01810 — THE RESEARCH EXPEDITION
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G3 — Numerical Scale Map
G1 established a precise geometric relation: the smaller tetrahedra have one-half the edge length of the original.
G2 showed that repeating the construction produces a sequence of scales related by powers of two.
G3 asks a different question:
What does that sequence look like when numerical physical scales are placed on it?
Status: M — Mathematical calculation from the scaling rule.
The calculation itself is straightforward. The physical interpretation is not.
From geometry to a numerical sequence
G2 established the outward scaling relation
[L_n=L_0,2^n]
where (L_0) is a chosen reference length and (n) is the number of scale steps.
Each increase of one in (n) doubles the characteristic length.
Each decrease of one halves it.
The sequence therefore has a simple structure:
| Notation | Relative scale |
|---|---|
| 0 | (L_0) |
| 1 | (2L_0) |
| 2 | (4L_0) |
| 3 | (8L_0) |
| 10 | (2^{10}L_0) |
| 20 | (2^{20}L_0) |
| (n) | (2^nL_0) |
Nothing physical has yet been asserted. This is simply what repeated doubling produces.
Choosing a physical reference
The 81018 investigation asks what happens if the lower end of this numerical hierarchy is associated with the Planck length.
Let [L_0=l_P]. Then[L_n=l_P,2^n].
Here an important distinction must be maintained.
The Planck length is a physical quantity from established physics.
Choosing it as the starting point of this particular discrete scale sequence is a modeling assumption.
The mathematics does not require that choice.
What does one step mean?
One scale step multiplies length by exactly two.
After two steps: [L_2=4l_P].
After ten: [L_{10}=1024l_P].
After one hundred: [L_{100}=2^{100}l_P].
The sequence rapidly spans enormous changes in scale.
This is the central numerical feature of the map:
a modest number of doublings can bridge an enormous range of physical length scales.
The scale coordinate
The sequence can also be read in reverse.
Given a physical length (L), its location in the doubling hierarchy is
[n=\log_2\left(\frac{L}{l_P}\right)].
This gives the investigation something more interesting than a list of increasingly large numbers.
It gives a possible scale coordinate.
A physical length can be represented by its position within the hierarchy.
The coordinate need not imply that physical space itself is made from discrete steps.
It may simply be a way of indexing scale.
That distinction is fundamental to the 81018 model.
The proposed cosmological span
The next question is numerical:
How many powers of two separate the Planck length from a cosmological length scale?
Using a characteristic observable-universe radius (R), the corresponding position in the hierarchy would be
[n_R=\log_2\left(\frac{R}{l_P}\right)].
The result is of order two hundred scale steps.
This is the origin of the approximately 202-step architecture that appears elsewhere in the 81018 investigation.
But the wording matters.
The calculation does not establish that nature contains exactly 202 discrete levels.
It establishes that a doubling hierarchy beginning at the Planck scale spans the enormous distance to cosmological scales in roughly two hundred doublings.
The proposed interpretation of those steps as a physical architecture remains a hypothesis.
The 202 question
The number 202 therefore has a precise status.
M: A Planck-to-cosmological ratio expressed as a base-2 logarithm produces a number of order 202, depending on the cosmological length chosen and the numerical conventions used.
A: The Planck length is selected as the lower reference scale.
A: A base-2 hierarchy is treated as a meaningful scale coordinate.
H: The resulting hierarchy may correspond to an underlying physical architecture.
These statements should not be collapsed into one another.
What the map actually shows
The numerical map reveals three things.
1. Exponential scale coverage
Because each step doubles length, the hierarchy covers enormous physical ranges with relatively few steps.
2. A natural logarithmic coordinate
The inverse relation [n=\log_2(L/l_P)] turns physical scale into a dimensionless coordinate.
3. A possible bridge
The same coordinate can be used to place very different physical scales on a common numerical axis.
That is potentially useful even if the deeper physical interpretation eventually fails.
A scale map is not yet a theory
This distinction becomes especially important here.
The numerical fact that a sequence of powers of two spans the range from Planck-scale lengths to cosmological lengths does not by itself demonstrate that the universe is organized according to powers of two.
Nor does it demonstrate that the intermediate notations correspond to distinct physical states.
The map is therefore best regarded as a coordinate scaffold.
The question is whether anything physically meaningful appears when known phenomena are placed upon it.
Q · The next question
If known physical scales are placed on the base-2 hierarchy, do meaningful structures, transitions, invariants, or correspondences appear?
Or is the apparent organization simply a consequence of choosing logarithmic coordinates?
That is the question G3 leaves open.
What G3 does not establish
- That the Planck length is the physical beginning of the hierarchy.
- That nature is fundamentally discrete.
- That physical scales occur only at powers of two.
- That there are exactly 202 physical levels.
- That the 202nd level represents the present universe.
- That intermediate levels correspond to physical epochs.
- That the scale coordinate has dynamical significance.
- That the hierarchy replaces standard cosmological models.
The numerical test
G3 therefore suggests a concrete next step.
Instead of asking whether the hierarchy looks interesting, place independently known physical scales on it.
For each scale (L), calculate [n=\log_2(L/l_P)]. Then ask:
Do physically significant scales cluster near particular values of (n)?
If they do not, that is important.
If they do, the pattern requires a further test to determine whether it is statistically or physically meaningful rather than an artifact of the chosen scales.
The investigation should therefore resist selecting only the examples that fit.
The scale map must be constructed first; interpretation comes afterward.
G3 — Numerical Scale Map
Subject: Powers-of-two scaling from a chosen reference length
Starting point: G2 geometric halving relation
Mathematical result: (L_n=L_0 2^n)
Inverse coordinate: (n=\log_2(L/L_0))
Physical reference: Planck length, used as a modeling choice
Cosmological observation: Planck-to-cosmological scales span roughly two hundred doublings
Epistemic status:
M — numerical consequence of the scaling rule
A — choice of Planck-scale reference
A — choice to use powers of two as a scale coordinate
H — possible physical significance of the resulting hierarchy
NEXT COORDINATE
The next investigation is no longer about constructing the coordinate.
It is about putting physical scales onto it—and asking what, if anything, the map reveals.
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