m19

Prompt

RESEARCH QUESTION (plane continuum theory / BFMOT variation and accesses; the door branch of RIG-nonXS (Route I of the L2 route map): the access side-condition of DOOR-OUT, restated as "no small mouth around the ball B"). 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 notes b_opus_door_d0, b_opus_rig_with_cp1, a_opus_nd and the microeval m-r2 / audit a161audsol; every project statement is 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). Then h, h^-1 are cm and h has no fixed point; m_1 := min_X |h(x) - x| > 0 (lab). L2 says no such (X,h) exists; we work INSIDE a hypothetical one. - F, G: plane -> plane are perfect extensions of h and of h^-1 (closed continuous maps with compact point inverses, F|X = h, G|X = h^-1; BFMOT Def 2.3.1, arXiv 1004.0214v2). G need not be F^-1. We never use G o F = id. - 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 two components; this is the lab's convention). var(f, q^+, X) is the BFMOT variation. - Access and I. An access of a point v is a proper ray rho in C with start v. For a path beta with ends off rho: I(beta, rho) := win(beta * delta, v), delta any path in C \ rho from the end to the start of beta. Properties: (I1) additive; (I2) reversal changes sign; (I3) I(beta, rho) = 0 if beta misses rho; (I4) invariant under homotopy in C \ {v} fixing ends; (I5) two paths with equal ends in an open convex set not containing v have equal I. - Lemma 4.3 (lab, DERIVED, uses only the BFMOT definition and S6): let f be F or G, c a crosscut with f(cl c) and cl c disjoint, v in c, rho an access of v with rho meeting X u cl c only in v, B an open convex set containing f(cl c) and not containing v. Then var(f, c^+, X) = I(beta, rho) for every path beta in B joining the f-images of the ends of c. - CP1 (lab, UNAUDITED): every small E-crosscut q of a map f has var(f, q^+, X) = -1 (first clause) and every small M-crosscut has var 0 (zero clause, flagged as possibly defective in the strip tail, A166). - SOURCE (BFMOT Thm 6.2.1, Lemma 6.1.3): F and G each have a unique geometric outchannel E_F, E_G, dense in X (Pr(E) = X), with external ray R_F, R_G. A crosscut q is an E_F-crosscut if the X-end of R_F lies in Sh(q). For a small chord g of a chain defining E_F, var(F, g, X) = -1. - Lemma 4.5 / Cor 4.6 (lab, DERIVED, CP1-free): for disjoint crosscuts c, d: c lies in Sh(d) or in Out(d); if c lies in Sh(d) then Sh(c) lies in Sh(d); if c is an E_F-crosscut and its point v_c lies in Sh(d), then d is an E_F-crosscut. - Strip and door (lab). A strip system has sides H1, H2: continuous, injective, disjoint images in X, and sections c_r crosscuts whose tails define E_F. A door is a crosscut s in U with ends s^- = H_i(a), s^+ = H_i(b) on a side; the pocket Pk(s) is the component of U \ s missing the open strip. Put u := h(s^-), v := h(s^+). - Prop 6 (b_opus_door_d0, "one-channel kill"): let C be a short E_F-crosscut with ends exactly u and v, v_C a point of C, rho_C an access of v_C. G(cl C) lies in a small ball B about s^- (uniform continuity of G, |u - v| small) and B misses v_C (dist(v_C, s^-) >= m_1/2). The lab then writes var(G, C^+, X) = I(G o C^+, rho_C) = I(s, rho_C) = 0, the last because v_C is not in Pk(s). The audit found the step I(G o C^+, rho_C) = I(s, rho_C) unjustified when s is long (s need not lie in B), and DOOR-OUT is the name of the missing side condition. CH-T (settled) makes C an E_G-crosscut when its ends lie in a proper subcontinuum; DOOR-PROPER (settled for K = h(H_i([a,b]))) supplies that subcontinuum. So T1-EX (no short E_F-crosscut with ends exactly u, v) needs only var(G, C^+, X) = 0, because the first clause of CP1 for G would give -1 for an E_G-crosscut. B. LEMMAS (status; where used) Known and new (all UNAUDITED). - (INFERRED, the drafter's reformulation; check it). ACC: C is a crosscut with ends in X, B an open ball with centre s^- of radius r_B, with G(cl C) in B and cl C disjoint from cl B. ACC(C, B) := some access rho_C of a point v_C of C meets X u cl C only in v_C and is disjoint from cl B. By Lemma 4.3 with beta in B (convex) and (I3): ACC(C, B) implies var(G, C^+, X) = 0. The door s does not enter at all. This is a one-line argument, so the whole difficulty of DOOR-OUT is the existence of the access. - (INFERRED) MOUTH. Because C is disjoint from cl B and is connected, C lies in one component of U \ cl B. If no access of any point of C avoids cl B, then C lies in a bounded component Q of U \ cl B (an access leaves C through Out(C) and C is connected). The boundary of Q in U is made of arcs of the circle bd(B), so some arc q of bd(B) with both ends in X is a crosscut with diam q <= 2 r_B and C in Sh(q). By Cor 4.6, if C is an E_F-crosscut then q is an E_F-crosscut: a small E_F-crosscut, with ends within r_B of s^-, whose shadow contains C and has diameter >= dist(u, s^-) >= m_1. So ACC fails for every r_B small exactly when there are small E_F-crosscuts at s^- with large shadow. Such crosscuts exist in general: the chain of a dense channel (Pr(E_F) = X) has diameters tending to 0 and shadows whose closures contain X, but where the chain sits relative to s^- is not given. - Lab remark (m-r2, a161audsol): the three claims "v_C in Out(s)", "I(G o C^+, rho_C) = I(s, rho_C)" and "var(G, C^+, X) = 0" are not the same; a ball estimate alone proves only the disjointness of G(cl C) and cl C (Lemma 4.3 hypothesis). C. THE TARGET Statement DOOR-ACC. In (S), with a strip system, let s be a door and C a short E_F-crosscut with ends exactly u = h(s^-) and v = h(s^+). Then var(G, C^+, X) = 0. We ask for the proof or the obstruction through ACC, not through Out(s). D. ASK Part 1 (equivalence). Prove or correct the two (INFERRED) items: (a) ACC(C, B) implies var(G, C^+, X) = 0 with every hypothesis of Lemma 4.3 checked (the disjointness, B convex and missing v_C, the access meeting X u cl C only in v_C); (b) the converse structure: if ACC(C, B) fails then C lies in Sh(q) for a crosscut q contained in bd(B) (state exactly which arcs of bd(B) in U can occur; bd(B) cap U may have infinitely many components). Say whether the choice of r_B can be shrunk with C fixed, and what that does to MOUTH. Part 2 (mouth). Decide whether a small E_F-crosscut q with ends within r_B of s^- and C in Sh(q) can exist when C is as in Prop 6. Use only (S), the strip system (the sections c_r define E_F and Pk(s) misses them), Cor 4.6, the first clause of CP1 for F (small E_F-crosscut: var(F, q^+) = -1) and F(cl q) near u. Either exclude it (state exactly which law excludes it) or give a planar configuration meeting every listed law: a nonseparating continuum with a dense channel, a door s, a small crosscut q at s^- whose shadow contains a short crosscut C at u, v. If it exists, say whether var(G, C^+, X) can then be -1 (so that T1-EX would not follow from this route) or whether the value is still forced to 0 by another argument. Part 3 (weaker statement and reduction). If Parts 1-2 do not close, name the weakest statement about C or about the point v_C (for example "v_C has an access avoiding a ball of radius r_B about every point of X at distance <= eta from u, v") that the proof of T1-EX needs, prove that it implies DOOR-ACC, and answer the gate: G1 statements side by side, G2 the implication, G3 strictly weaker than DOOR-ACC (a configuration where the new statement holds and DOOR-ACC is not yet available), G4 the hypothesis actually used (cm, no fixed point, the strip, CP1 first clause for F). Do not restate DOOR-OUT as "no door image is an E-crosscut". Each claim must be labelled PROVED, SOURCE, LAB, DERIVED, CONJECTURE, OPEN or RESTATEMENT. Do not claim L2 or PER. E. CONTROLS (all must hold) (C1) Planted negative: let X be a nonseparating continuum with a dense channel and q_j the chain crosscuts of that channel; they have diam q_j -> 0 and the closure of Sh(q_j) contains X. A proof of ACC must say why this does not give a small crosscut q at s^- with C in Sh(q). (C2) Hairpin door: a door whose pocket Pk(s) contains the point v_C; say whether ACC, v_C in Out(s) and the value var(G, C^+, X) agree there. (C3) Do not use E_F = E_G, G o F = id or G = F^-1; if a relation between F and G beyond F|X = h, G|X = h^-1 is needed, name it. (C4) Do not use the zero half of CP1; the first clause for F is allowed only in Part 2 and must be named. (C5) State where nonseparation of X, the exactness of the ends u, v and m_1 > 0 are used. F. DEAD ENDS (not progress) (a) E_G = E_F by a positive-displacement argument for G o F (fails: G o F is the identity on X). (b) A formal signed-module countermodel (0_B_research_inverse_channel). (c) D0 from density, the F-law, Lemma Z and BLOCK* (eliminated by the exact germ DD). (d) DOOR-PROPER and CH-T (settled). (e) Proving "v_C in Out(s)" and stopping: that does not give the access or the identity I(G o C^+, rho_C) = I(s, rho_C). (f) SHEET, paired hairpins, SO-, TRAP, Theorem R*, LEMMA_P, CP1 itself: other tasks. (g) Reading H_i([a,b]) as an arc in X (then Pk(s) does not exist). 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: ACC PROVED (var(G,C) = 0 for a short E_F-crosscut C with ends u, v, from stated hypotheses) | ACC FAILS (configuration with a small mouth, every standing law checked) | ACC CONDITIONAL (one named hypothesis on C or on the door, with proof of the rest) | 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_F = E_G; using G o F = id; calling a crosscut small when only its ends are close (the shadow of a small crosscut can be large: the chain of a dense channel has diameter to 0 and shadow closure containing X); using the zero half of CP1; replacing the door s by the short path G o C without a homotopy argument in C minus {v_C}; applying Lemma 4.3 with an access that meets B; treating "v_C in Out(s)" as the same statement as "an access of v_C misses B"; claiming L2 or PER