m16

Prompt

RESEARCH QUESTION (plane continuum theory / BFMOT outchannels and prime ends; door branch of RIG-nonXS (Route I of L2): the two named gaps DOOR-PROPER and DOOR-OUT left after CH-T was proved). 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, a_opus_sidechords, 206_two_outchannels_homeo; all 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, h^-1 are cm and h has no fixed point (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, compact point inverses, F|X = h, G|X = h^-1; BFMOT Def 2.3.1, arXiv 1004.0214v2). G need not be F^-1; G o F is the identity only 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. var(f, q^+, X) is the BFMOT variation. Lemma 4.3 (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 off rho. Property used: two paths with the same ends in a convex set B missing v have the same I. - SOURCE (BFMOT Thm 6.2.1, Lemma 6.1.3): for f = F and f = G there is a unique geometric outchannel E_F, E_G, dense in X, with Pr(E) = X; for a small chord g of a chain defining it, var(f, g, X) = -1. A crosscut q is an E-crosscut if the X-end of the ray R lies in Sh(q), an M-crosscut otherwise. - CP1 (lab, UNAUDITED; flagged in A166): every small E-crosscut q has var = -1 (first clause), every small M-crosscut has var 0 (zero clause). T1-EX needs ONLY the first clause for G. - Strip system XS (lab, b_opus_rig_with_cp1): H1, H2: [r_0, inf) -> X continuous, injective, disjoint images (sides), the sections c_r crosscuts from H1(r) to H2(r). A DOOR s of a side H_i (a_opus_sidechords, b_opus_door_d0) is a crosscut in U with ends s^- = H_i(a), s^+ = H_i(b) on the side; it has a pocket Pk(s) := the component of U \ s missing the open strip, and K_s := cl Pk(s) cap X. Put u := h(s^-), v := h(s^+), K' := h(K_s). Warning (the source of the gaps): the three readings of "s = H_i((a,b))" are (A) the arc H_i([a,b]) in X, (B) a crosscut s in U with ends H_i(a), H_i(b), (C) both objects, the door s and the arc A_i := H_i([a,b]) in X with the same ends. The lab text mixes A and B. Reading C is the consistent one (INFERRED). - Prop 6 (b_opus_door_d0, DERIVED; "one-channel kill"). Let C be a short E_F-crosscut with ends EXACTLY u and v, v_C its access start, rho_C its access. Then G o C lies in a small ball B about s^- with ends s^-, s^+ (continuity of G, |u - v| small); B misses v_C (dist(v_C, s^-) >= m' > 0 since h has no fixed point); so var(G, C^+, X) = I(G o C^+, rho_C) = I(s, rho_C) = 0 because s is a door chord and v_C is not in Pk(s). - CH-T (m-r2: PROVED by all five models; a161audsol: CONFIRMED after explicit prime-end and hereditary-nonseparation repairs). Let C be an E_F-crosscut whose ends lie in a proper subcontinuum K of X; then C is an E_G-crosscut (symmetric in F, G). K proper is used only once, in the final contradiction. B. LEMMAS (status; where used) Known (all UNAUDITED unless marked). - m-r2 (five models, unanimous): T1-EX not proved as written. Named missing step DOOR-PROPER: a proper subcontinuum K of X through the door ends s^-, s^+ (Astra and Sol 6.1 add DOOR-OUT: Prop 6's zero needs the access base v_C in Out(s) / explicit unbounded-side geometry; small diameter alone does not give var = 0). Argon proves T1-EX under Reading A, where Pk(s) does not exist. - a161audsol (Sol audit of A161, GAP). K = h(H_i([a,b])) is proper: it is an arc if H_i is injective on [a,b]; if K = X, conjugating h to a homeomorphism f of [0,1] gives f(0) - 0 >= 0 and f(1) - 1 <= 0, so f has a fixed point; without injectivity K is a Peano continuum and Borsuk's theorem (nonseparating plane Peano continuum has the fixed point property) gives the same. So DOOR-PROPER holds for K := h(H_i([a,b])) in reading C, when the ends of C are EXACTLY u and v. The gap left: to get zero from Lemmas 4.3 and 4.4 one must verify that G of the closure of C and the door s lie in one convex set B, that B misses v_C, that G o C^+ and s have the same ordered endpoints, that v_C lies in Out(s) (not merely outside an undefined Pk(s)), and that the relevant closures are disjoint. The near-end version (ends only within eta of u, v) needs the actual ends of C to lie in a common proper subcontinuum and is NOT covered: a proper continuum through u, v need not contain nearby endpoint pairs. - Corrections from a161audsol: ENDS-PROPER implies E_G = E_F is the only proved direction (CH-RED, one way); "some, equivalently every, chain" is unsupported; refuting ENDS-PROPER decides nothing; the F/G classification agree on every crosscut with ends in a common proper subcontinuum (E for both or M for both), not E for both. - A166 (Arena, unaudited): a possible defect in the zero half of CP1 in the strip tail (same-side bay with a deep section). Not used by T1-EX if the first clause for G is the only clause applied. - Door data (b_opus_door_d0): D_s(q) = I(P, rho_q) for E_F-crosscuts q far from u, v, P a short path from u to v; D0: all zero. Theorem 7 (trichotomy T0 M-joined, T1 E-joined with E_G != E_F by Prop 6, T2 infinite word). Exact germ DD shows D0 is not forced by density, T1, Lemma Z, BLOCK*. C. THE TARGET Statement T1-EX (exact ends). In (S), with CP1 (first clause) for G, in reading C no door s of a side of a strip system admits a short E_F-crosscut C with ends exactly u = h(s^-) and v = h(s^+). Sketch: K := h(H_i([a,b])) is a proper arc through u, v (DOOR-PROPER, a161audsol); CH-T makes C an E_G-crosscut; Prop 6 gives var(G, C^+, X) = 0, contradicting CP1's first clause (var = -1). Open as written: DOOR-OUT, the zero value in Prop 6. DOOR-OUT (the statement to settle). Let s be a door and C a short E_F-crosscut with ends exactly u, v, access start v_C in C. Show (1) v_C lies in Out(s) (the unbounded component of C \ (X u s)), and (2) with a convex set B containing G(cl C) and s, missing v_C, and G o C^+ and s having the same ordered ends, I(G o C^+, rho_C) = I(s, rho_C) = 0, and (3) the pair (G(cl C), cl C) are disjoint as Lemma 4.3 requires. Or give a configuration where one fails. D. ASK Part 1 (DOOR-PROPER, readings). Decide which reading of "door" the lab needs and confirm DOOR-PROPER in reading C with a complete short proof: K = h(H_i([a,b])) contains u, v; K is a proper subcontinuum (the arc case and the Peano/Borsuk case); say whether a nonseparating plane Peano continuum has the fixed point property is SOURCE or RECALL (not on disk, lab says RECALL). Then say whether the ENDS of C being exactly u, v (not nearby) is used and where. Part 2 (DOOR-OUT). Prove or refute (1)-(3). Hints, not claims: v_C is a point of C, C is a short E_F-crosscut with ends u, v at distance >= m_1 from s^-, s^+; s is a crosscut in U with ends s^-, s^+ and pocket Pk(s) missing the strip; the claim v_C in Out(s) says C does not lie in Sh(s). If Pk(s) = Sh(s) fails (the pocket can be the unbounded component?), say exactly what the definition of Pk(s) must be and whether it is the strip or the pocket that is bounded. (INFERRED: if C is inside Sh(s), then u, v lie in K_s, and CH-T applied to a different crosscut may give a contradiction; check.) Part 3 (the near version and reduction). State the near version (ends of C within eta of u, v, no K through the ends) and say what extra lemma it needs; name ONE lemma L, strictly weaker than T1-EX and not a restatement of RIG-nonXS or of "E_F != E_G is impossible", whose proof or refutation would finish the door branch. Apply the gate: G1 statements side by side, G2 L implies the target, G3 L strictly weaker, G4 the hypothesis actually used (CP1 first clause, cm, nonseparation, h fixed-point-free at a named step). 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) E_F = E_G must not be used (it makes T1-EX circular). (C2) Planted negative: C has ends a, b with X irreducible between a and b (K = X forced); CH-T must say nothing; your proof must show where K proper is used. (C3) Do not use G o F = id or G = F^-1; if you use a relation between F and G beyond F|X = h, G|X = h^-1, name it. (C4) Symmetry: CH-T for (F,G) and (G,F) must not contradict E_F != E_G when ENDS-PROPER fails. (C5) Reading control: apply your DOOR-OUT argument to a door with Pk(s) containing the start v_C (a hairpin door), and to a door far from C, and say which of (1)-(3) fails or holds in each. (C6) Do not use the zero half of CP1 (A166). 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; 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 do not decide the channel question. (c) D0 from density, F-law, M-crosscut law, Lemma Z and BLOCK*: eliminated by the exact germ DD. (d) ENDS-PROPER equivalent to E_F = E_G: false (a161audsol); only ENDS-PROPER implies E_F = E_G. (e) Using a ball B alone for DOOR-OUT: the ball estimate proves disjointness, not the side condition (Astra, m-r2). (f) SHEET, paired hairpins, SO-, TRAP, Theorem R*, LEMMA_P, CP1 itself: other tasks. (g) Argon's Reading A proof of T1-EX (H_i([a,b]) in X): then Pk(s) does not exist, so it is not the door statement. (h) Restating DOOR-OUT as "no door image is an E-crosscut" (RESTATEMENT of T1-EX). 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: T1-EX PROVED (DOOR-PROPER and DOOR-OUT settled, inputs listed) | T1-EX CONDITIONAL (one of the two settled, the other named with exact hypotheses) | T1-EX FAILS (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_F = E_G; using G o F = id; applying CH-T to a crosscut whose ends are not both in K; calling H_i([a,b]) a door (it lies in X; the door is a crosscut in U); using the zero half of CP1 (A166 flags it as possibly defective; T1-EX needs only the first clause for G); proving DOOR-OUT only for a ball B while the access start v_C is on C far from s; claiming L2 or PER