by ChatGPT building upon Grok4.

Overview & Assumptions
This is a global discrete-creation toy model that imagines the universe producing one “Planck sphere” per Planck time (t_P) across all of space. It’s not one sphere per Planck volume, but a single global event per tick. The goal is to translate that playful-but-ginormous rate into conventional Hubble units (km s^-1 Mpc^-1) to test whether it is anywhere near observations.

^{This file: https://81018.com/toy-model-derivation-of-the-hubble-constant/
Date posted: 6 October 2025}

Remarkably it is. The final calculation yields a present-day expansion rate numerically close to Hubble’s.

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1. Raw “thrust” frequency at the Planck scale

The Planck time is: t_P = 5.391 × 10^-44 so the raw creation frequency is: f_raw = 1 / t_P ≈ 1.85 × 10⁴³ s^-1 This is the foundational “drive” of the toy model — far larger than any observed expansion rate.

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2. Converting frequency to Hubble units

The Hubble constant H₀ is typically expressed in km s^-1 Mpc^-1. To convert a rate in s^-1 (call it H_s) to those units: H_km = H_s × (1 Mpc in km).

Since 1 Mpc ≈ 3.086 × 10¹⁹ km, a direct (naïve) substitution gives:

H_km^naïve = (1.85 × 10⁴³) × (3.086 × 10¹⁹) ≈ 5.7 × 10⁶² km s^-1 Mpc^-1

— an absurdly large value that makes clear a dilution factor is needed.

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3. Dilution by cosmic accumulation

Take the current cosmic age as: t ≈ 4.35 × 10¹⁷ s (≈ 13.8 billion years)

The number of Planck intervals that have elapsed is: N = t / t_P ≈ 8.1 × 10⁶⁰

We treat the effective expansion rate as the raw frequency diluted over those N accumulated events:

H_s^eff = f_raw / N = (1 / t_P) / (t / t_P) = 1 / t

Numerically: H_s^eff ≈ 2.30 × 10^-18 s^-1

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4. Conversion to km s^-1 Mpc^-1

H_km^eff = (2.30 × 10^-18) × (3.086 × 10¹⁹) ≈ 71 km s^-1 Mpc^-1

That value happens to fall right within the observed range (≈ 67–73 km s^-1 Mpc^-1).

Summary (at a glance)
f_raw = 1 / t_P → H_s^eff = 1 / t → H₀ = H_s^eff × 3.086 × 10¹⁹ ≈ 71 km s^-1 Mpc^-1

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5. Caveats and context

• This is a purely dimensional toy argument, not a physical derivation.
• The “one sphere per Planck time” rule is a speculative assumption.
• Cosmic expansion in standard cosmology refers to metric stretching, not literal creation of new spheres.
• Results depend directly on the assumed age of the universe: a 5–10 % change in age shifts the derived value similarly.

Still, the dimensional coincidence is intriguing: a Planck-time process diluted over cosmic history yields a present-day expansion rate numerically close to Hubble’s.

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Mathematics and geometries to expand upon the toy model:

1. Base-2 notation from the Planck base units to the cosmic scale: https://81018.com/chart/
2. Stabilizing geometries: The Four Primary Irrational Numbers Stabilizes Spheres from the Planck to the cosmological scale: https://81018.com/planck-polyhedral-core/ https://81018.com/ross/
3. Perfect Domains: Notations-0 to Notation-50: https://81018.com/starts-8/
4. A Lagrangian is slowly emerging.

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