batch 14

Prompt

RESEARCH QUESTION (P2EX siege, Step 7 under (R-A): the minimal-connector dichotomy). 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 PRE-DISPATCH CERTIFICATE (batch 14, owner microeval channel). (C1) R1 / E_p^-, arrow CAN <=> P2EX, Step 7, gate T. Target: for accessible non-fixed x and FP2, the HU connectors I_n = G^n(I_0), I_0 the least continuum containing x, G(x). YES: case (b) below is excluded, so (a) gives M2, then T, then a contradiction by tail exclusion (FP2 excluded under (R-A), audit owed). NO: an explicit relaxed configuration with limsup diam I_n > 0 in both time directions and no proper continuum holding a tail, with the clause it breaks. Neither alone is CLOSED. (C2) Controls: ARC-4^- (I_n are subarcs shrinking to a fixed point); canonical case Fix(h^2) = {xi_0, xi_1} (the argument must not exclude it); Oversteegen-Valkenburg Example 5.4; rotating p-od. (C3) New input: batch 13 showed every route to T needs a proper continuum or shrinking connector produced FROM FP2; the connectors I_n as one object were never attacked. (C4) Duplicate check: earlier work used arbitrary joining continua M_n (LCS); none used the HU-least connectors with G^n(I_0) = I_n. 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, HU: two subcontinua meet in the empty set or a continuum; every subcontinuum of a planar tree-like continuum is tree-like and nonseparating). 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 of x (union of all proper subcontinua containing x) partitions X into uncountably many pairwise disjoint dense composants. For a homeomorphism g: cm(g) means no proper nonempty g-invariant subcontinuum; cm_strong(g) is the same with singletons counted as subcontinua, so g has no fixed point. Per(g), Fix(g): periodic, fixed points. FPP (fixed point property): every continuous self-map has a fixed point; PPF: periodic-point-free; PFPP: the open plane fixed point problem (does every nonseparating plane continuum have FPP?). Cartwright-Littlewood (SOURCE): an orientation-preserving homeomorphism of the plane leaving a nonseparating plane continuum invariant has a fixed point in it (Bell, Brown extensions). Lab vocabulary: R1, E_p^- = lab route labels (setting below); siege 1 = the lab's current attack on P2EX. Statuses: PARTIAL/OBSTRUCTION = author's verdict; UNAUDITED = no audit yet; CONFIRMED-WITH-REPAIRS = audit passed, repairs written in, not yet closed; one GPT agent = one GPT run; never assign VERIFIED/CLOSED. Take your time, be extremely meticulous, I believe in you. SETTING (route R1, branch E_p^-; LAB; package UNAUDITED). X is a nondegenerate planar nonseparating tree-like indecomposable continuum in R^2, U = S^2 minus X (a simply connected domain containing inf), and X = boundary(U) (clause BD; automatic, as planar tree-like X has empty interior). p is an odd prime. b: X -> X is a homeomorphism with cm_strong(b) and anchor cycle A = {a_0,...,a_{p-1}}, b(a_i) = a_{i+1}; the a_i are called anchors. Sources disagree on the spectrum hypothesis: (R-A) Per(b) = A or (R-B) only Fix(b^p) = A; OPEN, state which reading each step uses. F: S^2 -> S^2 is an ORIENTATION-REVERSING homeomorphism with F(X) = X, F(inf) = inf and F|X = b^p; so F^2 preserves orientation, F^2|X = b^{2p}, and b^p, b^{2p} are NOT cm (anchors are fixed singletons): cm(F) is never a substitute for cm(b). Complement PPF: F has no periodic point in R^2 minus X. Prime ends. Let phi map D* = {|z|>1} + {inf} conformally onto U with phi(inf) = inf (normalisation only; F need not commute with phi, so F does NOT map radial rays to radial rays). Prime ends of U correspond to points zeta of the unit circle T. For a prime end e, the impression Imp(e) and the principal set Prin(e) are subcontinua of X with Prin(e) inside Imp(e). SOURCE (Pommerenke, Boundary Behaviour of Conformal Maps, 1992, Theorem 2.16, checked on the full text): Imp(e) is the unrestricted cluster set of phi at zeta, and Prin(e) equals the cluster set of phi along the radius ending at zeta, which also equals the intersection of the cluster sets along all curves in D* ending at zeta and the cluster set in any Stolz angle at zeta. e is accessible iff Prin(e) is one point (its landing point); e is full-principal if Prin(e) = X. F induces a homeomorphism h = h_F of T with Imp(h e) = F(Imp e), Prin(h e) = F(Prin e). Since F reverses orientation, h has exactly two fixed ends xi_0, xi_1, swaps the two open arcs of T between them, and h^2 preserves orientation; every h-periodic end has least period 1 or 2. A 2-cycle is a pair {e, e'} with h(e) = e' != e; its two ends lie on opposite arcs, so {e, e'} and {xi_0, xi_1} are LINKED (alternate on T). P2EX: no 2-cycle exists, i.e. Fix(h^2) = {xi_0, xi_1}. The lab's canonical prime-end law CAN is equivalent to P2EX (LAB, audited). LBACC (LAB standing package of this branch) includes: the two fixed ends are full-principal, Prin(xi_0) = Prin(xi_1) = X; it gives no prime-end action of b itself, only of F = b^p. Oversteegen-Valkenburg Example 5.4 (SOURCE, not rechecked): a planar pseudo-arc embedding with every impression the whole pseudo-arc and a dense second-category set of full-principal ends, so many full-principal ends carry no static contradiction. B. LEMMAS (status; where used) Step 3 (LAB, unaudited): (i) if Prin(e) = {x} then Prin(h e) = {F(x)}; (ii) at each point of an indecomposable X = boundary(U) at most one prime end has singleton principal set. Hence under (R-A) every 2-cycle {e, e'} has nondegenerate P = Prin(e), P' = Prin(e') with F(P) = P', F(P') = P. L-SC (sector collapse; LAB, 7/7 refuters no defect): if two ends e, e' are linked with two full-principal ends xi_0, xi_1, then NO proper subcontinuum C of X contains P u P'. Proof: collapse C to a point c (Moore; quotient is a 2-sphere); the images of the radial rays of e, e' form a Jordan curve J meeting q(X) only in c; the two fixed-end radial sectors lie in different components V_0, V_1 of S^2 minus J; full-principal rays cluster on all of q(X), so q(X) lies in cl(V_0) cap cl(V_1) = J, so q(X) = {c}, contradiction. DERIVED: a 2-cycle is (FP2) P = P' = X or (SPLIT) P, P' proper, disjoint, in different composants. Under (R-A) SPLIT is excluded (Cartwright-Littlewood for F^2 on P), so P2EX under (R-A) is EQUIVALENT to excluding FP2. Under (R-B), SPLIT: the b-orbit of P has length 2p, K = union b^j(P) is proper and disconnected, the component of K containing P has b-period q in {2, 2p}. Facts (several models, unaudited): a proper F^2-invariant continuum contains at most one anchor; proper b^p-invariant continua have pairwise disjoint b-translates; no proper subcontinuum contains two anchors; a subcontinuum meeting two composants is X. Under (R-A) every proper F^2-invariant subcontinuum contains exactly one anchor and lies in its composant (A6(a): it contains no accessible point). Every composant containing an accessible point is moved by F, so anchor composants are inaccessible. DISTINCT (4 models, 4 families, one third-family audit CONFIRMED): the p anchors lie in p pairwise distinct composants. Crosscut domains, one-sided cluster sets and the four open-arc landing closures at FP2 ends are all X (dead invariants); prime-end rotation of F^2 is 0; any reversing circle map extends PPF to the prime-end disk. Day 11-07 (unaudited unless noted). RIGIDITY: h is determined by F|X = b^p and the embedding, so FP2 is a property of (X in S^2, b^p). NO-GO: continua generated by one landing orbit have closure X; a valid exclusion must use h^2(e) = e, not only Prin(e) = X. Gate T (TAIL): a proper continuum containing a forward or backward G-orbit tail (G = F^2) of an accessible point. AP-UNIVERSAL: orbits of the orientation-preserving circle map h^2 are monotone in each complementary interval of Fix(h^2) and converge to its endpoints. LCS (valid): if continua M_n joining x_n, x_{n+1} = G(x_n) eventually lie in one closed ball not covering X, the tail lies in one proper continuum. M2 (a proper continuum M containing accessible x and G(x) with diam G^n(M) -> 0 as |n| -> inf) implies T (5 models). TAIL EXCLUSION (several models): no proper continuum contains an accessible-orbit tail under L-SC, LBACC and (R-A); hence under (R-A) 'FP2 => T' is the goal itself, not a weaker gate. Literal M1 anchoring is vacuous. SPLIT-QUOTIENT (3 answers, 2 families, audit owed): under (R-B) collapsing the components of K keeps xi_0, xi_1 full-principal and yields an ACCESSIBLE (R-B) 2-cycle on a tree-like quotient; that case is open. Day 11-08 (single GPT runs, unaudited). FOUNDATION (consolidated): h is the boundary map of the prime-end extension of F; exactly two fixed ends; unique access: each point of indecomposable X is the landing point of at most one accessible end; accessible ends are dense on T with full angular measure (Pommerenke Thm 1.7 via inversion). NP2_X: F|X has no point of least period 2. ACCESS-INTERIOR (cond. NP2_X + unique access + density): Fix(h^2) has empty interior in T and contains no accessible end other than xi_0, xi_1. COMMON-FIX (cond. Cartwright-Littlewood): every h^2-fixed end has an F^2-invariant nonseparating principal set containing an F-fixed point. R302 (one model): with the basins B_a of the anchors for G, the positive singular relation (x R y iff for all neighbourhoods V of x, W of y, G^n(V) meets W for infinitely many n > 0) is nonempty on X minus A; exits of R from a basin interior land in a closed nowhere dense set Z, so translated R-edges never concatenate into a cycle. REV-REAL (calibration, NOT the P2EX setting: F has fixed points in X and the host is arc-like, hence FPP): for every surjection f of [0,1] there is a reversing sphere homeomorphism F with invariant X = lim(I,f), F|X the shift, Per(F) = Per(F|X) + {inf}; for 0-attracted f (f(t) < t on (0,1)) every F^m-invariant subcontinuum is {a_0}, {a_1} or X, so any exchanged end pair is automatically full-principal; whether that h has a 2-cycle is OPEN. HU gives a unique least subcontinuum I(x,y) containing x, y; G maps I(x,y) onto I(Gx,Gy). C. THE TARGET Split (a) diam I_n -> 0 as n -> +inf or as n -> -inf; (b) limsup diam I_n > 0 in both directions. Under (b) take Hausdorff limits L+ and L- of subsequences. (1) Is each limit a proper subcontinuum? It contains the limit anchor of the orbit (AP-UNIVERSAL). (2) Use h^2(e) = e for the FP2 end e and L-SC with the linked pair to force L+ or L- to contain P u P' = X, or show it cannot. (3) Where does the canonical case escape? D. ASK Give the first failing step if not PROVED, with the exact quantifier that breaks. Separate (R-A) from (R-B). Test on the controls and name each first exit. No novelty/VERIFIED/CLOSED claims; no helper agents. 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 of [-1,1], Fix(f) = {-1,0,1}, no other periodic points), F(x,y) = (f(x), -2y) (orientation-reversing; R^2 minus X is PPF). Prime ends: x^+ and x^- (upper and lower sides) for -1 < x < 1 plus one end at each endpoint; h(x^+) = f(x)^-, h(x^-) = f(x)^+; the endpoint ends are the fixed ends (singleton principal sets, NOT full-principal) and {0^+, 0^-} is a genuine 2-cycle landing at (0,0) from two sides. X is decomposable and chainable. Interval reflection: b(t) = -t on [-1,1]; Fix = {0}, all other points have period 2; every symmetric [-c,c] is invariant. Circle reversal: z -> conj(z) has two fixed points and period-2 pairs, so reversal alone cannot give P2EX. Rotating p-od: p arcs meeting at a centre o, b rotating the arms; translates of the terminal half of one arm are pairwise disjoint; the only b-invariant point is o (forbidden only by cm_strong). Pseudo-arc: hereditarily indecomposable chainable; it has FPP, so it is never a PPF host, but it is the static calibration for full-principal ends (Example 5.4). Pseudo-circle and SPIRAL(3) separate the plane; a disk with a whisker fails BD. State each added control's first exit. A proof that works verbatim on ARC-4^-, on a disk with whisker, or on a chainable host is suspect; name the step where indecomposability, tree-likeness, nonseparation or cm_strong(b) is used and show it fails on the control. F. DEAD ENDS (not progress) Orbit-generated continua (closure X); crosscut domains; index differences; cm(F) or cm(b^p); LCS restated. 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; UNKNOWN if unsure a cited result exists (give exact theorem numbers or say UNSURE). RED FLAGS: using cm(F) or cm(b^p) in place of cm(b); treating F as commuting with phi or mapping radial rays to radial rays; confusing impression with principal set; assuming Fix(b^{2p}) = A under (R-B); assuming P u P' is b-invariant; dropping the canonical full-principal fixed ends xi_0, xi_1 as a negative test.