collatz real example #1 – the log-drift fingerprint

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Here is a real example of #1: The Log-Drift Fingerprint, using the orbit that starts at n = 7.

R(n)=\frac{b(n)\log 2-a(n)\log 3}{\log n}

Collatz orbit for n = 7

7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

Now count the two competing forces:

Odd steps, where we apply 3n + 1:

7 → 22
11 → 34
17 → 52
13 → 40
5 → 16

So:

a(n) = 5 odd steps

Even steps, where we apply n / 2:

22 → 11
34 → 17
52 → 26
26 → 13
40 → 20
20 → 10
10 → 5
16 → 8
8 → 4
4 → 2
2 → 1

So:

b(n) = 11 halving steps

Now compute the log-drift:

11 log(2) − 5 log(3)

Using natural logs:

11 × 0.6931 − 5 × 1.0986

= 7.6241 − 5.4930

= 2.1311

Now divide by log(7):

R(7) = 2.1311 / 1.9459

R(7) ≈ 1.095

Interpretation

For n = 7, the log-drift ratio is:

R(7) ≈ 1.095

That means the orbit had slightly more downward halving pressure than was minimally needed to erase the upward pressure from the odd 3n + 1 steps.

In plain English:

The number 7 rises to 52, but the orbit eventually “overpays” its entropy debt with enough divisions by 2 to collapse to 1.

A ratio near 1 means the orbit descends in a fairly expected way. A ratio much above 1 means it falls efficiently. A ratio below 1, especially for a long stretch, would signal a stubborn orbit that resists collapse and may climb high before finally falling.

For n = 7, the Collatz path is:

7 rises up to 52, then falls down to 1.

The “log-drift fingerprint” is just a way of asking:

Did the downward steps beat the upward steps strongly enough?

In the Collatz rule, two things happen:

When the number is odd, it gets pushed upward:

7 becomes 22
That is the “up force.”

When the number is even, it gets cut in half:

22 becomes 11
That is the “down force.”

For the number 7, there are:

5 upward odd moves
and
11 downward halving moves

So even though 7 climbs as high as 52, it gets cut in half often enough afterward that the downward pressure wins.

The final score is about:

1.095

In plain English, that means:

7 had a little more falling power than rising power.

Not a huge amount. Just enough.

So the story of 7 is:

It climbs, peaks at 52, then the repeated halvings overpower the climbs and pull it down to 1.

That is what #1 is measuring:
the tug-of-war between climb and collapse.