Toy Model of our Universe, thought experiment for all: “Start with a Sphere.”

First, we’ll have to understand the basics of an infinitesimal perfect sphere. We’re still learning. But we do know this. It is defined by the most fundamental units of measurement we can know. It helps define the most basic equations in the universe. If you start with the smallest possible units of space and time — the Planck base units –and double them, then double that, and do it 202 times, you’ll have everything — every particle, every force, every galaxy — in a quiet, geometry-driven unfolding that leaves no singularity, no arbitrary initial conditions, and no unexplained mysteries (of course, there will still be plenty of questions).

That is the idea this project has been exploring since 2011.

Also, before we began, we learned about a 7.356° gap in the geometry of sphere-packing — an irreducible imperfection that Aristotle and those who followed him missed for 1800 years. Our physical models of that gap jittered and bounced; we immediately thought it might have something to do with quantum fluctuations. And now, in this new environment of 202 base-2 notations, we began to imagine how it could also be the engine behind everything: entropy, expansion, the cosmic microwave background, even one of physics’ deepest unsolved mysteries, the fine-structure constant α ≈ 1/137.

We will submit this article to arXiv. We are looking for co-authors, critics, and curious minds who want to begin to grasp the continuity-symmetry-harmony of the sphere that is shared with the nature of infinity.

If you say:

→ I’m curious but not a physicist

Welcome. You don’t need any physics background for this page.

Here is the idea in one paragraph:

The universe is incomprehensibly large — some say “46.5 billion light-years across” while others say 93 billion light years. There is general concurrence that the universe is about 13.8 billion years old. The most-commonly accepted, smallest-meaningful physical scale — the Planck base units — gives us a length of about 10-35 meters and first-and-smallest unit of time at 5.39×10-44 seconds. If you start at the Planck length and double it 202 times, you arrive at the size of the observable universe. That is not an approximation. It is accurate to better than one percent.

That simple fact — 202 doublings — is the foundation of this project. It means the universe has a natural address system. Every physical scale has a notation number, from 0 (the Planck scale) to 202 (the current horizon). The proton lives near Notation 65. The atom near Notation 80. The first second of cosmic time falls within Notation 143.

Within this address system, something unexpected appeared: a 7.356° gap in the geometry of sphere-packing — an imperfection that Aristotle and his followers missed for 1800 years — may leave a specific fingerprint on the oldest light in the universe. Telescopes currently under construction are sensitive enough to find it or rule it out.

That is the story. The fingerprint page explains the prediction in plain language. The paper explains the mathematics.

→ Read the fingerprint explanation → Read the full paper → See the 202-notation map

If this makes you want to send the link to someone — a teacher, a curious friend, a physicist you know — please do. We are looking for exactly that kind of engagement.

Spheres to
tetrahedrons

→ I’m a student

If you are studying physics, mathematics, or astronomy, here is where this project connects to things you may already know.

The framework uses nothing beyond high school geometry and logarithms. Every physical scale, L, has a notation address, N = log₂(L / ℓ_P), where ℓ_P is the Planck length. The calculation is elementary. The implications are not.

Three results worth checking yourself:

First: log₂(0.8412 × 10⁻¹⁵ / 1.616 × 10⁻³⁵) = 65.496. The proton charge radius falls almost exactly at the geometric mean of Notations 65 and 66. Pull out a calculator and verify it. It takes thirty seconds.

Second: log₂(2.8179 × 10⁻¹⁵ / 1.616 × 10⁻³⁵) = 67.24. The classical electron radius falls at Notation 67.24. From there, 70 doublings reach Notation 137 — the integer nearest to α⁻¹ ≈ 137.036.

Third: δ = 2π − 5arccos(1/3) ≈ 7.356°. Five regular tetrahedra sharing a common edge always leave this gap. It is exact, irrational, and incommensurate with 2π. Aristotle thought tetrahedra tiled space perfectly. They don’t. Nobody noticed for 1,800 years.

The central open problem: Can the connection between δ and α⁻¹ be derived rigorously through the base-2 doubling structure? That derivation does not yet exist. If you find it, or find a proof that it cannot exist, that is a publishable result. Interested? Then, the more specific question is…

→ Read the artricle for arXiv → See the 202-notation chart → Read the plain-language CMB explanation

256 spheres to 18.5 tredecillion spheres/ sec.

→ I’m a physicist

The framework makes three classes of falsifiable predictions, organized by immediacy.

The most immediate: the proton charge radius (0.8412 ± 0.0009 fm, CODATA 2018) falls at Notation 65.496 in a base-2 grid anchored at the Planck length — almost exactly the geometric mean of two consecutive notation scales. The classical electron radius falls at Notation 67.24. From there, 69.76 doublings reach Notation 137. We conjecture a resonant minimum in cumulative gap-induced geometric tension at that notation, corresponding to α⁻¹ ≈ 137.036.

We are not claiming a derivation. We are claiming the observation is precise enough to warrant either finding one or demonstrating why it is coincidental.

The second class: the ratio of Planck charge to elementary charge (factor ≈ 11.7, log₂ ≈ 3.55 notations) should be derivable from the Aristotle gap operating over the first few notations — a calculation in elementary geometry requiring no new assumptions.

The third class: The gap’s harmonic imprint across all 202 notations predicts a specific non-Gaussianity and B-mode correlation pattern in CMB polarization data, distinguishable from inflationary predictions and accessible to CMB-S4.

The framework fits naturally in gr-qc with hep-th cross-listing. We are seeking arXiv endorsers and welcome co-authors willing to work on the open derivations.

→ Read the full paper → Read a two-page technical summary → Contact Bruce Camber

A geometric illustration of a quadrilateral pyramid with a hexagonal base, showing colored sections that represent angles and segments within the solid.

→ I want to see the full 202-notation map

The map is here: https://81018.com/chart/

It shows all 202 doublings from the Planck scale to the current observable horizon, with columns for Planck Length, Planck Time, Planck Mass, Planck Charge, and Planck Temperature at each notation. Every known physical scale has an address in this map.

The map was first outlined in December 2011. It has been refined continuously since. It is the foundation from which everything else in this project follows.

A few addresses worth noting:

A Key Question We are Trying to Answer:

  • Does the position of a composite structure at the geometric mean of two consecutive approximation layers in a Goodwillie tower have a known characterization, and if so, what does 2-excisive mean for a doubling functor on Euclidean spheres?

→ Open the map

References

  1. Thomas Goodwillie, Brown University
  2. More to come…
  3. Resource page for dimensionless constants (2017)

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URL (this page): https://81018.com/operad/