Last Update: 22 September 2026
The Expedition: G7 · G6 · G5 · G4 · G3 · G2 · G1 · G1 Analysis
This inventory is a living supplement to G7.
It records independently established physical scales and their corresponding 81018 coordinates. Entries are classified according to what the underlying physics actually establishes. The inventory is not constructed to validate notation 67; it is constructed to test whether anything distinctive occurs there.
The inventory to date
Status: T / Q — test under construction, not a claim that 67 is physical.
Rule used: physics populates the map. The map does not manufacture the physics.
Coordinate: (n=\log_2(L/\ell_P)) with CODATA 2022 (\ell_P=1.616255\times10^{-35},\mathrm{m}).
Choosing (\ell_P) as (n=0) remains an A — modeling choice.
This sheet does three things G7 asked for:
- Correct one mis-labeled length on the live G7 table.
- Widen the inventory with independently established lengths, each classified.
- Run the five G5 controls far enough to see whether 67 still looks special.
It does not add the Aristotle gap, Quiet Expansion, or the sphere-rate → (H_0) conversion.
1. Corrections to the G7 table within the G7 article
| G7 label | Length used on G7 | What that length actually is | Correct reduced Compton | Correct (n) |
|---|---|---|---|---|
| Neutron reduced Compton wavelength | (1.32\times10^{-15},\mathrm{m}), (n\approx66.2) | Unreduced Compton wavelength of the neutron/proton | (\bar\lambda_n=2.100194\times10^{-16},\mathrm{m}) | (n\approx63.50) |
| Muon reduced Compton | (1.87\times10^{-15},\mathrm{m}), (n\approx66.6) | Correct | same | (n\approx66.65) |
Proton unreduced Compton is (\lambda_{C,p}=1.32141\times10^{-15},\mathrm{m}) ((n\approx66.15)).
Proton reduced Compton is (\bar\lambda_p=2.10309\times10^{-16},\mathrm{m}) ((n\approx63.50)).
The live cluster at 65.5–67.2 is therefore a mix of:
- one measured nuclear charge radius,
- unreduced nucleon Compton lengths,
- muon reduced Compton,
- classical electron radius.
Those are four different kinds of quantity. They may share a neighborhood. They do not share a meaning.
2. Working inventory
Lengths from CODATA 2022 / NIST and standard conversions. QCD and compositeness rows are inferred or constrained, not measured radii.
| Physical entry | (L) (m) | (n=\log_2(L/\ell_P)) | Classification |
|---|---|---|---|
| Planck length (\ell_P) | (1.616255\times10^{-35}) | 0.000 | Reference / model anchor (A) |
| Stoney length (\ell_S=\ell_P\sqrt{\alpha}) | (1.381\times10^{-36}) | −3.55 | Defined EM–gravity scale (control reference) |
| (\hbar c/37,\mathrm{TeV}) | (5.33\times10^{-21}) | 48.23 | Experimental constraint (energy → length) |
| Quoted compositeness ~(10^{-20},\mathrm{m}) | (1.00\times10^{-20}) | 49.14 | Experimental constraint (quoted on G7) |
| (\hbar c/1,\mathrm{TeV}) | (1.97\times10^{-19}) | 53.44 | Defined conversion of 1 TeV |
| Tau reduced Compton (\bar\lambda_\tau) | (1.111\times10^{-16}) | 62.58 | Defined quantum scale |
| Proton / neutron reduced Compton | (2.103\times10^{-16}) / (2.100\times10^{-16}) | 63.50 | Defined quantum scale |
| (\hbar c/\Lambda_\mathrm{QCD}) at 300 MeV | (6.58\times10^{-16}) | 65.14 | Inferred QCD scale |
| Tau Compton (\lambda_{C,\tau}) | (6.978\times10^{-16}) | 65.23 | Defined quantum scale |
| Proton rms charge radius (r_p\approx0.841,\mathrm{fm}) | (8.41\times10^{-16}) | 65.50 | Measured characteristic scale |
| (\hbar c/\Lambda_\mathrm{QCD}) at 200 MeV | (9.87\times10^{-16}) | 65.73 | Inferred QCD scale |
| Proton / neutron Compton (\lambda_C) | (1.321\times10^{-15}) / (1.320\times10^{-15}) | 66.15 | Defined quantum scale |
| Alpha-particle charge radius | (1.678\times10^{-15}) | 66.49 | Measured characteristic scale |
| Muon reduced Compton (\bar\lambda_\mu) | (1.868\times10^{-15}) | 66.65 | Defined quantum scale |
| Deuteron rms charge radius | (2.128\times10^{-15}) | 66.84 | Measured characteristic scale |
| Classical electron radius (r_e) | (2.818\times10^{-15}) | 67.24 | Derived EM characteristic scale |
| Muon Compton (\lambda_{C,\mu}) | (1.173\times10^{-14}) | 69.30 | Defined quantum scale |
| Electron reduced Compton (\bar\lambda_e) | (3.862\times10^{-13}) | 74.34 | Defined quantum scale |
| Electron Compton (\lambda_{C,e}) | (2.426\times10^{-12}) | 76.99 | Defined quantum scale |
| Bohr radius (a_0) | (5.292\times10^{-11}) | 81.44 | Atomic characteristic scale |
| 1 Å / hydrogen (2a_0) | (1.00\times10^{-10}) / (1.06\times10^{-10}) | 82.36 / 82.44 | Atomic / molecular scale |
| Rydberg wavelength (1/R_\infty) | (9.113\times10^{-8}) | 92.19 | Atomic spectral scale |
| 500 nm | (5.00\times10^{-7}) | 94.64 | Optical control |
| Earth radius | (6.371\times10^{6}) | 138.18 | Macro control |
| AU | (1.496\times10^{11}) | 152.70 | Astro control |
| (c\times13.8,\mathrm{Gyr}) | (1.306\times10^{26}) | 202.33 | Cosmo control (age-distance) |
| Particle horizon ~46.5 Gly | (4.4\times10^{26}) | 204.08 | Cosmo control (comoving) |
The table is still incomplete. It is large enough to test 67.
3. What the first widening already shows
There is a real concentration of hadronic and muon-scale lengths between about (n=65.1) and (n=67.2). That is not imaginary.
It is also not a single physical object. In that two-step window sit:
- QCD’s inferred length,
- the proton’s measured charge radius,
- nucleon Compton (not reduced) wavelengths,
- light-nucleus charge radii,
- the muon’s reduced Compton wavelength,
- the classical electron radius (r_e=\alpha,\bar\lambda_e=a_0\alpha^2).
Those last two are electromagnetic definitions, not nuclear measurements. (r_e) lands at 67.24 because (\alpha) and the electron mass put it there, not because an experiment resolved a 2.82 fm electron.
Below that window, reduced nucleon Comptons sit at 63.5 and compositeness limits sit near 48–49. Above it, the electron’s own quantum lengths sit at 74.3 and 77.0, and the atom at 81.4.
G7’s outward control survives: the coordinate does not dump all familiar physics onto 67.
4. The five G5 controls
Control 1 — neighbors 65, 66, 67, 68, 69
Rounded occupancy from this inventory:
| Integer (n) | Occupants in this sheet |
|---|---|
| 63 | proton/neutron reduced Compton |
| 65 | (\Lambda_\mathrm{QCD}), tau Compton, proton charge radius |
| 66 | nucleon Compton, alpha radius, (\bar\lambda_\mu), deuteron radius |
| 67 | classical electron radius only |
| 69 | muon Compton |
| 74 | (\bar\lambda_e) |
| 77 | (\lambda_{C,e}) |
| 81 | Bohr radius |
67 is not an isolated spike. 65–66 is the denser bin, and it is the nuclear/QCD bin. 67 is the electromagnetic derived scale (r_e). If the question is “where does experimentally grounded spatial size of composites appear?”, the answer in this inventory is closer to 65.5 ((r_p)) than to 67.
Control 2 — wider band 60–74, plus atomic control
60–74 contains at least three different stories:
- lepton/quark mass → Compton lengths (τ, p, n, μ, then e much later),
- measured nuclear charge radii (p, α, d),
- one derived EM radius ((r_e)).
The atomic control at 81.4 is cleanly separated. A “boundary at 67” has to explain why 65.5 and 66.6 are in the same breath as 67.2, and why 74.3 (the electron’s actual quantum length) is not.
Control 3 — change the reference (L_0)
Same lengths, different zero point:
| Entry | (n) if (L_0=\ell_P) | (L_0=\ell_S) | (L_0=r_p) | (L_0=a_0) |
|---|---|---|---|---|
| (r_p) | 65.50 | 69.05 | 0.00 | −15.94 |
| (\bar\lambda_\mu) | 66.65 | 70.20 | 1.15 | −14.79 |
| (r_e) | 67.24 | 70.79 | 1.75 | −14.20 |
| (\lambda_{C,p}) | 66.15 | 69.70 | 0.65 | −15.29 |
| (a_0) | 81.44 | 84.99 | 15.94 | 0.00 |
The integer “67” is an artifact of (\ell_P) plus base 2.
The relative gaps survive: (r_e/r_p\approx3.35) is 1.75 doublings; (a_0/r_e=\alpha^{-2}) is 14.20 doublings. Those ratios are physics. The label 67 is bookkeeping.
Control 4 — change the base
With (L_0=\ell_P):
| Entry | (\log_2) | (\ln) | (\log_3) | (\log_{10}) |
|---|---|---|---|---|
| (r_p) | 65.50 | 45.40 | 41.32 | 19.72 |
| (\bar\lambda_\mu) | 66.65 | 46.20 | 42.05 | 20.06 |
| (r_e) | 67.24 | 46.61 | 42.42 | 20.24 |
| (a_0) | 81.44 | 56.45 | 51.38 | 24.52 |
The cluster remains a cluster under any log. It does not remain “the number 67.”
If the claim is “base-2 notation 67 is privileged,” this control fails unless an independent reason is given for base 2 and (\ell_P). G1 supplies a geometric reason to try base 2. It does not yet supply a reason that nature uses it.
Control 5 — blind reconstruction (first pass)
Without being told to look at 67, the inventory suggests these transitions, in order of defensibility:
- Compositeness / resolution wall, (n\sim48)–(53): experimental limits, not sizes.
- Onset of measured finite size for QCD bound states, (n\sim65.5): proton charge radius. This is the strongest measured entry.
- A pack of hadronic and muon-scale characteristic lengths, (n\sim65)–(67): mixed classes.
- Electron quantum lengths, (n\sim74)–(77): defined, not measured radii.
- Atomic structure, (n\sim81.4): Bohr radius.
A blind reader would not circle 67 first. They would circle 65.5 as the first measured composite size, and 81.4 as the first atomic size. 67 is where (r_e) happens to fall.
5. Relations that are real — and do not need 67
These are identities, not 81018 results:
[ r_e=\alpha,\bar\lambda_e=a_0\alpha^2 ]
[ n(a_0)-n(r_e)=2\log_2(1/\alpha)\approx14.197 ]
[ n(\lambda_{C,p})-n(\lambda_{C,e})=-\log_2(m_p/m_e)\approx-10.842 ]
A logarithmic map will always make (\alpha) and mass ratios look like integer-ish steps. That is what logarithms do. It is not evidence that those integers are physical floors.
6. Present verdict (null still standing)
G5’s null hypothesis:
Nothing fundamental happens at notation 67. The apparent transition is produced by the chosen reference, base-2 indexing, experimental resolution, and the historical list of named particle-physics lengths.
On this first widened inventory the null is not eliminated.
What survives as an observation about the map:
- Independently named nuclear and muon lengths concentrate between (n\approx65) and (n\approx67) when (\ell_P) and base 2 are chosen.
- The first measured spatial size in the list is the proton charge radius at (n\approx65.50), not 67.
- The integer 67 is not stable under change of (L_0) or base.
- Several occupants of the 67 neighborhood are defined or derived scales, not measured radii.
- The live G7 table inflated the neighborhood by listing the nucleon unreduced Compton wavelength as a reduced Compton wavelength.
What would still be interesting, and is not yet tested:
- A change in the character of evidence (measured size vs pointlike constraint) near the first nuclear radius.
- Whether that character-change remains after a much larger, pre-registered list of PDG/CODATA lengths, including many that do not sit near 67.
That is the next increment of G7: more rows, same rules, still trying to make 67 disappear.
7. Proposed next rows (we will not invent sizes)
Add only if a number already exists:
- deuteron / helium-3 / triton charge radii (measured),
- nuclear rms radii for a few closed-shell nuclei (measured, model-dependent extraction),
- (\hbar c/M_W), (\hbar c/M_Z), (\hbar c/M_H) as defined conversions, labeled as such,
- electron and muon g-2 are not lengths; do not force them onto the map,
- no preons, inflatons, dark-matter candidates, or Planck spheres.
G8 should not start until Control 5 is repeated on a list an independent reader could have assembled from NIST + PDG without seeing the number 67.
G7 is not presenting a finished physical map.
It is opening a working inventory page.
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