m35

Prompt

RESEARCH QUESTION (plane continua / periodic points of homeomorphisms of nonseparating plane continua; PER (the main target) after the reduction to R-instances with NON-INVERTIBLE perfect extensions: what survives when F is not a plane homeomorphism?). 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 research/ROUTE_MAP_L2.md and the lanes b_opus_ssdyn, b_opus_imptransport, b_opus_sscof_trace; all UNAUDITED unless a status says otherwise). - PER: every homeomorphism of a nondegenerate nonseparating plane continuum X has a periodic point. Lab reduction (VERIFIED, D2011, D2021; sharpened CONFIRMED-WITH-REPAIRS in ROUTE_MAP_L2 Section 2a): PER follows from L2_PPF (for a planar tree-like continuum C and a continuum-minimal homeomorphism g of C without periodic points, g^2 is NOT continuum-minimal) together with NOT-BLOCK*(2) (no planar block of period 2, applied to a quotient C/(K u gK)). So the PER question for a periodic-point-free h is: does an R-instance (below) exist? - R-instance: X is a nondegenerate nonseparating plane continuum, U := C minus X; h: X -> X is a homeomorphism with h^2 continuum-minimal (cm: no nonempty proper subcontinuum K with h^2(K) inside K) and m_1 := min over x of |h x - x| > 0. F, G are PERFECT plane extensions of h and h^-1: continuous proper maps of the plane with F|X = h and G|X = h^-1. They are NOT homeomorphisms in general, G is not F^-1 off X, and F(U) may meet X. (Every h has a proper extension by Tietze; the lab's local invariants of F near X do not depend on the extension, by a straight-line homotopy.) A crosscut q is an open arc in U whose closure is an arc with distinct ends in X; Sh(q) is the bounded component of C minus (X u q). Prime ends of U are classes of null chains of crosscuts; Theta_X is the set of prime ends whose impression is X. By BFMOT (arXiv 1004.0214) F and G have unique outchannels E_F, E_G: prime ends with principal set X (dense), assumed here to be different. - Plane-extendability (SOURCE-lab F030, D136 audit, quoting Oversteegen-Valkenburg 2011 Theorem 3.4 and Remark 3.2; RECALL-level, not checked on disk here): h extends to a homeomorphism of the plane iff it is compatible with prime ends in the sense of H*_pe; for such h, Bell 1978 and Cartwright-Littlewood give a fixed point (both orientations), so that case is EMPTY. The real residual is the NON-compatible case (N1/N2 of F030, TARGET-V): h does not extend to any plane homeomorphism. Hence in an R-instance NO perfect extension is a homeomorphism. Further consequences for a periodic-point-free h (DERIVED, D1892/D1894, secondary SOURCE: BFMOT lines 198-202, 361-364; primary CL51, Bel78, Bel67 not on disk): X contains an h^m-invariant indecomposable subcontinuum for every m (Bell 1967), X is not chainable, not arcwise connected, not hereditarily decomposable, and no iterate h^m extends to a plane homeomorphism. Non-extendability alone is far from aperiodicity (D1895: a planar inverse limit of stars with no landing-compatible iterate, every point periodic). The known periodic-point-free tree-like example of Minc 1996 is NONPLANAR (D1486), so planarity must be used. B. LEMMAS (status; where used) What the lab reduction already gives for an R-instance (DERIVED, single agents; named statements). - Prop NX (b_opus_ssdyn): at a small crosscut s whose shadow misses a tail of the E_F ray (an M_F door) the F-variation is 0, and if the image-end crosscut C of a failing configuration exists, the shadows of s and C are disjoint (order (D)). Prop DEG: deg(F, Sh(q), w) = a locally constant integer function of the image ends (h a, h b) and w, so in a failing configuration F maps the M_F pocket D = Sh(s) with degree +-1 over the far part of the E_F pocket Sh(C), and G maps Sh(C) over the far part of D. - Prop HOM (b_opus_ssdyn): if F were a plane homeomorphism with F(X) = X and G = F^-1, that is a contradiction, so no failing door admits C. The obstruction that invertibility removes: F(s) is then again a crosscut and F maps pockets homeomorphically, so a degree +-1 map from D onto the far part of Sh(C) with Sh(C) inside the complement of D is impossible. For a perfect extension F(s) may meet X, so F(s) need not be a join of (h a, h b) and the argument stops at that one point; perturbations of F by compactly supported bumps (A176) keep every local invariant and make F(s) meet X. - b_opus_imptransport section 6 (HOM case, Tier 0): for a plane homeomorphism F with F(X) = X, F induces a circle homeomorphism phi of prime ends, phi(Theta_X) = Theta_X, every impression-transport law and TT hold, yet an explicit circle model has a failing door at every depth. So no argument using only impression or composant transport (Theorem IMP*, JOIN, EXC, LIM-FT, TT) with NX, DEG, UJ can prove SS-COF; in an actual homeomorphism the contradiction comes from the GLOBAL Cartwright-Littlewood/Bell theorem, which no listed law reproduces. b_opus_sscof_trace: h does not act on prime ends of a perfect extension; Prop LIM: the pocket limit is X itself, so cm gets no proper continuum (MICROEVAL m24, 5/5). - Nearest unproved node (SS-COF): there is n_0 with no failing door (full trace K_s = X, both M_F and M_G) of depth >= n_0 in a canonical strip near E_F. SS-COF <=> no full-impression prime end in deep door caps <=> deep door ends lie in one composant (Theorem IMP*). Consensus of the lane and of m24: the missing input must be GLOBAL and fixed-point-sensitive, a Cartwright-Littlewood type index argument for perfect extensions. C. THE TARGET Find a method that gives a periodic point of h (equivalently, excludes an R-instance, or proves SS-COF-type statements) and survives NON-INVERTIBLE perfect extensions. D. ASK Part 1 (method). Name ONE method and explain in a paragraph: (a) the global object it uses; (b) the exact hypothesis of the classical theorem it replaces (Cartwright-Littlewood, Bell, Brouwer's plane translation theorem, Franks' free-disk chains, Le Calvez/Handel foliation methods, prime-end rotation theory) and why that hypothesis (an invertible F, or a homeomorphic action on prime ends) is not needed; (c) where cm, m_1 > 0, E_F != E_G and the properness of F enter; (d) how it is independent of the choice of the perfect extension F (quantify over all of them, or prove invariance). Ideas you may use or reject, with reasons: replace F by a map on the circle of prime ends via the connected cluster set of image angles (D2560: a canonical connected set); pass to the space of planar embeddings of X (i and i o h^n are embeddings; h is non-extendable exactly when they are inequivalent); a free-disk or Brouwer argument on an invariant open region of U where F is a homeomorphism; Lefschetz-Nielsen or Conley index of F^2 on an isolating neighbourhood of X (careful: F may have fixed points off X; say how you isolate); an entropy or measure argument for h on X. A method needing invertibility of F is to be reported REJECTED with the line where it fails. Part 2 (decisive sub-case). Name a sub-case of the R-instance class on which the method decides, for example: F a homeomorphism on a neighbourhood of X but not globally; F a branched covering near X; X tree-like or indecomposable; finitely many accessible prime ends of E_F; or the circle model of b_opus_imptransport 6.3 realised by an actual X. Prove the exclusion, or reduce to ONE named lemma STRICTLY weaker than PER (or than SS-COF) with the gate below. Part 3 (decisive test). Take the circle model (failing doors at every depth, all transport laws satisfied) or a concrete non-extendable h of your choosing, and compute the invariant of your method on it; report the FIRST law that is violated, or that the invariant does not see the model (then the method is REJECTED; say what that teaches). Each claim labelled PROVED, SOURCE (exact reference), DERIVED, CONJECTURE, OPEN or RESTATEMENT. UNKNOWN if unsure a cited result exists. Restatement gate: for any reduction give (G1) L and the target side by side with the same hypotheses, (G2) a proof that L implies the target, (G3) a configuration meeting L and not yet excluded, (G4) where L's own hypotheses are used. Take your time and be extremely meticulous: check every step, every hypothesis and every quantifier before you write it down. This is a hard open problem, and I believe in you. E. CONTROLS (all must hold) (C1) The extendable case: if h extends to a plane homeomorphism, PER holds (Bell 1978, Cartwright-Littlewood; RECALL). Your method must either reduce to them there or say which hypothesis it uses that they do not; it must also fail to apply, or say why it still applies, when h is non-extendable. (C2) The circle model of b_opus_imptransport 6.3: all transport laws hold with failing doors at every depth; your global object must not be definable in that model, or must be violated there. (C3) Minc 1996 constructs a tree-like continuum with a periodic-point-free homeomorphism; it is NONPLANAR (lab D1486, VERIFIED by two cross-family audits); your method must use planarity of X at a named line. (C4) The disk with an irrational rotation (center fixed, h^2 not cm) fixes sign conventions. (C5) A perfect extension that agrees with a homeomorphism outside a compact set but folds inside: your method must say whether it applies. F. DEAD ENDS (not progress) (1) Impression, composant or channel transport under F, TT, Theorem IMP*, LIM-FT: eliminated for SS-COF by the HOM circle model (b_opus_imptransport). (2) Variation, index or degree counts of crosscuts (Prop NX, Prop DEG): stop at F(s) meeting X; blind at deep doors (codex_sscof). (3) cm applied to Hausdorff limits of pocket traces: the limit is X (MICROEVAL m24). (4) Prime-end rotation and accessibility for h itself: h is intrinsic, and F does not act on prime ends. (5) Image I as a universal lemma: false (A176). (6) Nielsen or coincidence numbers of contractible spaces (FAILED_APPROACHES F012). (7) Restating the extendable case as the general one. 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: METHOD CARRIES (a named method that works for non-invertible perfect extensions, with a decisive sub-case settled by proof) | METHOD REDUCED (to one named lemma, gate G1-G4 passed, test included) | METHOD REJECTED (it needs invertibility of F or of the prime-end action; first line where it does) | PARTIAL (one named sub-claim settled) | 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: assuming F or G is a plane homeomorphism, or that h acts on the prime ends of a perfect extension; assuming h extends to a plane homeomorphism (then the case is classical and EMPTY); a method that depends on the choice of extension F without quantifying over all of them; repeating impression or composant transport, variation counts, or cm on limits of pockets; naming a classical theorem you cannot state with its hypotheses (write UNKNOWN); claiming PER or L2 unless the proof is complete