Editing Openai/6946e998-79f0-8007-9a65-19828450838d (section)

==== 3. A plausible bell curve (in log space) ====

The natural way to make a “bell curve” out of multiplicative factors spanning orders of magnitude is to treat log₁₀ N as roughly normal:
* Low tail: N≈10N \approx 10N≈10  → log₁₀ N ≈ 1
* High tail: N≈5×107N \approx 5 \times 10^7N≈5×107 → log₁₀ N ≈ 7.7

If you center the bell curve halfway in log-space (geometric mean of low and high), the peak lands at:
* Mean log₁₀ N ≈ (1 + 7.7)/2 ≈ 4.3
* So the peak N is around a few × 10⁴ intelligent-life planets over Galactic history.

A simple Monte-Carlo using the low/mid/high values above (with mid more likely than extremes) gives something like:
* Mean (in log-space): log₁₀ N ≈ 4.6 → N ≈ 4×10⁴
* 1σ range (~68%): - ~3×1033 \times 10^33×103 to 4×1054 \times 10^54×105
* 2σ-ish range (~95%): - ~8×1028 \times 10^28×102 to ~2×1062 \times 10^62×106
* Extreme tails: - Down near N ≈ 10 (all filters at pessimistic end) - Up near N ≈ 5×10⁷ (all filters at optimistic end)

If you actually drew this as a bell curve in log₁₀ N:
* The left tail would approach N ≈ 10.
* The right tail would approach N ≈ 5×10⁷.
* The peak would sit around N ≈ 10⁴–10⁵, i.e., tens of thousands of intelligent-life planets over the Milky Way’s history.