m12

Prompt

RESEARCH QUESTION (plane continuum theory / Cantor fans in planar tree-like indecomposable continua; the FAR-MIX node of NOT-BLOCK*(2) case beta (route (ii), not-EXIST-F_pl)). 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 kfan_planar_opus_1102 and the notes block2_opus_1102 and fp2f_opus_1102; every DERIVED item is a single-agent claim, UNAUDITED here; lines marked (INFERRED) are my bridging). - Y is a compact metric space (a continuum). A Cantor fan Phi in Y with vertex p is a subset homeomorphic to (C x [0,1])/(C x {0}) for a Cantor set C; the blades beta_c (c in C) are arcs with beta_c cap beta_c' = {p} for c != c'. f: Y -> Y is a homeomorphism with f(p) = p, f(Phi) = Phi, f(beta_c) = beta_{sigma c}, sigma: C -> C a homeomorphism (the blade action). EXIST-F asks for Y tree-like and indecomposable, with sigma minimal (hence aperiodic). EXIST-F_pl is EXIST-F with Y planar (embeds in the plane). The point of the node: not-EXIST-F_pl implies FP-2F implies NO-FAN (fp2f_opus_1102 Prop E, DERIVED), a branch of NOT-BLOCK*(2), which the lab needs for the periodic point theorem PER. - Vertex transfer (fixed heights). (a, b) in T iff a != b and there are x_a in beta_a \ {p}, x_b in beta_b \ {p} such that for every eta > 0 and all blades d, d' not in {a, b} there is a continuum Z in Y \ (beta_d u beta_d') with Z inside the eta-neighbourhood of beta_a u beta_b meeting both balls B(x_a, eta), B(x_b, eta). Bridges (block2_opus_1102): c ~> c' iff for every eps there are initial segments of beta_c, beta_c' of diameter < eps and continua T_k in Y \ ({p} u other blades) converging in the Hausdorff metric to their union. Both notions are f-equivariant and intrinsic. - Planar circular order O on C: for four distinct blades, {d, d'} separates {a, b} if after straightening the 4-od (Schoenflies, RECALL) the directions interleave. O is a circular order on C; a pair (a, b) is O-adjacent if one of the two open O-intervals between a and b is empty. B. LEMMAS (status; where used) Known (kfan_planar_opus_1102, DERIVED, one agent, UNAUDITED; the same agent family proved Corollary K and Lemma FL independently in block2_opus_1102, which is NOT an independent audit). - Theorem 1. If Y is planar, every (a, b) in T and every bridge is O-adjacent. - Lemma ST and Theorem M. If Y is planar, the order topology of O equals the topology of C, every open O-interval is open in C, each blade has at most ONE O-adjacent partner, and the adjacent pairs form a countable partial matching (the gaps of a Denjoy-type Cantor subset of S^1). Hence T is a countable matching (degree <= 1). - Theorem BA. If Y is planar and sigma aperiodic, every (a, b) in T is bi-asymptotic: d(sigma^n a, sigma^n b) -> 0 as n -> +infinity and as n -> -infinity. Corollary D: for distal sigma (every odometer) T is EMPTY in every planar host; for Denjoy sigma bridges lie on gap orbits only; for Toeplitz sigma only on asymptotic pairs. - Corollary K (block2_opus_1102 and kfan, DERIVED): the known tree-like indecomposable host K_sigma (a_opus_prf_beta, with all pairs (d, sigma d) in T) is NONPLANAR for every aperiodic sigma, Denjoy included. - Lemma FL (block2_opus_1102, DERIVED): planar level-construction Cantor fans are decomposable. Proposition LOC (block2_fanbridge_prove_opus_1102, DERIVED): every local datum at p is realised by a planar equivariant hairpin model with no bridges, so the local picture at p cannot decide the node. C. THE TARGET FAR-MIX (the sharpened residual of not-EXIST-F_pl). Is there a planar tree-like indecomposable continuum Y with a homeomorphism f, f(p) = p, and an f-invariant Cantor fan Phi at p with minimal blade action sigma, such that T is either empty or a countable union of sigma-orbits of bi-asymptotic O-gap pairs? By Theorem M and BA every planar EXIST-F host has this form, so the question is whether such a host exists: indecomposability must mix the blades away from the vertex. Two sub-tests named by the lane (INFERRED reading of "mixing away from p": a nested sequence of subcontinua of Y whose union is dense and which meet Phi only in initial segments or in p). (1) sigma an odometer. Then T is empty, so no blade is ever joined to another near p. Show that indecomposability + tree-like + planarity still forces a contradiction, or build such a Y. (2) sigma Denjoy: the same with bridges allowed on gap orbits. D. ASK Part 1 (static case, no f). Decide whether a planar indecomposable (tree-like if you can) continuum Y can contain a Cantor fan Phi at p with T(Phi) empty, or with T a countable matching. Give an explicit planar construction (a nest of continua, or an inverse limit with the embedding checked) or prove that indecomposability forces an uncountable T or a bridge at a non-O-adjacent pair. Say which of the planar facts Theorem 1, M you use. A proof that T = empty is impossible would already settle sub-test (1) for odometers; a planar example shows the obstruction is equivariance only. Part 2 (equivariant, odometer). If Part 1 gives an example, decide whether it can carry f with f(p) = p, f(Phi) = Phi and sigma an odometer. The mixing continua away from p must be permuted by f; state which finite or countable family of f-invariant subcontinua you need and whether a minimal sigma on C is compatible with the circular order O (Theorem M says O is a Denjoy-type order, f is intrinsic and need not preserve it). Part 3 (reduction). If neither part is settled, state ONE lemma L strictly weaker than FAR-MIX, prove that L implies FAR-MIX or implies not-EXIST-F_pl, and answer the restatement gate below. Each claim must be labelled PROVED, SOURCE-lab, DERIVED, CONJECTURE, OPEN or RESTATEMENT. No claim of PER, NOT-BLOCK*(2) or NO-FAN unless the proof is complete and uses no unaudited lab lemma beyond those listed. E. CONTROLS (all must hold) (C1) Cantor fan over a Denjoy set K with f an ambient Denjoy rotation (the cone): planar, T empty, decomposable. Your argument must use indecomposability, or it proves too much. (C2) The hook dendroid of block2_opus_1102 (Denjoy cone with V-strands on one gap orbit): planar, bridges exactly on that orbit, decomposable; it shows Theorem BA is sharp. (C3) K_sigma, abstract: tree-like, indecomposable, EXIST-F holds, uncountable T, nonplanar; a planar argument must fail there at planarity and nowhere else. (C4) Knaster bucket handle: planar, indecomposable, no fan, so nothing is said. (C5) Do not assume any extension of f to the plane; the prime-end and Cartwright-Littlewood routes are vacuous for intrinsic f. F. DEAD ENDS (not progress) (a) Showing K_sigma is nonplanar, for any aperiodic sigma (known: Corollary K, Lemma FL). (b) Level-fan presentations of Phi (Lemma FL: decomposable). (c) Local data at p alone (Proposition LOC). (d) Prime-end rotation, accessibility of blade tips, Bell, Kuperberg (need an ambient extension). (e) Restating NO-FAN, NOT-BLOCK*(2) or EXIST-F_pl itself. (f) Hoping Denjoy order buys planarity of K_sigma (it does not, kfan section 5.1). RESTATEMENT GATE (mandatory; answered BEFORE the headline, in its own section). If your verdict reduces the node N to a residual lemma L (with or without other proved pieces), you must give ALL of: (G1) the exact statement of L and of N side by side, with the same standing hypotheses; (G2) a proof that L implies N (the reduction); (G3) a reason L is STRICTLY weaker than N: either an explicit configuration (or consistent abstract data meeting every standing law) where L holds and the conclusion of N is not yet available, or the named hypothesis of L that the negation of N does NOT supply; (G4) the line of your argument where L's own hypothesis (the defect, the witness, the scale) is actually used; if it is never used, L is a restatement. If G3 or G4 fails, the headline word is RESTATEMENT, not REDUCED, and the answer reports the proved building blocks only. A headline REDUCED without G1-G4 counts as a failed answer. G. FORMAT Line 1 exactly: VERDICT: FAR-MIX DECIDED (planar host built, or impossibility proved; inputs listed) | PARTIAL (odometer case or static case settled, rest named) | REDUCED (to one named lemma, proof of the reduction, G1-G4 answered) | 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: using a level-fan presentation (Lemma FL kills it); using an ambient extension of f to the plane or prime-end rotation at p (f is intrinsic); proving only that K_sigma is nonplanar (known); letting the blade action be periodic; claiming PER or NOT-BLOCK*(2); counting a decomposable host as an example (it must be indecomposable)