M R6

Prompt

RESEARCH QUESTION (plane topology / BFMOT variation; do the crosscut laws (variation = intersection, sign well defined, laws -1 / 0) need Theorem CP1 once the strip system XS is assumed (node LAWS of Route I of the L2 route map)). 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. C. THE TARGET Node LAWS: Lemma 4.2 (laws) with the sign convention of Lemma 4.4 and the nesting of Corollary 4.6, together with the downstream uses in Lemma 5.1 and Theorem 5.2 (master identity V_2(c) = 1 + E(c)), where the laws are applied to the components d of short paths near h(a), h(b), h^2(a), and to c itself. D. ASK Part 1 (Lemmas 4.3, 4.4, 4.5, Corollary 4.6 without CP1). Check, step by step, that these four statements use only S5, S6, S16, Lemma 3.4 (T of a thin neighbourhood of X stays near X), the thresholds r', r_1, delta_0 of DEFS (without the CP1 requirement on r_1), and the hypothesis F^j(cl c) disjoint from cl c. List any other input. If one is hidden (for example: S6 assumes S2 for the map, or the completing arc A of S16 must meet a junction clause), say exactly which. Part 2 (laws in the strip tail). Under XS plus R-KP and Lemma W, prove or refute: for every small crosscut q inside the tail, var(F, q^+, X) = -1 if the ends of q lie on opposite sides and 0 if on the same side (and the F^2 version in {0,-1}, -1 exactly when q crosses R_s essentially). The two steps to settle are (a) from a small opposite-side q to a section c_r: the access start v_q on q and the access start on c_r differ, and Lemma 4.3 compares only F^j(q) with a path beta in a convex set B avoiding v; give an explicit B and beta, or an explicit obstruction, and (b) the same-side case q with both ends on H1: show var = 0 from Lemma W-type index arguments, or produce a configuration where it fails. Part 3 (where CP1 is really used). The master identity (Lemma 5.1, Theorem 5.2) applies Lemma 4.2 to arbitrary small pieces d of paths near h(a), h(b), which need not lie in the strip tail. Decide: does Theorem 5.2 (V_2(c) = 1 + E(c), E in {-1,-2}) survive under XS without CP1 for c a section, or must the laws be known for arbitrary small crosscuts near h(a)? Give the minimal sub-statement of Lemma 4.2 that is needed, as a precisely stated lemma, and either prove it from KP-chords (BFMOT 6.2.1 plus a completing-chain argument) or reduce it to one named, checkable statement. Expected value: a clean list of which of the four laws-dependent steps survive without CP1. Each answer must state which inputs are SOURCE (quoted), which are DERIVED here, which are INFERRED. E. CONTROLS (all must hold) (C1) In Part 1, the proof of Lemma 4.3 must show where F^j(cl c) and cl c disjoint is used and where it comes from (threshold r_1, m > 0); this hypothesis must not be assumed from S2 for a non-KP crosscut. (C2) In Part 2(a), the sign of the result for a section c_r must agree with Lemma W (var = -1 for F); an argument giving +1 or 0 contradicts a DERIVED lemma and must be rechecked. (C3) Any claimed reduction must say whether the ends of the crosscut lie on X or in the interior of the KP-neighbourhood: Lemma W needs ends on the KP-arc. (C4) The tame case: if X is the closed unit disc, no homeomorphism with h^2 cm has the structure above; say so rather than testing on it. F. DEAD ENDS (not progress) (a) Proving Theorem CP1 for arbitrary small crosscuts: the conjugation/lens route (PROOF_CP1) is unaudited and Lemma W's remark records that completing a crosscut with an end hidden inside the KP-neighbourhood by KP-chords fails. (b) Re-proving Theorem R* or RIG (other tasks). (c) Counting signed crossings with transversality: the calculus I is defined without any regularity. (d) Using the sums of var(F,d) and I(F(d),sigma_d) separately in Lemma 5.1: they need not have finite support; only the combined sums are finite. G. FORMAT Line 1 exactly: VERDICT: LAWS CP1-FREE UNDER XS (proof) | CP1-FREE EXCEPT ONE NAMED STEP (step, proof of the rest) | NEEDS CP1 AT (named step, 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; no theorem beyond its hypotheses; UNKNOWN if unsure a cited result exists. RED FLAGS: assuming Lemma 4.2 for crosscuts outside the strip tail; treating a small tail E-crosscut as equal to a section without proving the variation is unchanged; using F(cl q) disjoint from cl q for diam q < delta_0 and then quoting a threshold that was built from delta_t; quoting BFMOT 6.2.1 for a crosscut that is not a KP-chord; claiming CP1 is not needed because the proof of Lemma 4.3 does not mention it, without checking the hypothesis F^j(cl c) disjoint from cl c