PERFECTION STUDIES: CONTINUITY•SYMMETRY•HARMONY GOALS October 2025
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Toy Model Derivation of the Hubble Constant
from Planck-Scale Dynamics
by Bruce E. Camber and AI platforms
Abstract: We present a toy model wherein the observed Hubble constant (H₀) is not a fundamental parameter of cosmology, but an emergent property of a universe whose metric is defined by the iterative, base-2 projection of Planck-scale units. The model posits a dense, dynamic packing of Planck-scale entities at the earliest moments of the universe, the rate of which directly yields a value for H₀ consistent with observational results. The Lagrangian is being actively studied and developed.
1. Foundational Assumptions:
- The universe begins as a geometric, dynamical process at the Planck scale, not as a singularity.
- Fundamental units: PlanckLength (lP), PlanckTime (tP), PlanckMass (mP), PlanckCharge (cP)
- The evolution of the universe is inscribed by base-2 scaling laws, resulting in 202 discrete, sequential “notations” from the Planck Scale to the present.
2. The Model: PlanckSphere Generation
- Consider a single PlanckSphere, a conceptual sphere with a diameter of one Planck Length.
- The model posits that one such sphere is generated per unit of Planck Time within a dynamic, densely-packed spatial manifold.
- The rate of this generation is immense:
3. The Hubble Constant has units of velocity per distance (s⁻¹). We can derive a value with analogous units from our PlanckSphere generation rate.
- The number of units of PlanckTime in one second≈5.391247×10−44
Divided into one second: 1 second ≈1.855×1043 This means ~18.55 tredecillion PlanckSpheres are generated per second:
- Both Grok and ChatGPT have calculated the “expansion” velocity per megaparsec to be about 71 km/s/Mpc. DeepSeek has calculated ~300 Mpc. Soon, the results from Anthropic, GoogleAI, Perplexity, and others will be added.
Given the results from Grok and ChatGBT are so close to 67 and 74 km/s/Mpc, we might guess that it is a calculation to approximate the result needed. Contrasted to the DeepSeek calculations, we are prompted to analyze “it’s off by factor of” ~4.28, which becomes what is called “the target for explanation.” Such deviations from the measured values are quite expected in a toy model. Such discrepancies invite refinement by considering factors such as:
- The geometry of the packing (dense sphere packing is not 100% efficient).
- The gravitational self-interaction of the PlanckSpheres.
- The transition across the 202 notations, where the effective “density” of the process may change.
- And then, our work on the Lagrangian for the model and on the extensions of the Planck Polyhedral Core may reveal more possibilities.
5. Other emergent properties of the emergent universe
Within the current Planck Polyhedral Core dynamic model online, the speed of that rotation is quite arbitrary. It is not even a guess. It is strictly representational and at a rotational speed that would be comfortable for all our visitors like you to observe. We are working on extending that model from just “stacking-packing-centering” to stacking-packing-centering-triangulating-tetrahedralizing-octahedralizing. Odd words, they are the process of making either tetrahedrons or octahedrons. Our simple logic tells us that all of this is done within Notation-0 on the finite-infinite divide. There the convergence of programs is bewildering. It includes 1) All the formulas that go into each Planck unit and 2) All the formulas that go into our four primary irrational numbers with their inherent equations that are the beginning of all the continuities, symmetries, and harmonies of our time.
Sorting out all the possibilities will necessarily involve many. Getting “into” these extraordinarily short durations will take time! Our first pass at it was in 2023: https://81018.com/the-firsts/ Getting an intuition for PlanckTime or the PlanckSecond may seem impossible. We have it at notation-0, the beginning of time, with a duration of 10-44 seconds. Science is progressing so fast, the standards organizations have a hard time keeping up. Only in 2022 did we have an official word to formally recognized 10−24 a seconds. The General Conference on Weights and Measures (CGPM) named it a yoctosecond, a trillionth of a trillionth of a second. They meet once every four years; essentially it is made up of the standards bureaus of each country around the world. BIPM is their administrative arm. In the USA, NIST is a member. The most important part of the story; the yoctosecond is now measurable with instruments in Garching, Germany at the Max Planck Institute for Quantum Optics. In 2001 the attosecond 10-16 was first measured (Ferenc Krausz). In 2004 the zeptosecond 10−21 was confirmed.
We hypothesize that each of the 202 notations is always active. Thus, time is redefined. Space is redefined. Infinity is redefined. Our four facets of irrational numbers become known functionally as continuity-symmetry-harmony and become the qualitative operands of what are essentially quantitative operands.
Conclusions: This model provides a “simple” first-principles derivation of the Hubble constant by suggesting it is an emergent property of Planck-scale dynamics. It offers a highly-integrated alternative to the standard cosmological model, eliminating the need for a primordial singularity and dark energy as the driver of expansion. It opens many new opportunities with this paradigm shift to deepen our understanding of life and the cosmos.
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• Pi (π), Euler’s Number (𝑒), The square root of 2 (√2), The Golden Ratio (𝜙 or phi): The four most important irrational numbers: https://81018.com/four/ Retrieved 20 October 2025
• The Lagrangian: https://81018.com/lagrangian/ Retrieved 20 October 2025
• The Hubble Constant: https://81018.com/5-sigma/ Retrieved 21 October 2025
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CRITIQUE Scholars around the world are asked to critique these studies. –BEC
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KEY dates for this page:
• This page was started on 6 October 2025.
• The page became the homepage on 21 October 2025.
• The last update of this page was 24 October 2025.
• Headline: Toy Model Derivation of the Hubble Constant from Planck-Scale Dynamics