M R5

Prompt

RESEARCH QUESTION (plane topology; countable bridging of a bouquet of continua at a point (the continuum version M4 of Theorem FT, from the BLOCK*(2) beta lane)). Self-contained. Label every claim PROVED (you prove it), SOURCE (exact reference; unchecked?), LAB (from this prompt, status as given), CONJECTURE, OPEN, or RESTATEMENT (a gap reworded). If a definition is ambiguous, state your reading in one line and continue. Say where your argument stops. A. DEFINITIONS Setting (compressed from the lab note block2_opus_1102, DERIVED there, one agent, unaudited; lines marked (INFERRED) are my own additions). This is a static planar question; no dynamics is used. - Lab context (one line, not needed for the proof). BLOCK*(2) is the lab's residual case of the planar periodic-point question: a planar tree-like continuum B with a homeomorphism b that has exactly two periodic points p, q = b(p). A planar construction needs many subcontinua of B through p. Theorem FT (below) is a counting law for arcs at p. The lab asks whether it survives for continua at p, because members of the lab's families through p need not contain arcs. - Bouquet of continua at p. A family K = {K_c : c in C} of nondegenerate continua in the plane R^2 with K_c cap K_c' = {p} for c != c'. For c != c' put Omega(c,c') := {p} cup the union of K_d over d not in {c,c'}. - Bridge. c bridges to c' (c ~> c', symmetric) if for every eps > 0 there are subcontinua A of K_c and A' of K_c' with p in A cap A', 0 < diam A < eps, 0 < diam A' < eps, and continua T_k contained in R^2 minus Omega(c,c') with T_k -> A cup A' in the Hausdorff metric. (T_k may meet K_c cup K_c' away from p, and may cross the plane freely otherwise. A bridge inside a subset Y of the plane is a bridge in R^2, so it suffices to work in R^2; a counterexample should additionally be placed in a planar continuum Y, ideally tree-like, as a secondary demand.) The bridging graph Gamma(K) has vertex set C and an edge {c,c'} when c ~> c'. For arcs, the subcontinua of K_c through p are exactly the initial segments, so this is the definition of the arc case. - Hit sets (INFERRED, notation for the question below). For 0 < r small let D_r be the open disk of radius r about p, S_r its circle. Q_c(r) := the component of K_c cap closure(D_r) that contains p. By boundary bumping, if K_c is not inside D_r then Q_c(r) meets S_r; put H_c(r) := Q_c(r) cap S_r (compact, nonempty); these sets are pairwise disjoint since K_c cap K_c' = {p}. Two indices c, c' are r-ADJACENT if there is an open arc J of S_r with one endpoint in H_c(r), the other in H_c'(r), and J cap (union of all H_d(r)) contained in H_c(r) cup H_c'(r). B. LEMMAS (status; where used) - (K1) Theorem FT (arc case; DERIVED in block2_opus_1102 section 4, one agent, unaudited; here restated in ASCII and shortened). Let beta_c (c in C) be arcs with common endpoint p, beta_c cap beta_c' = {p}, in the plane; the bridging graph is the one above with K_c = beta_c (then A, A' are initial segments). Then every vertex has degree <= 2 and there are countably many edges. Proof. Fix r. Let C_r be the set of c with beta_c not inside D_r; for c in C_r let P_c be the first point of beta_c on S_r and kappa_c the subarc from p to P_c, so kappa_c minus {P_c} lies in D_r and c -> P_c is injective. Step 1: if P_d, P_d' lie in different components of S_r minus {P_c, P_c'}, then lambda := kappa_d cup kappa_d' is a crosscut of D_r; by the theta-curve theorem D_r minus lambda has two components V, V' bounded by lambda and one of the two circle arcs each, and kappa_c minus {p, P_c} is connected, misses lambda and ends on the boundary arc of V, so lies in V, while kappa_c' minus {p, P_c'} lies in V'. Step 2: if c ~> c' and c, c' in C_r, then one open arc of S_r between P_c and P_c' contains no P_x. Else pick d, d' with P_d, P_d' in the two arcs; take eps < r/2, I, I', T_k from the bridge; the far endpoints z of I and z' of I' lie in V and V' respectively (I lies in kappa_c as diam I < r/2), T_k lies in D_r for large k, misses lambda (lambda is in Omega(c,c')), meets small balls around z and z' inside V and V': a connected set in D_r minus lambda = V disjoint-union V' meeting both, a contradiction. Step 3 (degree): three neighbours of c would give three disjoint empty open arcs from P_c, two of which start in the same direction and are nested, putting an endpoint inside an empty arc. Step 4 (countable): the empty arcs of distinct adjacent pairs at radius r are pairwise disjoint (else an endpoint lies inside another arc), so countably many pairs are adjacent at radius r; every edge is adjacent at r = 1/n for large n. - (K2) The degree clause fails for continua (INFERRED, checked by hand here). Let K_0 be a triod: three segments from p at angles 0, 120, 240 degrees, and K_1, K_2, K_3 segments from p at 60, 180, 300 degrees. Then 0 bridges to each of 1, 2, 3: for K_1, take A = the initial segment of the 0-degree arm of K_0 of length eps/2, A' = an initial segment of K_1, and T_k = the polygonal arc from the end of A to a point at distance 1/k from p on the 30-degree ray and on to the end of A'; it misses p, K_2, K_3. The pairs {1,2}, {2,3}, {1,3} are not bridged (an arm of K_0 separates them). So Gamma has a vertex of degree 3 (a star). Hence only a counting clause can be asked for continua. Where the arc proof breaks: it uses that the spoke K_c minus p is connected, so a point z of the bridged part is joined to the far hit point P_c inside K_c minus p, away from the crosscut. For a continuum, K_c minus p can have several components (the three arms of a triod), and z and the hit points can lie in different ones. - (K3) Counting lemma for hit sets (INFERRED proof sketch; I believe it is right). If H is a compact subset of a circle and the H_c are pairwise disjoint nonempty compact sets with union H, then the set of adjacent pairs (adjacent as defined above for these sets) is countable. Sketch: an adjacency arc J with J cap H empty is a gap of H: countably many, and a gap determines its pair. Otherwise J cap H is nonempty and relatively open in H, so it contains a member B_n of a fixed countable base of H, with B_n inside H_c cup H_c'. If B_n meets both, B_n determines the pair. If B_n lies in H_c only, then c is determined by B_n, and c' is the label of the first point of H minus H_c met when leaving B_n in each of the two directions, so at most two pairs per n. So for fixed r only countably many pairs are r-adjacent. - (K4) Not known: that a bridge forces r-adjacency for a sequence of r tending to 0. Arcs: yes (Step 2 of FT). Continua: the natural attempt replaces the crosscut kappa_d cup kappa_d' by Q_d(r) cup Q_d'(r) (two continua meeting only at p, each joining p to the circle; two continua in a closed disk that join interleaved boundary points must meet), and replaces the arc segments by A, A'. It breaks as in (K2): a point of A minus p need not be joined to a hit point of Q_c(r) inside Q_c(r) minus p. - (K5) M3 (the lab's OPEN dynamical conjecture, context only): in a reduced planar beta block with an invariant Cantor fan at p of minimal blade action, uncountably many blades are bridged. M4c plus M3 would exclude such blocks. The planar Denjoy hook dendroid of block2_opus_1102 (section 7 of that report) has minimal blade action and only countably many bridges, so M3 needs indecomposability, not only dynamics. C. THE TARGET **M4c (OPEN).** For every bouquet K of nondegenerate continua at a point p of the plane, the edge set of the bridging graph Gamma(K) is countable. (A vertex with infinitely many edges is allowed, see (K2); the claim is about the total number of edges.) D. ASK Decide M4c. Either (a) prove it, with the step "a bridge forces adjacency" (K4) done honestly for continua; or (b) give an explicit planar bouquet (describe K_c for all c, the point p, and the bridging continua T_k for uncountably many pairs) with uncountably many bridges; or (c) prove it under a named extra hypothesis H on the K_c (for example: p is a non-cut point of Q_c(r) for a sequence of r tending to 0; or every K_c is a dendrite, hereditarily unicoherent, arcwise connected, or locally connected at p) and say whether M4c fails without H; or (d) reduce M4c to one named lemma strictly weaker than M4c and prove the reduction. If you use the counting lemma (K3), prove it in full. State exactly where planarity enters. E. CONTROLS (all must hold) (C1) The arc case: your argument must contain FT as a special case, and each step that uses an arc must be replaced by a step valid for continua. (C2) The triod example (K2): your argument must give a vertex of degree 3 there and still countably many edges. (C3) A planar Cantor fan of segments at p: bridged pairs are exactly the endpoint pairs of the gaps of the angle set. (C4) Nested tangent circles at p with a Cantor set of radii: each K_c has two germs at p; your argument must handle one germ bridging while T_k crosses the other germ. F. DEAD ENDS (not progress) (1) Straightening K_d cup K_d' to a segment by a plane homeomorphism (needs tameness; not available for continua). (2) Counting arcs of the bouquet instead of continua: the K_c are the vertices, and T_k may cross the other germs of c and c'. (3) Equivariance or minimality of a dynamics on the blades: it gives only a union of orbits of bridges, not uncountably many bridges. (4) An earlier Cantor-fan nonplanarity proof was audit-refuted in the lab because it argued from a drawing; use first-hit points on round circles and the theta-curve theorem, not pictures. (5) Moore's triod theorem as a black box: it is blind to hosts whose proper subcontinua are arcs and is not known to give the bridging count. G. FORMAT Line 1 exactly: VERDICT: M4c PROVED | M4c FALSE (explicit planar bouquet with uncountably many bridges) | REDUCED (to a named lemma, with proof of the reduction) | OPEN. Then SUMMARY (<= 100 words), the argument (<= 2,000 words), STILL OPEN EXACTLY. Short of effort: line 1 + summary; an honest OPEN is valid. No finite computation as proof; no theorem beyond its hypotheses; UNKNOWN if unsure a cited result exists. RED FLAGS: straightening the arcs by a plane homeomorphism (not available for continua); assuming Q_c minus p is connected without a reason; reproving the arc case FT and calling that progress; counting adjacent pairs of hit sets without showing that a bridge forces adjacency; claiming the degree clause for continua (it is false, see (K2))