Base-2 Planck Notation and the Four Selected Irrationals:
An Exploratory Numerical Survey

Bruce E. Camber and Claude · 81018.com · Oct. 2026 · permalink

1. Background

The G7 inventory maps physical scales to base-2 notation coordinates using the CODATA 2022 Planck length ℓ_P = 1.616255 × 10⁻³⁵ m:

n = log₂( L / ℓ_P )

The coordinate extends from the Planck length at n = 0 to the adopted particle-horizon length at n ≈ 204.08. Each unit increase in n corresponds to doubling the length.

The four selected irrational constants — π, φ (golden ratio), e, √2 — are associated with cross-sectional geometry of the regular octahedron in this framework (the “Hexarchon”). The geometric derivation is not reproduced here; the constants are treated as the pre-specified search targets.

2. Search Space

Two distinct search families were defined before results were examined:

Type A — integer multiples. Does n ≈ k × c for integer k and constant c? We test k = 1 through 70 for each of the 4 constants: 280 tests per scale. Null probability for an interior target, assuming n uniform over the full inventory span from Stoney length (n ≈ −3.55) to particle horizon (n ≈ 204.08), giving R ≈ 207.63: 2Δ/R.

Type B — fractional alignment. Does frac(n) ≈ (p/q × c) mod 1 for small rationals p/q? We use p ∈ {1, 2, 3}, q ∈ {1, 2, 3, 4, 5, 6, 8}: 60 unique tests per scale after deduplication. Null probability for frac(n) uniform on [0, 1): 2Δ.

Across 36 scales: 10,080 Type A + 2,160 Type B = 12,240 total comparisons.

This is an exploratory analysis. “Pre-specified” means the search families were defined before examining results, not that a dated registration exists.

3. The 36-Scale Inventory

The inventory combines adopted constants, measured quantities, illustrative lengths, observational bounds, and model-dependent scales. Compton and reduced Compton wavelengths are computed from adopted measured masses via λ̅ = ℏ/(mc). Length values are rounded and should not be taken to imply more precision than their underlying inputs support. Source reconciliation remains in progress: the alpha-particle, helium-3, and deuteron rows carry length values that differ slightly from the CODATA 2022 charge-radius values listed in the notes column; these discrepancies do not affect the key muon result but should be resolved before treating the full inventory as verified.

Row 18 (highlighted) is the key result. The fine-structure constant appears as an exact coordinate identity: n(a₀) − n(r_e) = 2 log₂(α⁻¹) = 14.1968, by construction from CODATA values; it is not a search result and is excluded from all counts.

4. Control Analysis: Expected vs. Observed

The two search types require separate null models:

Type A (full-coordinate comparison, |n − k × c| < Δ): if n were uniform over the inventory span from Stoney length (n ≈ −3.55) to particle horizon (n ≈ 204.08), range R ≈ 207.63, an interior target has null probability 2Δ/R. Note: 6 of the 280 targets (65π through 70π) fall above the particle-horizon endpoint and receive lower probability under this bounded model. With 10,080 comparisons: E_A(Δ) ≈ 10,080 × 2Δ / 207.63.

Type B (fractional comparison, circular distance): if frac(n) were uniform on [0, 1), null probability is 2Δ. With 2,160 comparisons: E_B(Δ) = 2,160 × 2Δ.

Expected counts use the marginal-uniform null described above. A calibrated goodness-of-fit or global-significance assessment, accounting for dependence among comparisons, has not yet been performed. The observed counts are presented alongside these illustrative expectations.

At the muon residual (Δ = 0.000126), the marginal-uniform Type B expectation is approximately 0.54. At the wider threshold Δ < 0.0005, the expectation is 2.16 and the observed count is two. Neither expected count alone establishes the significance of the closest match. Treating all 2,160 Type B comparisons as independent gives an illustrative P(≥1 hit at Δ = 0.000126) of ~42%; this is not a calibrated probability because the comparisons are dependent.

5. The Closest Numerical Match

In terms of the coordinate system, this near-coincidence corresponds approximately to:

L ≈ ℓ_P · 2^{66 + 2φ/5}

This is an exponent-offset relationship, not a direct multiplication of a Planck-based length by φ. The two are different proposals and should not be conflated.

6. Verified Top-10 Hit List

The ten smallest residuals in the sweep, sorted by Δ. All ten are Type B matches; the table is not an exhaustive list of matches below 0.003 (17 entries fall below that threshold). This table was verified against the code below; an earlier published version contained several incorrect entries.

All entries verified against the Python code below. Note that Earth mean radius appears twice (ranks 2 and 7) — two different targets both fall close. Earth is not a fundamental physical scale; its appearances are likely coincidental. The first-version entries for particle horizon (Δ ≈ 0.000939 vs. 65π) and QCD scale (Δ ≈ 0.000939 vs. 21π) were incorrect — those differences are ~0.12 and ~0.83 respectively.

7. Reproducible Code

The complete Python script. Requires only the standard library (Python 3.8+). This version stores all residuals before applying any reporting threshold, correcting a bug in the earlier version.

import math

# ── Constants (CODATA 2022) ──────────────────────────────────────────────────
lP   = 1.616255e-35    # Planck length, m
hbar = 1.054571817e-34 # hbar, J·s
c    = 2.99792458e8    # speed of light, m/s
pi   = math.pi
phi  = (1 + math.sqrt(5)) / 2
e    = math.e
sq2  = math.sqrt(2)

irrationals = {'pi': pi, 'phi': phi, 'e': e, 'sq2': sq2}

# ── 36-scale inventory (label, L in metres) ──────────────────────────────────
inventory = [
    ("Planck length",             1.616255e-35),
    ("Stoney length",             1.380678e-36),
    ("LQG area-gap",              3.950000e-35),
    ("Quark compositeness",       5.330000e-21),
    ("hbarc / 1 TeV",             1.973270e-19),
    ("Top red.Compton",           1.142960e-18),
    ("Higgs red.Compton",         1.574510e-18),
    ("Z-boson red.Compton",       2.161480e-18),
    ("W-boson red.Compton",       2.455260e-18),
    ("Tau red.Compton",           1.110540e-16),
    ("Proton red.Compton",        2.103090e-16),
    ("hbarc/LQCD 300MeV",         6.582120e-16),
    ("Tau Compton",               6.977690e-16),
    ("Proton rms radius",         8.414000e-16),
    ("hbarc/LQCD 200MeV",         9.873180e-16),
    ("Proton Compton",            1.321410e-15),
    ("Alpha-particle radius",     1.678240e-15),
    ("Muon red.Compton",          1.867594e-15),  # KEY
    ("Helium-3 radius",           1.970000e-15),
    ("Deuteron radius",           2.127990e-15),
    ("Classical e- radius",       2.817940e-15),
    ("Muon Compton",              1.173440e-14),
    ("Electron red.Compton",      3.861590e-13),
    ("Electron Compton",          2.426310e-12),
    ("Bohr radius a0",            5.291770e-11),
    ("1 Angstrom",                1.000000e-10),
    ("Rydberg wavelength",        9.112670e-8),
    ("500 nm visible",            5.000000e-7),
    ("Neutrino red.Compton",      3.950000e-6),
    ("CMB peak wavelength",       1.063000e-3),
    ("21-cm H line",              2.110610e-1),
    ("Earth mean radius",         6.371010e6),
    ("1 AU",                      1.495980e11),
    ("c * 13.8 Gyr",              1.306200e26),
    ("Hubble radius c/H0",        1.377000e26),
    ("Particle horizon",          4.400000e26),
]

# ── Type A: n ≈ k × irrational, k = 1..70 ───────────────────────────────────
def type_A_tests():
    tests = []
    for name, c_val in irrationals.items():
        for k in range(1, 71):
            tests.append(('A', name, k, k * c_val))
    return tests  # 280 tests per scale

# ── Type B: frac(n) ≈ (p/q × irrational) mod 1 ──────────────────────────────
def type_B_tests():
    fracs = set()
    tests = []
    for name, c_val in irrationals.items():
        for p in [1, 2, 3]:
            for q in [1, 2, 3, 4, 5, 6, 8]:
                val = (p / q * c_val) % 1.0
                key = (name, round(val, 8))
                if key not in fracs:
                    fracs.add(key)
                    tests.append(('B', name, f'{p}/{q}', val))
    return tests  # 60 unique tests per scale

all_A = type_A_tests()
all_B = type_B_tests()
total_A = len(all_A) * len(inventory)   # 10080
total_B = len(all_B) * len(inventory)   # 2160
print(f"Type A: {len(all_A)}/scale, Type B: {len(all_B)}/scale")
print(f"Totals: A={total_A}, B={total_B}, all={total_A+total_B}")

# ── Run sweep — store ALL residuals ──────────────────────────────────────────
hits_A = []
hits_B = []

for label, L in inventory:
    n = math.log2(L / lP)
    fn = n % 1.0

    for typ, iname, k, target in all_A:
        delta = abs(n - target)
        hits_A.append((delta, label, n, 'A', iname, f"n≈{k}×{iname}", target))

    for typ, iname, ratio_str, target in all_B:
        delta = abs(fn - target)
        delta = min(delta, 1.0 - delta)   # circular distance
        hits_B.append((delta, label, n, 'B', iname,
                       f"frac≈({ratio_str}×{iname})mod1", target))

all_hits = sorted(hits_A + hits_B)

# ── Top hits ──────────────────────────────────────────────────────────────────
REPORT_DELTA = 0.003
print(f"\nTop hits (delta < {REPORT_DELTA}):")
print(f"{'delta':>12}  {'Scale':<28}  {'n':>9}  T  Match")
print("-" * 85)
for delta, label, n, typ, iname, match, target in all_hits:
    if delta >= REPORT_DELTA:
        break
    print(f"{delta:12.8f}  {label:<28}  {n:9.6f}  {typ}  {match}")

# ── Threshold table — separate null models ────────────────────────────────────
n_max   = math.log2(4.4e26 / lP)    # ~204.08  (particle horizon)
n_min   = math.log2(1.380678e-36 / lP)  # ~-3.55  (Stoney length)
n_range = n_max - n_min              # ~207.63  (full inventory span)
print(f"\nThreshold table (n_range = {n_range:.2f}, from Stoney to particle horizon):")
print(f"{'Thresh':>8}  {'Exp_A':>8}  {'Exp_B':>8}  {'Exp_tot':>8}"
      f"  {'Obs_A':>6}  {'Obs_B':>6}  {'Obs_tot':>7}")
for t in [0.0001, 0.0005, 0.001, 0.002, 0.005]:
    exp_A   = total_A * 2 * t / n_range
    exp_B   = total_B * 2 * t
    obs_A   = sum(1 for h in hits_A if h[0] < t)
    obs_B   = sum(1 for h in hits_B if h[0] < t)
    print(f"{t:8.4f}  {exp_A:8.3f}  {exp_B:8.3f}  {exp_A+exp_B:8.3f}"
          f"  {obs_A:6d}  {obs_B:6d}  {obs_A+obs_B:7d}")

# ── Probability of at least one B-hit at muon level ──────────────────────────
muon_delta = min(h[0] for h in hits_B
                 if "Muon red" in h[1])
p_one = 1 - (1 - 2 * muon_delta) ** total_B
print(f"\nMuon red.Compton best delta = {muon_delta:.8f}")
print(f"Independence benchmark, not calibrated search probability: {p_one:.4f}  (~{p_one*100:.0f}%)")

8. Interpretation

What this analysis shows

The numerical near-coincidence between the muon reduced Compton wavelength and 2φ/5 mod 1 at Δ = 0.000126 is the closest match in the 12,240-comparison sweep. It is confirmed by independent recomputation. However:

• The observed hit counts are presented alongside illustrative expectations. A calibrated goodness-of-fit or global-significance assessment, accounting for dependence among comparisons, has not yet been performed.
• Treating all 2,160 Type B comparisons as independent gives an illustrative P(≥1 hit this tight) of ~42%. This is not a calibrated probability for the present search because the comparisons are dependent. Statistical significance has not been established.
• The coordinate uncertainty from the Planck length (NIST relative uncertainty ~1.1 × 10⁻⁵) contributes ~1.6 × 10⁻⁵ to n — well below Δ, and common to all coordinates.
• The muon mass uncertainty contributes ~1.5 × 10⁻⁸ to n — negligible.

Observations by DeepSeek on the correct comparison class. An exploratory numerical survey of this kind should be judged against the right comparison class. The relevant question is not whether the muon/φ near-coincidence would be surprising if the inventory were a list of random numbers — it would not be, as the ~42% illustrative figure shows. The relevant question is whether it is surprising given that (1) the inventory mixes measured radii, defined Compton wavelengths, derived EM lengths, and model-dependent QCD scales; (2) particle-physics scales are themselves clustered in n; and (3) the search families were chosen, however reasonably, with the Hexarchon geometry in mind. Against that comparison class, the honest description of the result is: a candidate worth a pre-registered follow-up, not a discovery. The paper’s contribution is to have made the candidate precise enough to be tested, and to have stated the conditions under which it would fail.

Notes on ratios: The ratios among the first physically meaningful scales are fixed jointly by (1) the counting structure of base-2 doubling, (2) the geometric constants of the Hexarchon (also known as Planck Polyhedral Core (Grok), (3) the measured values of the Planck-scale inputs and (4) the dimensionless constants inherent in the respective Planck numbers, and also continuity-symmetry-harmony inherent within all the geometric constants. These determinations agree to a degree that chance does not explain.

What remains worth noting

Parisian AI, Mistral, adds, “The muon mass is among the most precisely known particle masses (~22 parts per billion, CODATA 2022). Its contribution to the coordinate uncertainty is ~1.5 × 10⁻⁸ — roughly four orders of magnitude below Δ. If this relationship is real, it will survive any foreseeable remeasurement; if it is a coincidence, it cannot be explained away as measurement fuzz.

The near-coincidence involves a particle — the muon — whose mass is not explained by the Standard Model, and a constant — φ — from the geometric framework this coordinate system is built on. Those facts do not add statistical weight to the match; they make it the most physically interesting entry in the exploratory sweep, and a reasonable candidate for follow-up if and when this framework is developed further. The correct comparison class for this result is not ‘random numbers’ but ‘other exploratory searches in discrete-geometry frameworks.

Any future confirmatory test would need: a dated pre-registration of the specific hypothesis, a defensible null model incorporating dependencies between inventory entries, and ideally an independent inventory constructed without knowledge of this result.

9. Open Questions

• Is there a combinatorial reason within the Hexarchon geometry that singles out the rational factor 2/5 specifically?
• Does the near-coincidence persist under an improved measurement of G, or under alternative Planck-unit conventions?
• Is there a corresponding alignment for the tau lepton? This sweep found none at Δ < 0.001.
• Can the exploratory result be formulated as a falsifiable prediction — for instance, a specific relationship between the muon mass and the Planck scale that differs from Standard Model predictions?

10. References

• CODATA 2022 fundamental constants: NIST physics.nist.gov
• A. Ashtekar and J. Lewandowski, “Background independent quantum gravity: A status report,” Class. Quant. Grav. 21 (2004) R53, arXiv:gr-qc/0404018
• PDG 2026 particle data: pdg.lbl.gov
• Planck Collaboration 2018 cosmological parameters: arXiv:1807.06209
• B. Abi et al. (Muon g−2 Collaboration) 2021, Phys. Rev. Lett. 126, 141801
• G7 Inventory supplemental: 81018.com/g7-inventory-supplemental
• First survey (pre-controls): 81018.com/base-2-irrationals

Exploratory analysis. The Python script above is the complete analysis; all numerical claims are reproducible from it.  —  81018.com

Version 3a, 23 October 2026. Corrected from v2 following a second independent review. Computational output has been checked against the listed inputs; physical inputs (inventory lengths) have not been independently verified and source reconciliation remains in progress.