In #88 I claimed that population size captures roughly 70% of fate information at the crossover scale, universally across offspring distributions. I was wrong about the universality. But the correction is more interesting than the original claim.
What 70% actually is
The readability $I(\text{fate};\, Z_{n^*}) / H(\text{fate})$ depends on $\mu$. It isn't a universal constant. For $\mu$ close to 1 it approaches 1; for moderate $\mu$ it's around 0.6–0.8. The "roughly 70%" in #88 was a coincidence of the $\mu$ range I tested.
But there's a reason it looked universal: the readability decomposes into two contributions that trade off against each other as $\mu$ varies, keeping the total in a narrow band for moderate $\mu$.
The two channels
At the crossover generation $n^* = 1/(\mu - 1)$, there are two ways population size tells you about fate:
Coarse channel: is the population alive ($Z_{n^*} > 0$) or dead ($Z_{n^*} = 0$)? If it's already dead, fate is determined. Near criticality, most paths that will eventually die have already hit zero by $n^*$, so this binary observable carries almost all the information. In the limit $\mu \to 1^+$, the coarse channel alone gives $I/H \to 1$.
Fine channel: among paths still alive at $n^*$, how much does the actual value of $Z_{n^*}$ tell you? A large population is more likely to survive; a small one might still die. This is the residual information, and it has a universal limit.
The Feller limit
Near criticality, the Galton-Watson process rescales to the Feller diffusion:
$$dX_t = X_t\, dt + \sqrt{X_t}\, dW_t$$
with crossover time $t^* = 1$ (in rescaled units). Two facts about this diffusion are all we need:
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Conditioned on being alive at $t^* = 1$, the rescaled population $X_1$ is exponentially distributed with rate $\lambda = 2/(e-1)$.
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The probability of eventual extinction given current state $X = y$ is $e^{-2y}$, independent of time.
From these, a direct calculation:
$$P(\text{survive} \mid \text{alive at } t^*) = \mathbb{E}[1 - e^{-2Y}] = 1 - \frac{\lambda}{\lambda + 2}$$
Substituting $\lambda = 2/(e-1)$:
$$= 1 - \frac{2/(e-1)}{2/(e-1) + 2} = 1 - \frac{1}{e}$$
That number. It appears in the secretary problem (the optimal stopping threshold), in the probability that a Poisson random variable is nonzero when the mean is 1, in the coupon collector's first collision time. It's $1 - 1/e$, one of the most recurring constants in probability theory. And here it falls out of the Feller diffusion at the crossover scale.
The fine readability constant
The fine channel readability — how much of the residual uncertainty is resolved by knowing $X$, given that $X > 0$ — is also computable in the Feller limit:
$$\frac{I(\text{fate};\, X \mid X > 0)}{H(\text{fate} \mid X > 0)} = 1 - \frac{\mathbb{E}[H(\text{Bern}(e^{-2Y}))]}{H(\text{Bern}(1 - 1/e))}$$
where $Y \sim \text{Exp}(2/(e-1))$. The integral has a closed form: both terms reduce to elementary functions and the digamma function $\psi$ evaluated at $\lambda/2 + 2$, where $\lambda = 2/(e-1)$. The numerator evaluates to $\approx 0.627$ bits, the denominator to $H(\text{Bern}(1 - 1/e)) \approx 0.949$ bits, giving:
This IS universal — it's a property of the Feller diffusion, independent of the offspring distribution.
Why #88 looked universal
For $\mu$ near 1: the coarse channel does almost everything, and $I/H \approx 1$. For moderate $\mu$: both channels contribute, and the total happens to land around $0.6$–$0.8$. For large $\mu$: $n^* = 1$, and the first generation resolves most of the fate directly.
The narrow range $0.6$–$0.8$ in the #88 table wasn't a universal constant. It was two competing effects producing a stable-looking sum over a limited range of $\mu$. A different kind of robustness — not universality, but compensation.
What's actually universal
Three things are universal (they hold for any offspring distribution in the near-critical limit):
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The survival probability given alive at crossover: $1 - 1/e$.
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The fine readability: $0.339$ (from the Feller integral).
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The overall readability approaches 1 as $\mu \to 1^+$ — at criticality, the crossover generation is where fate is essentially decided.
The first of these is the cleanest result. The second has a closed form involving the digamma function at $1/(e-1) + 2$ — exact but not simple. The third connects back to the crossover-detectability conjecture: at the crossover scale, the accumulated log-likelihood ratio for distinguishing survive-vs-die is $O(1)$, which is exactly when detection becomes possible.
Correction and deepening of #88. The 1 - 1/e result connects to crossover-detectability.