction consistency, clustering behavior, negative results, and methodological observations that emerged during the reconstruction process.

Taken together, the publications in the series trace a progression from abstract mathematical structures toward empirical reconstruction spaces:

linear infinity → asymptotic contraction → metric multiplicity → temporal stratification → cyclic space → spectral phase structure → reconstruction manifolds.

These stages are not interpreted as physical phases of nature. Rather, they represent increasingly rich organizational regimes that appear across mathematical models, geometric constructions, informational systems, and observational datasets.

All project publications, datasets, supplementary materials, and machine‑readable metadata are maintained through the project portal: https://linearcyclic.eu

which serves as the central repository for the Linear Infinity → Cyclic Space → Expansive Chaos research program.

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References

  • M. Freiberger, Pi appears in a crop circle, https://plus.maths.org/content/ pi-appears-crop-circle, plus Magazine, University of Cambridge. Accessed: 2026- 01-08 (2008)
  • S. MacLane, G. Birkhoff, Algebra, AMS Chelsea, 1999
  • D. Knuth, The Art of Computer Programming, Vol. 1, Addison-Wesley, 1997
  • R. Penrose, The Road to Reality, Jonathan Cape, 2004
  • E. Ghys, Spirals and the dynamics of curves, Notices of the AMS 53 (3) (2006) 344–353
  • . Stewart, Nature's Numbers: The Unreal Reality of Mathematics, Basic Books, 1995
  • B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, 1982
  • A. Hatcher, Algebraic Topology, Cambridge University Press, 2002
  • V. I. Arnold, Topological Methods in Hydrodynamics, Springer, 1998
  • H. S. M. Coxeter, Introduction to Geometry, Wiley, 1961
  • J. Kappraff, Connections: The Geometric Bridge Between Art and Science, World Scientific, 2001
  • R. Thom, Structural Stability and Morphogenesis, Benjamin, 1975
  • M. Livio, The Golden Ratio: The Story of Phi, Broadway Books, 2003
  • Pi crop circle archive, https://temporarytemples.co.uk, accessed: 2026-01-08 (2008)
  • Crop circle connector archive, https://cropcircleconnector.com, accessed: 2026-01- 08 (2008)
  • D. K. Washburn, D. W. Crowe, Symmetries of Culture: Theory and Practice of Plane Pattern Analysis, University of Washington Press, 1988
  • M. Ascher, Ethnomathematics: A Multicultural View of Mathematical Ideas, Chapman & Hall, 1991
  • A. Aveni, Skywatchers of Ancient America, University of Texas Press, 2001
  • J. Gleick, Chaos: Making a New Science, Viking, 1987
  • I. S. Shklovskii, C. Sagan, Intelligent Life in the Universe, Holden-Day, 1966
  • S. Shostak, Seti institute technical publications, https://www.seti.org, accessed: 2026-01-08
  • G. Lakoff, R. Núñez, Where Mathematics Comes From, Basic Books, 2000
  • C. Sagan, The Cosmic Connection, Cambridge University Press, 2000
  • W. I. Newman, C. Sagan, Galactic civilizations: Population dynamics and interstellar diffusion, Icarus 46 (1981) 293–327
  • A. Frank, M. Lingam, The anthropocene generalized: Technosignatures in planetary context, Astrobiology 18 (4) (2018) 503–518
  • F. C. Adams, The astrobiology of technosignatures, Astrobiology 10 (5) (2010) 491–498
  • J. T. Wright, Prioritizing technosignature searches, International Journal of Astrobiol- ogy 17 (2) (2018) 177–188