m followup

Prompt

RESEARCH QUESTION (plane topology / BFMOT variation; node KP-REPL of Route I of the L2 route map: the law for small crosscuts outside the strip tail whose X-endpoint is hidden inside a KP-neighbourhood). 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 file PROOF_L2_v5 Sections 1 and 4, the lanes b_opus_rig_with_cp1 and b_opus_ndrepair; every DERIVED line is UNAUDITED; lines marked (INFERRED) are my bridging). - X is a nondegenerate nonseparating plane continuum, U := C \ X (connected, unbounded), T(A) the filled hull. h: X -> X is a homeomorphism with h^2 continuum-minimal (no proper subcontinuum K has h^2(K) = K). F is a plane extension of h (BFMOT sense). m_j := min_X |h^j(x) - x| for j = 1,2; m := min(m_1,m_2) > 0 (DERIVED from h^2 cm, Corollary 3.7). - A crosscut is an open arc q in U whose closure is an arc with two distinct endpoints a, b in X. Sh(q) is the bounded component of C \ (X u q), Out(q) the unbounded one. q^+ is the orientation in which q is traversed by a counterclockwise simple closed curve S = q u A (A a completing arc near X, X in T(S)). - var(F^j, q^+, X) is the BFMOT variation (S5) of q^+ for the map F^j, defined when F^j(cl q) and cl q are disjoint. SOURCE BFMOT arXiv:1004.0214v2: S5 = Definition 2.2.2 (var(f,A,S) = win of f(A) closed up by a path in T(S) \ {v}, v a junction point); S6 = Proposition 3.4.3 (the value does not depend on the completing curve S); S16 = Lemma 7.1.1 (completing arcs arbitrarily near X exist). - Access: a proper ray rho in C with rho disjoint from X; start v = rho(0). For a path beta off v with endpoints 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) additivity, (I2) reversal changes sign, (I3) I = 0 if beta misses rho, (I4) homotopy invariance in C \ {v}, (I5) two paths with equal endpoints in an open convex set B not containing v have equal I. - E_t: the unique geometric outchannel of BFMOT Theorem 6.2.1 for F, with external ray R_t; E_s, R_s the same for F^2. A crosscut q crosses R_t essentially (is an E_t-crosscut) if a bounded-side end of R_t lies in Sh(q) (BFMOT Def 3.6.4). v_q is the first point of cl q met by R_t from infinity, rho_q the rest of R_t. - SOURCE BFMOT 6.2.1, for KP-chords only (chords of the Kuratowski-Pelczynski neighbourhood Y_delta of X): for small delta, a KP-chord g has var(F, g^+, X) = -1 if g crosses R_t essentially and 0 otherwise (R-KP, the uniform-threshold reading is itself INFERRED). The same for F^2, R_s. Standing hypothesis S2 (printed form): cl g and F(cl g) are disjoint, for KP-chords g. - Thresholds (DERIVED, section 3.8 of v5): r' <= m/6; r_1 <= r' with F(B(y,r_1)) in B(h(y),r') for y in X; delta_0 <= r_1 with F(B(x,delta_0)) in B(h(x),r_1). For diam c < delta_0, c = (a,b): cl c in B(a,delta_0), F(cl c) in B(h(a),r_1), F^2(cl c) in B(h^2(a),r'), and the three disks B(a,m/6), B(h(a),m/6), B(h^2(a),m/6) are pairwise disjoint, so F^j(cl c) and cl c are disjoint (j = 1,2). In v5 the constant r_1 was also required to satisfy 2 r_1 < min(delta_t,delta_s), the CP1 thresholds; that extra requirement is not needed for disjointness. - XS (strip system, lane b_opus_rig_with_cp1): phi: [r_0,inf) x [-1,1] -> cl U a continuous map, injective on the open strip, with sections c_r := phi({r} x (-1,1)) crosscuts, sides H1 = phi(.,-1), H2 = phi(.,1) continuous into X, injective with disjoint images, widths w_r := diam cl c_r -> 0, and the c_r form a chain defining E_t (so Sh(c_r) = phi((r,inf) x (-1,1)), Lemma 2 of that lane). Hosts without arcs and hosts with doors on the sides are not covered. B. LEMMAS (status; where used) Statements to be settled (v5 numbering; DERIVED, UNAUDITED). - Lemma 4.3 (v5). Let j in {1,2}, c = (a,b) a crosscut with F^j(cl c) and cl c disjoint, v in c, rho an access with start v and rho meeting X u cl c only in v, B an open convex set containing F^j(cl c) and not containing v. Then var(F^j, c^+, X) = I(beta, rho) for every path beta in B from h^j(a) to h^j(b). Proof sketch in v5: complete c by A near X missing rho (S16), apply S5 to the curve S = c u A with a path alpha in T(S) \ {v}, then (I5) and S6. It never mentions CP1. - Lemma 4.4 (v5). For a crosscut d, w in U off cl d, any access rho_w with start w: I(d^sigma, rho_w) = sigma [w in Sh(d)], sigma = +-1. In particular the orientation d^+ is well defined and does not depend on the access. - Lemma 4.5 and Corollary 4.6 (v5). For crosscuts c, d with c and d disjoint: c lies in Sh(d) or in Out(d); if c lies in Sh(d) then Sh(c) lies in Sh(d); if neither lies in the other's shadow the shadows are disjoint. If c is an E_t-crosscut and v_c is in Sh(d), then Sh(c) lies in Sh(d) and d is an E_t-crosscut. - Lemma 4.2 (v5, the only item that uses CP1 there). There is delta_t > 0 such that every crosscut q with diam q < delta_t has var(F, q^+, X) = -1 if q crosses R_t essentially and 0 otherwise; the same for F^2, R_s, delta_s. Proof in v5: Theorem CP1 (conjugation and lens argument, DERIVED, UNAUDITED, conditional on BFMOT 6.1.2-6.1.5 and the uniqueness part of 6.2.1 for all triples satisfying S2). - Lemma W (lane b_opus_ndrepair, DERIVED, UNAUDITED, assumes R-KP). A section c_r is tau-walled if there is an arc L_r with the ends of c_r, L_r disjoint from c_r, the deep strip inside S_r := L_r u cl c_r, and every component of L_r \ X a KP-chord of diameter < tau (W1-W3). If r is tau_2-walled and w_r < tau_1 then var(F, c_r^+, X) = -1 and var(F^2, c_r^+, X) is in {0,-1}. Proof: ind(F^j, S_r) = 0 (no fixed point of F^j on T(S_r)) splits into the section term and finitely many KP-chord terms, each evaluated by R-KP. Needs the identification (i) (INFERRED) that the strip is a channel of a KP-neighbourhood with sides on the KP-arc. - Lemma 2 of lane b_opus_rig_with_cp1 (under XS, DERIVED): a small crosscut q inside the tail has its ends on H1 u H2; it is an E_t-crosscut iff the ends lie on opposite sides; then every section between the ends meets q, and H1[s1,s2], H2[s1,s2] lie in a w-neighbourhood of q ("q is a section up to small side arcs"). The lane then asserts "equal variation by Lemma 4.3 and (I5)" to transfer the laws from sections to q, and concludes that under XS Corollary J, Theorem AP and Theorem B need no CP1 beyond Lemma W. - LAB (previous 5-model round, unaudited consensus): Lemmas 4.3-4.6 are CP1-free from S5, S6, S16 and the m > 0 thresholds; for small finite-end strip crosscuts inside the tail the transfer from sections works (opposite side var(F) = -1, same side 0). All five answers stopped at the SAME step, stated below as KP-REPL. C. THE TARGET (KP-REPL) In Lemma 5.1 / Theorem 5.2 (V_2(c) = 1 + E(c), E in {-1,-2}, c a walled section) the law of Lemma 4.2 is applied to the components d of short paths near h(a), h(b) (and h^2(a)). Such a d is a small crosscut whose ends lie on X, but need not lie in the strip tail; an end may lie in X inside a KP-neighbourhood N (BFMOT Section 6), not on the exposed KP-arc. Statement KP-REPL: there is tau > 0 such that every crosscut d of diameter < tau arising this way has var(F, d^+, X) = -1 if d crosses R_t essentially and 0 otherwise, using only BFMOT 6.2.1 on KP-chords (S2 triples), S5, S6, S16, Lemma W, R-KP and the thresholds, but NOT Theorem CP1. D. ASK Part 1 (replacement). Given d with an endpoint e inside N, construct an explicit replacement: a chain of KP-chords k_1..k_n of diameter < tau plus arcs in X (an X-connector) from e to exposed KP endpoints, such that d u (connector) u (chords) bounds a region with no fixed point of F, so that index additivity gives var(F, d^+) = sum of KP-chord variations (each known by R-KP). Either give the construction with every hypothesis checked (in particular: does an arc of X from e to an exposed endpoint exist inside a small ball? X need not be locally connected), or give an explicit planar configuration where no small connector exists. Part 2 (weaker sufficient law). If Part 1 fails, find the weakest statement that Theorem 5.2 actually needs: for instance only the SUM over the components d of the path near h(a) (combined with the I(F(d), sigma_d) terms), or only d whose ends are both exposed, or only after replacing the path by another path with the same endpoints chosen inside the KP-neighbourhood. Prove that weaker statement, or reduce it to one named, checkable statement. Part 3 (path choice). Theorem 5.2 lets us CHOOSE the short paths near h(a), h(b). Can they always be chosen so every component d has both ends exposed (on KP-arcs) or inside the strip tail? Prove or give an obstruction (e.g. h(a) itself hidden in N with no exposed point of X nearby on the needed side). E. CONTROLS (all must hold) (C1) Sign check: on a walled section c_r the result must agree with Lemma W (var(F) = -1). (C2) Say for each crossing curve whether its ends lie on X or in the interior of N; Lemma W and R-KP need ends on the KP-arc. (C3) No local connectedness of X may be assumed; an X-connector must be justified (a subcontinuum of small diameter joining two points is acceptable only if its existence is proved). (C4) Tame case: if X is the closed unit disc no h with h^2 cm exists; say so rather than testing on it. F. DEAD ENDS (not progress) (a) Proving Theorem CP1 (conjugation/lens route) for all small crosscuts. (b) Re-proving Lemmas 4.3-4.6 or the strip-tail transfer (done). (c) Counting signed crossings with transversality. (d) Using sums of var(F,d) and I(F(d),sigma_d) separately (no finite support; only combined sums are finite). (e) Quoting BFMOT 6.2.1 for a crosscut that is not a KP-chord. G. FORMAT Line 1 exactly: VERDICT: KP-REPL PROVED | WEAKER LAW SUFFICES (statement, proof) | PATH CHOICE AVOIDS IT (proof) | OBSTRUCTION (configuration) | 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; UNKNOWN if unsure a cited result exists. RED FLAGS: assuming Lemma 4.2 outside the strip tail; an X-connector without proof (X not locally connected); treating a hidden endpoint as exposed.