m R2

Prompt

RESEARCH QUESTION (plane continuum theory / BFMOT outchannels and prime ends; door branch of RIG-nonXS (Route I of the L2 route map): T1 with two channels, channel transfer through a proper subcontinuum). 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 (lab notes b_opus_door_d0, b_opus_rig_with_cp1, 0_B_research_inverse_channel, 206_two_outchannels_homeo; project statements are DERIVED and UNAUDITED unless marked SOURCE; lines marked (INFERRED) are the drafter's bridging). - (S) X is a nondegenerate nonseparating plane continuum, U := C \ X (connected, unbounded). h: X -> X is a homeomorphism with h^2 continuum-minimal (cm: no proper nonempty subcontinuum K has h^2(K) in K). Then h and h^-1 are cm and h has no fixed point (lab, DERIVED). The project conjecture L2 says no such (X,h) exists; we work INSIDE a hypothetical one, so a "counterexample" can only be a germ or a lemma-level configuration. - F: plane -> plane is a perfect extension of h, G a perfect extension of h^-1 (closed continuous, compact point inverses, F|X = h, G|X = h^-1; BFMOT Def 2.3.1, Blokh-Fokkink-Mayer-Oversteegen-Tymchatyn, arXiv 1004.0214v2). G need not be the inverse of F, and G o F is only known to be the identity on X (lab: 0_B_research_inverse_channel). - Crosscut: an open arc q in U whose closure is an arc with two distinct ends in X. Sh(q) is the bounded component of C \ (X u q), Out(q) the unbounded one (U \ q has exactly these two components). var(f, q^+, X) is the BFMOT variation (Def 2.2.1-2.2.2). Lemma 4.3 of the lab (DERIVED): if f(cl q) and cl q are disjoint, v in q and rho is an access (proper ray from v meeting X u cl q only in v), then var(f, q^+, X) = I(f o q^+, rho), the winding about v of f o q^+ closed up by a path off rho. Property used: if two paths beta, beta' with the same ends lie in a convex set B missing v, then I(beta, rho) = I(beta', rho). - SOURCE (BFMOT Thm 6.2.1 and Lemma 6.1.3): for f = F and for f = G (G satisfies the standing hypothesis since h^-1 is cm) there is a unique geometric outchannel E_F, E_G, dense in X, with Pr(E) = X (principal continuum). For a sufficiently small chord g of a chain defining the channel, var(f, g, X) = -1, and 0 for a small chord not crossing the ray essentially. R_F, R_G are the external rays. A crosscut q is an E-crosscut (for that map) if the X-end of the ray lies in Sh(q) (BFMOT Def 3.6.4), an M-crosscut otherwise. Prime-end form: with a conformal map of the disc onto U u {infinity}, q lands at two circle points and has a shadow arc I_q; q is an E-crosscut iff the channel's circle point lies in the interior of I_q. - CP1 (lab, UNAUDITED): every small E-crosscut q (not only KP-chords) has var(f, q^+, X) = -1, every small M-crosscut has var 0. The first clause is what turns "var = 0" into "M-crosscut". - Strip and doors (lane b_opus_rig_with_cp1, b_opus_sidechords): a strip system has sides H1, H2 continuous into X, injective, disjoint, and sections c_r crosscuts. A door is a chord s = H_i((a, b)) with ends s^- = H_i(a), s^+ = H_i(b) in X; Pk(s) is the component of U \ s missing the open strip, K_s := cl Pk(s) cap X. Put u := h(s^-), v := h(s^+), K' := h(K_s). Door data: D_s(q) := I(P, rho_q) for E_F-crosscuts q far from u, v, with P a short path from u to v. D0 at s: D_s(q) = 0 for all far deeper E_F-crosscuts q. - Lemma 3 (empty bay; lab, re-derived from Lemma M of a_opus_nd). Let m be an M-crosscut for a map whose channel is dense, and K a continuum in X containing both ends of m. Then T(K u cl m) = K u cl m u Sh(m); in particular X cap cl Sh(m) is in K. Lab proof: Out(m) is connected, unbounded, misses K u cl m, so a bounded domain Gd of C \ (K u cl m) has Gd cap U inside Sh(m); if x in X lay in Gd, the chain crosscuts of the dense channel near x would lie in Sh(m), so m would be an E-crosscut. (INFERRED: the proof shows that no point of X lies in a bounded domain; the step "Sh(m) itself lies in a bounded domain, so cl Sh(m) cap X is in K" is not written there. Check it, using that U \ m has exactly two components and that X has empty interior if you need it; say where.) B. LEMMAS (status; where used) - Prop 6 (lane b_opus_door_d0, DERIVED, UNAUDITED; "one-channel kill"). Let C be a short E_F-crosscut with ends EXACTLY u and v (accessible case). Then G o C lies in a small ball B about s^- with ends s^-, s^+ (continuity of G, |u - v| small), B misses the start v_C of the access rho_C (dist(v_C, s^-) >= m' > 0 because h has no fixed point), so by the property above var(G, C^+, X) = I(G o C^+, rho_C) = I(s, rho_C) = 0, the last because s is a door chord and v_C is not in Pk(s). Hence, with the first clause of CP1 for G, C is not an E_G-crosscut. So if E_F = E_G, no short E_F-crosscut joins u and v. - Theorem 7 (same lane, DERIVED, UNAUDITED; trichotomy at scale rho for the door s, using Lemma C, AB-circle, Lemma P of b_opus_rig). (T0) u, v are near-joined by a short M_F-crosscut, door data 0. (T1) u, v are near-joined by a short E_F-crosscut C, data +-1 beyond C, K' eta-dense, and E_G != E_F by Prop 6. (T2) the word of the path P is infinite, hence perfect; K' u cl P separates two points of X or D_s jumps. D0 at s fails at scale rho iff T1 or T2. The lane lists as open: "T1 with two channels: can a door image be a short E-crosscut when E_G != E_F?" and "E_F = E_G is not known". - Lane 206_two_outchannels_homeo (DERIVED, UNAUDITED): if h induces a compatible homeomorphism of the prime-end circle (hypothesis H*_pe) then E(h) != E(h^-1) in the orientation-reversing case, and E(h) != E(h^-1) is proved under local accessibility along one chain of E(h) plus the general-crosscut variation law. So equality E_F = E_G is NOT expected; the useful statement is a weaker one. - Prop 8 (same lane): X indecomposable and Y a proper subcontinuum containing K' u (cl P cap X) imply D_s constant on X \ Y, so in T2 the mouth strands lie in more than one composant. C. THE TARGET Three statements, all new to the lab (the drafter's candidates, INFERRED; the first is the one to settle). CH-T (channel transfer). Let C be an E_F-crosscut whose two ends lie in a proper subcontinuum K of X. Then C is an E_G-crosscut. The same holds with F and G exchanged. Drafter's sketch. (1) The channel E_F lies in the interior of the shadow arc of C, and the chain crosscuts of E_F lie in Sh(C) from some index on, so Pr(E_F) is in cl Sh(C); Pr(E_F) = X, so X is in cl Sh(C), i.e. X cap cl Sh(C) = X. (2) Suppose C were an M_G-crosscut. Lemma 3 for G and K gives X cap cl Sh(C) in K. (3) So X = K, contradicting K proper. T1-EX (T1 with two channels, exact ends). In (S), with CP1 (first clause) for G, no door s of a strip system admits a short E_F-crosscut C with ends exactly u = h(s^-) and v = h(s^+), whatever E_G is. Sketch: K := h(H_i([a, b])) is a continuum through u and v; it is proper, since X = K would be a path image, hence a Peano continuum, and a nonseparating plane Peano continuum has the fixed point property (RECALL, not on disk), against h fixed-point-free. By Prop 6, var(G, C) = 0, so C is an M_G-crosscut (CP1); CH-T says C is an E_G-crosscut. Contradiction. CH-RED (reduction of E_F = E_G). If some chain defining E_F consists of crosscuts whose ends lie in proper subcontinua of X (ENDS-PROPER), then by CH-T each of them is an E_G-crosscut, the X-end of R_G lies in every Sh(g_i), so E_G = E_F. In an indecomposable X: ENDS-PROPER means the two ends of g_i lie in one composant. D. ASK Part 1 (CH-T). Prove or refute CH-T with exact hypotheses, step by step: (1) the claim "chain crosscuts of E_F lie in Sh(C) eventually and Pr(E_F) is in cl Sh(C)" (name the prime-end fact used, including the case when an end of C is not accessible); (2) Lemma 3 for G, in particular the missing step that cl Sh(C) cap X is in K (a bounded domain of C \ (K u cl C) containing Sh(C)); (3) where K proper is used. If CH-T fails, give the configuration (a crosscut, a continuum K, two channels) and name the failing step. Part 2 (T1-EX). Given Part 1, decide whether T1-EX is a proof or needs a named extra step. Check Prop 6 for exact ends (the ball B and the access rho_C, the value I(s, rho_C) = 0), the properness of K, and exactly which clause of CP1 for G is used. State whether a KP-chord C would avoid CP1. Then state the "near" version (ends of C only within eta of u, v, no K through the ends) and say what extra lemma it needs (a proper continuum through the ends of C), or give a reason it cannot be reduced. Part 3 (reduction). Using CH-T, state ONE named lemma whose proof or refutation would decide E_F = E_G (candidate ENDS-PROPER above), prove CH-RED, and say whether ENDS-PROPER is consistent with lane 206 (E_F != E_G under H*_pe): if CH-T holds, E_F != E_G forces the chain crosscuts of E_F to have ends in no common proper subcontinuum; say whether that is contradictory in the indecomposable case (ends in different composants) or merely a restriction. Do not claim L2. E. CONTROLS (all must hold) (C1) E_F = E_G must not be used anywhere (it would make T1-EX circular). (C2) Planted negative: let C have ends a, b such that X is irreducible between a and b (K = X forced). CH-T must say nothing there; your proof must show where "K proper" is used. (C3) Do not use G o F = id or G = F^-1; if you use any relation between F and G beyond F|X = h and G|X = h^-1, name it and say whether lane 0_B_research_inverse_channel (two proper extensions with different return intersection numbers) is consistent with it. (C4) Symmetry: CH-T for (F, G) and for (G, F) together must not produce a contradiction with E_F != E_G when ENDS-PROPER fails; check. (C5) State where nonseparation of X and the exactness of the ends are used. F. DEAD ENDS (not progress) (a) Proving E_G = E_F by a positive-displacement argument for G o F: it fails (G o F is the identity on X, BFMOT Def 2.2.2 needs the image arc to miss the source arc). (b) A formal signed-module countermodel (0_B_research_inverse_channel): the four negative scalar laws for h, h^-1, h^2, h^-2 do not decide the channel question, so do not rebuild it. (c) Proving D0 from density, the F-law, the M-crosscut law of cdfail, Lemma Z and BLOCK*: eliminated by the exact germ DD of b_opus_door_d0. (d) SHEET, paired hairpins, SO-, TRAP, Theorem R*, LEMMA_P, CP1 itself: other tasks. G. FORMAT Line 1 exactly: VERDICT: T1-EX PROVED (inputs listed) | CH-T TRUE, T1-EX NEEDS A NAMED EXTRA STEP (step, proof of the rest) | CH-T FALSE (configuration, failing step) | 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 E_t(G) = E_t(F); using G o F = id or G = F^-1; applying Lemma 3 to a crosscut whose ends are not both in the continuum K; using K proper without a reason; quoting BFMOT 6.2.1 (var = -1) for a crosscut that is not a KP-chord without naming CP1; treating an end of the ray as a point of X; claiming L2 or PER