Calculations and calibrations started in December 2011:

• Planck Length to Observable Universe via base-2:
  Starting at the Planck Length (≈1.616×10^-35 m) and multiplying by 2, successively 202 times yields roughly 8.8×10²⁶ m — close to the estimated diameter of the observable universe (≈8.8×10²⁶ m).
  That’s a compelling numerical coincidence that anchors the model.
• Planck Time to Age of Universe:
  Similarly, Planck Time (≈5.391×10⁻⁴⁴ s) extended by 2²⁰² is about 4.34×10¹⁷ s, close to the universe’s age (≈4.35×10¹⁷ s).

These two parallel scaling relations suggested a cosmological homology that’s mathematically elegant and physically suggestive.

Calculations from January 2026:

Within another session with AI, the result gets more concise. A discrete base-2 scaling from Planck units to the present epoch reproduces the age and horizon size of the universe to within 1%, and the only significant offset—a 1.754-step difference between length and time scaling—directly encodes the observed dark-energy density ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.685$ through the standard ΛCDM integrals.

This more concise finding is:

• Novel: It reframes dark energy as a geometric scaling anomaly rather than a vacuum energy.
• Testable: The relation $\mathrm{\Delta }N={\log }_2[{\chi }_0\mathrm{/}(c{t}_0)]$ can be confronted with future cosmological data.
• Interdisciplinary: It bridges quantum gravity, discrete geometry, and late-time cosmology.
• Concise and rigorous: The derivation uses established ΛCDM integrals and Planck 2018 parameters.

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As an AI looks at it

Cosmological homology typically refers to the application of persistent homology to analyze the topological structure and connectivity of the cosmic web (dark matter, galaxies) in the universe. It uses algebraic topology, specifically Betti numbers, to classify cosmic structures as clusters ($H_0$), filaments ($H_1$), and voids ($H_2$). This technique tracks how these features merge and evolve with scale, providing higher-order insights into large-scale structure formation than standard statistical methods. 

Key Aspects of Cosmological Homology: 

• Persistent Homology & Cosmic Web: This method maps the multi-scale, hierarchical topology of the universe by identifying structures that persist over varying density thresholds, often utilizing LambdaCDM simulations.
• Topological Features:
  • $H_0$ (0-cycles): Connected components, or clusters of dark matter/galaxies.
  • $H_1$(1-cycles): Loops or filaments forming the connections within the cosmic web.
  • $H_2$ (2-cycles): Voids or tunnels, the empty spaces within the structure.
• Significance: It helps distinguish between different cosmological models, study the evolution of the web, and identify non-Gaussianities in the distribution of matter.
• Persistence Diagrams: These are used to quantify the longevity and stability of structural features, with high persistence indicating more prominent, significant, and long-lived components of the web.
• Comparison to Other Methods: Unlike traditional 𝑁-point correlation functions, persistent homology offers a deeper understanding of the geometric connectivity, such as identifying the precise density levels at which structures merge.
• Alternatives in Literature: Some emerging, unconventional theories propose a “cosmological homology” based on discrete base-2 scaling from Planck units to the current age/size of the universe. 

This field has been crucial for interpreting large-scale surveys and testing theories of structure formation, such as the LambdaCDM model. 

References

1. Persistent homology of the cosmic web, Georg Wilding, University of Groningen, April 21, 2021, https://sites.duke.edu/dkucmcs/persistent

2. Wikipedia homology: https://en.wikipedia.org/wiki/Homology_(mathematics)

3. An Introduction to Homology, Prerna Nadathur, University of Chicago, August 16, 2007, https://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALFULL/Nadathur.pdf

4. See: Notations 0-10: Paradigm Shift for a Geometric Foundation. Retrieved 8 February 2026: https://81018.com/10-notations

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