Base-2 Planck Notation and the Four Primary Irrationals: A First Survey
A systematic search through 36 physical scales — from the Planck length to the particle horizon — finds three non-trivial coincidences between their base-2 notation coordinates and the irrationals π, φ, e, and √2. One is accurate to five decimal places. None has been reported before.
The Coordinate System
The G7 inventory assigns every physical scale a single number: how many base-2 doublings separate it from the Planck length. Formally, for a scale of characteristic length L,
where ℓP = 1.616255 × 10−35 m is the CODATA 2022 Planck length. The notation number n is in general irrational — a real number carrying both an integer part (the Notation) and a fractional part.
The inventory spans: n = 0 (Planck length) through n ≈ 204 (particle horizon, 4.4 × 1026 m). The 202 integer Notations in the table correspond to 202 base-2 doublings of length from the Planck scale to the edge of the observable universe.
The four primary irrationals — π, φ (golden ratio), e (Euler’s number), and √2 — appear explicitly in the geometry of the regular octahedron through its four distinct hexagonal cross-sections. They are the Hexarchon constants. The question explored here is whether they appear again, as exact or near-exact values, in the notation coordinates of known physical scales.
Three Findings
1. The Muon Full Compton Wavelength and 49√2
λμ = h / mμc = 1.17344 × 10−14 m (CODATA 2022 muon mass: 105.6583755 MeV/c²)
The full Compton wavelength of the muon — the natural quantum length scale of the muon — sits at 49 doublings of √2 above the Planck length, to better than three parts in 10,000.
49 = 72. The factor √2 is the diagonal of a unit square — the most elementary irrational in three-dimensional geometry. The coincidence encodes both the squareness of 7 and the irrationality of the diagonal, expressed in units of Planck-scale doublings.
No adjustment was made. The muon mass is a measured constant. The integer 49 and the constant √2 were selected by the search, not pre-chosen.
2. The Muon Reduced Compton Wavelength and 2φ/5
λ̄μ = ℏ / mμc = 1.867594 × 10−15 m (reduced Compton, n = 66.647088)
The fractional part of the notation number — the sub-doubling position of the muon reduced Compton wavelength within Notation 66 — matches 2φ/5 (mod 1) to within 1.25 × 10−4. This is the tightest agreement with any simple irrational expression found across all 36 inventory entries.
φ = (1 + √5)/2 = 1.6180339… The expression 2φ/5 = 0.6472136… agrees with the measured fractional coordinate to within 0.019%.
Notably, both muon results concern the same particle. The full Compton wavelength (Finding 1) and the reduced Compton wavelength (Finding 2) differ by a factor of 2π, yet each independently produces a clean irrational relation — with different irrationals (√2 and φ) and different expressions (integer multiple vs. fractional remainder).
3. The QCD Confinement Scale and π
ℏc/ΛQCD at ΛQCD = 300 MeV: L = 6.582 × 10−16 m (n = 65.141533)
The QCD confinement scale ℏc/ΛQCD at 300 MeV sits in Notation 65. Its fractional position within that doubling matches the decimal part of π — 0.14159… — to within 6 × 10−5. This is the most precise agreement with a bare irrational constant found in the survey.
ΛQCD ≈ 300 MeV is the canonical round-number value of the QCD scale; the precise value is scheme-dependent and ranges roughly 200–400 MeV. The coincidence holds most cleanly at 300 MeV and weakens at 200 MeV (Δ grows to 0.0016). This introduces some dependence on the choice of ΛQCD.
π appearing as the sub-doubling position of the strong-force confinement scale — within Notation 65, just above the proton charge radius at n = 65.50 — is not obviously accidental.
Summary Table
| Physical scale | n | Irrational relation | Value | Δ |
|---|---|---|---|---|
| Muon full Compton λμ | 69.2986 | n = 49 × √2 | 69.2965 | 0.0021 |
| Muon reduced Compton λ̄μ | 66.6471 | frac(n) = (2φ/5) mod 1 | 0.6472 | 0.0001 |
| ℏc/ΛQCD (300 MeV) | 65.1415 | frac(n) = π mod 1 | 0.1416 | 0.0001 |
| Bohr radius a0 | 81.4374 | frac(n) ≈ (2e) mod 1 | 0.4366 | 0.0008 |
| Earth mean radius | 138.1779 | frac(n) ≈ (3π/8) mod 1 | 0.1781 | 0.0002 |
Δ is the absolute difference between the notation coordinate and the irrational expression. Green: Δ < 0.003. The Earth radius entry is included for completeness; it carries no physical motivation beyond the geometric coincidence.
Context: Why These Four Irrationals?
The search was not open-ended. The four constants π, φ, e, √2 were pre-selected for a specific geometric reason: they are the four cross-sectional irrationals of the regular octahedron, encoded in its four distinct hexagonal cross-sections. This structure — called the Hexarchon — is the focus of ongoing work on the interior dynamics of the octahedron at the foundations of the tetrahedral-octahedral space-filling tiling.
The notation coordinate system was also pre-existing, not constructed to find these results. The G7 inventory was assembled from CODATA 2022 values and standard particle physics constants before any irrational search was conducted.
The combination of a pre-specified set of constants and a pre-existing coordinate system means the search is genuinely systematic rather than post-hoc. The three primary findings emerged from a Python sweep of all 36 entries against simple expressions in {π, φ, e, √2}.
The Known Baseline: α in the Notation System
For comparison, the deepest known irrational relationship in the notation system involves the fine-structure constant α = 1/137.036…:
The gap between the Bohr radius and the classical electron radius equals exactly twice the base-2 logarithm of the inverse fine-structure constant. This follows from the quantum mechanical relation a0 = re/α2 and is exact by construction. The three findings above are of a different character: they are not known to follow from any established relation.
Caveats and What Would Strengthen This
What these findings are not
- Derived results — no theory predicts that muon scales should relate to √2 or φ in Planck notation
- High-significance statistical claims — a proper look-elsewhere correction would require specifying the full search space in advance and computing p-values across all 36 × (all simple irrational expressions) comparisons
- Reproduced by independent methods — this is a first pass
What would make this compelling
- A derivation — showing from first principles why 49√2 doublings above Planck corresponds to the muon mass
- A third muon-family relation (the tau lepton has n ≈ 62.575 — does it hit a clean expression?)
- A physical mechanism connecting the Hexarchon irrationals to particle mass scales
- Independent reproduction of the Python sweep with a different inventory and different base point
Open Questions
Why the muon? Both primary findings (√2 and φ) involve the muon and no other particle in the inventory. The muon is anomalous in the Standard Model — its mass is unexplained, its magnetic moment shows a persistent deviation from theory (the (g−2)μ anomaly), and it is otherwise identical to the electron. If the notation system is encoding something real about particle physics, the muon is the place to look first.
Why Notations 65–66? The QCD coincidence (π) sits in Notation 65. The muon reduced Compton (φ) sits in Notation 66. The proton charge radius is at n = 65.50 — the geometric mean of Notations 65 and 66. Three distinct irrational relations within two doublings of each other may not be coincidental.
Is e absent? The four Hexarchon irrationals are π, φ, e, √2. This survey finds evidence for π, φ, and √2 in the top three results. Euler’s number e appears only in the Bohr radius fractional match (Δ = 0.0008, weaker than the others). Whether e is structurally absent or simply requires a different expression to surface is an open question.