Batch 17: A2LOC-AP on J, Theta singleton, PRIN-Q normal forms

Prompt

RESEARCH QUESTION (prover or refuter: is hypothesis HA with |Theta| >= 2 impossible?). Self-contained. Label every claim PROVED (you prove it), SOURCE (exact reference), 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. Take your time and be extremely careful; no helper agents. A. DEFINITIONS AND SETTING PRE-DISPATCH CERTIFICATE (batch 17, microeval channel). Shared map: Siege 1 CAN <=> P2EX; gate G2 <=> ACCANC (R*): an accessible anchor forces a contradiction; PER arrow R1 / E_p^-. (C1) R1 / E_p^-. Target: under HA with |Theta| >= 2, derive a contradiction (equivalently A2LOC-AP for the J-gaps). YES: the proper-impression branch of ACCANC with |Theta| >= 2 is excluded. NO: an explicit circle-level or arc-level model of a tame J-gap with an endpoint outside Fix(h) whose carrier is the F-INVARIANT composant K_0, naming the exact property planarity or cm_strong(b) must forbid. (C2) Controls: positive = Theta = T (Julia-type, J does not exist, statement vacuous); negative = ARC-4^- (Theta empty, decomposable). (C3) New input: ACCFP2 facts (3)-(7): the carrier K_0 is F-invariant, unlike every earlier packet case. (C4) Duplicate: one Opus lane runs the same statement; this is the cross-family prover/refuter pass. Definitions: a continuum is a nonempty compact connected metric space; subcontinua are closed and connected; nondegenerate means more than one point. Planar means embeddable in R^2, hence in the sphere S^2 = R^2 + {inf}; nonseparating means S^2 minus X is connected. Tree-like: for every eps>0 an onto map to a finite tree with all fibres of diameter < eps (tree-like continua are hereditarily unicoherent). Indecomposable: not a union of two proper subcontinua; then every proper subcontinuum is nowhere dense, the union of two proper subcontinua that meet is proper, and the composant Comp(x) (union of all proper subcontinua containing x) partitions X into uncountably many pairwise disjoint dense composants; a proper subcontinuum meeting Comp(x) lies in Comp(x). For a homeomorphism g: cm_strong(g) means no proper nonempty g-invariant subcontinuum, singletons included (so g has no fixed point). Per(g), Fix(g): periodic, fixed points. PFPP: the open plane fixed point problem. Cartwright-Littlewood (SOURCE, via Fokkink-Mayer-Oversteegen-Tymchatyn Thm 10.3): an orientation-preserving homeomorphism of the plane leaving a nonseparating plane continuum invariant has a fixed point in it. Statuses: LAB = earlier lab answer, unaudited unless noted; DERIVED = lab proof, audit owed. Never assign VERIFIED or CLOSED. CLASS A_p+ (LAB). (K1) p is an odd prime. (K2) X is a nondegenerate planar nonseparating tree-like indecomposable continuum, U = S^2 minus X contains inf, X = boundary(U) (X has empty interior). (K3) b: X -> X is a homeomorphism with an anchor cycle A = {a_i : 0 <= i <= p-1}, b(a_i) = a_{i+1} (indices mod p). (K4) cm_strong(b). (K5) R-A: Per(b) = A. (K6) The anchors lie in pairwise distinct composants K_i = Comp(a_i). (K7) F: S^2 -> S^2 is an ORIENTATION-REVERSING homeomorphism with F(inf) = inf and F|X = b^p (so every anchor is F-fixed; F and b^p are NOT cm). (K8) h is the prime-end map of F on U. Write G = F^2 (orientation preserving), H = h^2, E = Fix(H). There is NO assumption that the h-fixed ends are full-principal. Prime ends. phi maps D* = {|z|>1} + {inf} conformally onto U, phi(inf) = inf; F need not commute with phi (it does NOT map radial rays to radial rays). Prime ends = points of the unit circle T. Imp(e) (impression) and Prin(e) (principal set) are subcontinua of X with Prin(e) inside Imp(e). SOURCE (Pommerenke 1992, Thm 2.16): Imp(e) is the unrestricted cluster set of phi at e; Prin(e) is the radial cluster set, also the cluster set along any curve in D* ending at e. e is accessible iff Prin(e) is a point L(e) (the radial curve lands there). Full-principal: Prin(e) = X. Theta = {e : Imp(e) = X}. INPUTS (LAB/DERIVED, all usable here). I1: h is a homeomorphism of T with Imp(h e) = F(Imp e), Prin(h e) = F(Prin e), L(h e) = F(L e). h reverses orientation of T, so it has exactly two fixed ends, and H preserves orientation. I3 (Moore, SOURCE): collapsing one nonseparating subcontinuum C of X to a point gives a sphere and a map that is a homeomorphism off C. I7: accessible ends are dense on T. Unique access (Brechner 1978 Thm 3.2 via F, LAB): each point is L(e) for at most one accessible end e. STEP 2: every subcontinuum of X is nonseparating. GENERALIZED L-SC (STEP 3): if ends alpha, beta alternate on T with two ends zeta_0, zeta_1 of Prin = X (fixed or not), no proper subcontinuum of X contains Prin(alpha) u Prin(beta) (proof: collapse C, the two radial images form a Jordan curve meeting q(X) only at the collapse point, each side contains a full-principal ray, so q(X) is in the curve). ARC IMPRESSIONS: for a closed arc J of T, Imp_all(J) = intersection over eps of the closures of phi of the eps-neighbourhood of J in D*; it is a subcontinuum equal to the union of Imp(e), e in J. BIS: if a closed arc J misses Theta, Imp_all(J) is a PROPER continuum containing Imp(e) for every e in J. Theta is closed, nonempty and h-invariant. E is closed; H fixes the endpoints of each gap (component of T minus E), is increasing there, and its orbits in a gap converge to the two endpoints. ACCFP2 (LAB, one author, Sol audit GAP only on a headline: wording 'not forced full-principal' should read 'not derivable by separation'; the numbered facts below were not refuted). HYPOTHESIS HA: a tuple of A_p+ in which a_0 is accessible, its access end eta is h-fixed, Prin(eta) = {a_0}, and Imp(eta) != X. Then (1) eta is not in Theta; J := the component of T minus Theta containing eta satisfies h(J) = J. (2) If |Theta| >= 2, J is an open arc with endpoints q-, q+ in Theta, h(q-) = q+, h(q+) = q-, Imp(q+-) = X, eta is the only h-fixed end in cl J and the second fixed end zeta lies in J' = T minus cl J. (3) Every end of J has its impression in K_0 (BIS + INDEC), and K_0 is F-invariant. (4) Every H-fixed end alpha of J other than eta has Prin(alpha) proper and containing a_0 (Cartwright-Littlewood for G on Prin(alpha) gives a G-fixed point, R-A makes it an anchor, it lies in K_0, so it is a_0); no end of E n J is full-principal. (5) E n J is nowhere dense in J; the J-gaps (components of T minus E inside J) are dense in J, H(I) = I for each. (6) Every J-gap is TAME (misses Theta) and has an endpoint outside Fix(h). (7) Lemma P: if P := Prin(q-) != X, then P contains an anchor a_k, C = P u F(P) is a proper F-invariant continuum with C n A = {a_k}; collapsing C splits X = O u C u O' (O over J, O' over J'); if k != 0 then O' is empty, Theta = {q-, q+} and every end of J' has impression in C, inside K_k (case Pk); if k = 0, C lies in K_0 (case P0). (8) Separation arguments (L-SC with any separators) cannot decide Prin(q-) = X, because a separator would need an end of Theta inside J. (9) A2LOC-AP (OPEN): for every tuple of A_p+ and every gap I with I n Theta empty, both endpoints of I are h-fixed. By (6), A2LOC-AP implies HA with |Theta| >= 2 is impossible. C. THE TARGET Assume HA and |Theta| >= 2. Prove a contradiction, or build the obstruction model. Suggested mechanisms (test, do not assume): (i) take a J-gap I = (u, v) with v not h-fixed; h(I) is a J-gap on the other side of eta (h reverses J); accessible ends of I and of h(I) all land in K_0 (fact 3), and the H-orbit of an accessible e in I converges to u and v; Imp_all of [e, H e] is a proper continuum through L(e) and G L(e) inside K_0; ask whether the increasing union over the H-orbit stays proper and whether its closure is an F-invariant or G-invariant proper continuum, and whether cm_strong(b) applied to the b-orbit of such a continuum (b permutes composants: b(K_0) = K_1) gives a contradiction; (ii) use eta itself: the radial curve lands at a_0, so any proper continuum through a_0 and F(a_0) = a_0 is in K_0; (iii) the order reversal of h on J and the two gaps adjacent to eta. Say exactly where R-A, K6 and indecomposability are used. D. ASK Prove or refute. If you prove, give a numbered linear argument citing each input above by name. If you refute, give the model and check every class clause K1-K8 on it. Then run each control and say where it exits. E. CONTROLS (all must hold) ARC-4^- (lab control): X = [-1,1] x {0}, f the time-one map of the flow of v(x) = (x^3 - x)/(1 + x^4) (increasing homeomorphism, Fix(f) = {-1,0,1}, no other periodic points), F(x,y) = (f(x), -2y) (orientation-reversing). Prime ends: x^+ and x^- for -1 < x < 1 plus one end at each endpoint; h(x^+) = f(x)^-, h(x^-) = f(x)^+; the endpoint ends are the h-fixed ends (accessible), {0^+, 0^-} is a 2-cycle landing at (0,0); X is decomposable and chainable, Theta is empty. Circle reversal: z -> conj(z) has two fixed points and period-2 pairs, so orientation reversal alone gives nothing. Rotating p-od: p arcs meeting at o, b rotating the arms; the only b-invariant point is o (forbidden only by cm_strong). Pseudo-arc: hereditarily indecomposable chainable, has the fixed point property (never a host), but Oversteegen-Valkenburg Ex. 5.4 embeds it with every impression the whole continuum and many full-principal ends: impressions alone give no contradiction. A disk with a whisker fails empty interior. State each control's first exit. A proof that works verbatim on ARC-4^-, on a chainable host, or without cm_strong(b) is suspect: name the step where indecomposability, tree-likeness, nonseparation or cm_strong(b) is used. F. DEAD ENDS (not progress) L-SC with separators inside J (fact 8: there are none); KEY-STRONG; assuming q+- full-principal; restating A2LOC-AP. G. FORMAT Line 1 exactly: VERDICT: PROVED | COUNTEREXAMPLE-CANDIDATE | PARTIAL | OPEN-OBSTRUCTION | RESTATEMENT. Then SUMMARY (at most 100 words), the argument (at most 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; give exact theorem numbers or say UNSURE. RED FLAGS: using cm(F) or cm(b^p) in place of cm_strong(b); treating F as commuting with phi or mapping radial rays to radial rays; confusing impression with principal set; assuming the h-fixed ends are full-principal (that is the P7 packet assumption, NOT available in A_p+); using KEY-STRONG ('a proper continuum through an accessible point misses its F-image'), which needs full-principal fixed ends; assuming P u F(P) is b-invariant.