m14

Prompt

RESEARCH QUESTION (plane continua / intersection numbers of paths; the wall-limit step of the relative form L-MPN-REL of WRAP-MPN (does the limit wall of a bad sequence carry h^2-invariance or a proper-subcontinuum structure?)). 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 (S) (compressed from the lab notes D2608 and D2612 and the Arena answer A164; every DERIVED item is a claim of a single agent and is UNAUDITED; lines marked (INFERRED) are my bridging). - X is a nondegenerate nonseparating plane continuum, indecomposable with empty interior, U := C \ X (connected, unbounded). h: X -> X is a homeomorphism with h^2 continuum-minimal (cm: no nonempty proper subcontinuum K of X has h^2(K) inside K). Then m_j := min over x of d(x, h^j x) > 0 for j = 1, 2. F: C -> C is a perfect extension of h (continuous, closed, compact fibres). The lab target L2 is that (S) is impossible. - A crosscut q is an open arc in U whose closure is an arc with two distinct endpoints q^-, q^+ in X. Sh(q) is the bounded component of C \ (X u q), Out(q) the unbounded one. The wall Z_q := cl Sh(q) cap X is a continuum containing q^- and q^+ (Janiszewski). In (S), F has a unique outchannel E_t with external ray R_t and Pr(E_t) = X (SOURCE-lab, BFMOT as used in PROOF_L2_v5). A crosscut q is an E_t-crosscut if a tail of R_t lies in Sh(q); v_q is the first point of cl q met by R_t from infinity, rho_q the part of R_t from v_q to infinity. - Intersection number: for a path beta missing the start v of an access rho, with ends off rho, I(beta, rho) := winding number about v of beta followed by a path in C \ rho. (I3) a path in C \ rho has I = 0; (I4) I is invariant under homotopy rel endpoints in C \ {v}; (I5) two paths with the same endpoints in an open convex set missing v have the same I. - Thresholds: r' <= m/6 with m := min(m_1, m_2); r_1 <= r' with F(B(y, r_1)) inside B(h(y), r') for y in X; delta_0 <= r_1 with F(B(x, delta_0)) inside B(h(x), r_1). omega_h is a modulus of continuity of h. - The node. c = (a, b) is an E_t-crosscut with diam c < delta_0 and access rho_c from v_c. gamma = gamma_c := the straight segment [h(a), h(b)], inside B(h(a), r_1); Lgam := |h(a) - h(b)| > 0. The pieces of gamma are the components of gamma \ X: crosscuts, pairwise disjoint subsegments of gamma. An M-piece m is a piece that is not an E_t-crosscut. n_c(m) := I(F o m, rho_c). Fact T1: I(m, rho_c) = 0 for an M-piece. - L-MPN-REL: for every kappa in (0, 1] there is delta(kappa) > 0 such that for every E_t-crosscut c with diam c < delta(kappa) and every M-piece m of gamma_c with diam m >= kappa Lgam one has n_c(m) = 0. WRAP-MPN-REL is its negation: there are kappa > 0, E_t-crosscuts c_n with diam c_n -> 0 and M-pieces m_n of gamma_{c_n} with diam m_n >= kappa Lgam_n and n_{c_n}(m_n) != 0. (The absolute form eta(c) = omega_h(diam c) is vacuous: diam m <= Lgam <= omega_h(diam c).) B. LEMMAS (status; where used) Known in (S) (DERIVED by single agents, UNAUDITED unless marked). - Theorem WRAP. For a path m with ends m^-, m^+ in X, a continuum Q in X containing them, and an access rho with start v not in F(cl m): if v lies in the unbounded component of C \ (h(Q) u F(cl m)), then I(F o m, rho) = 0. So n_c(m) != 0 forces h(Q) u F(cl m) to separate v from infinity for every continuum Q in X through m^-, m^+. - Theorem BP and Cor BP1. For a crosscut m with diam m small and w in U \ F(cl m) with I_w(F o m) != 0: dist(w, h(Z_m)) <= omega_1(diam m). Hence for a bad M-piece m of gamma_c: every point of cl c is within omega_1 of h(Z_m); Z_m meets B(h^-1(a), omega_2) and B(h^-1(b), omega_2) with omega_2 = omega_{h^-1}(omega_1); diam Z_m >= m_2 - omega_2 - omega_h(diam c). All errors tend to 0 as diam c -> 0. - Cor BP2 (dichotomy). Either (BP-dense) a subray of the X-end of R_t lies in one component W' of U \ F(cl m) with I_{W'}(F o m) != 0 and then h(Z_m) is omega_1-dense in X; or (BP-cancel) the R_t-segment from v_c to a deep subray crosses F o m with algebraic number -n_c(m). BP and BP2 need ND-M (no non-E_t small crosscut has an eta_0-dense wall); ND-M is FALSE in every host with a second total channel E' != E_t. - Pull-back identity (no CP1). With tau := [h^-1 m^-, h^-1 m^+]: n_c(m) = I(F^2 o tau, rho_c). Note tau is a segment with ends at distance <= omega_{h^-1}(diam m). - A164 (Arena, one agent, unaudited, PARTIAL). PROVED there: (1) L-MPN-REL is equivalent to the existence of a uniformly little-o relative threshold eta(c) = o(Lgam), but a prescribed smaller rate is a formally stronger statement (quantifier counterexample, abstract ratios only). (2) For fixed kappa at most floor(1/kappa) pieces of gamma have diam >= kappa Lgam. (3) For a bad sequence (c_n, m_n), pass to a subsequence with a_n -> x0 and Z_{m_n} -> Z_infty in the Hausdorff metric; then Z_infty is a continuum in X with h^-1(x0) and h(x0) in Z_infty (the second because the ends of m_n lie in gamma_n -> h(x0) and in Z_{m_n}), diam Z_infty >= m_2, and x0, h^2(x0) are in h(Z_infty). (4) If h^2(Z_infty) were inside Z_infty then cm would force Z_infty = X, so a proper h^2-invariant limit is impossible in (S); but the limit argument yields neither invariance nor properness: if Z_infty = X, invariance is automatic and no contradiction arises. cm does NOT give wall invariance. OPEN even for walls = X. C. THE TARGET Statement WALL-INV. In (S), for every bad sequence (c_n, m_n) as in WRAP-MPN-REL, the limit wall Z_infty (after passing to a subsequence) satisfies one of: (i) Z_infty is a proper subcontinuum of X with h^2(Z_infty) inside Z_infty; (ii) Z_infty = X and a contradiction with (S) follows from the data (BP2 dichotomy, pull-back identity, Theorem WRAP, cm); (iii) a contradiction with (S) otherwise. Any of (i)-(iii) proves L-MPN-REL (WRAP-MPN-REL is impossible). The task is to prove WALL-INV, to give a germ where the wall limit is neither invariant nor X while all the known constraints (WRAP, BP1, BP2, pull-back, finiteness) hold, or to isolate the missing hypothesis exactly. D. ASK Part 1 (proper walls). Suppose Z_infty is a proper subcontinuum of X. Decide whether h^2(Z_infty) is inside Z_infty, using the pull-back identity: n_{c_n}(m_n) = I(F^2 o tau_n, rho_{c_n}) with tau_n -> h^-1(x0)-side, the walls of the pieces of tau_n, and the fact that Z_{m_n} converges to Z_infty. State the quantifier order you use (a fixed kappa, a subsequence, a diagonal choice) and say whether tau_n, as a short segment near h^-1(x0), has wall data comparable with Z_{m_n}. (INFERRED: apply the same wall analysis to the pieces of tau_n, one level down, to get a chain Z_infty, Z', Z'' ..., and ask whether the chain closes up.) Part 2 (full walls, Z_infty = X). Decide the sub-case Z_{m_n} -> X separately. Here BP-dense is automatic and BP1 gives no contradiction. State exactly what extra hypothesis is needed (a second total channel E' with a bay at x0? CP1? nonseparation of X used at a named step?), or give a local germ at three scales (a bad piece m, its wall, its F-image and rho_c) that satisfies Theorem WRAP, Cor BP1 and BP2 and the pull-back identity, saying which part of (S) is not checked. Part 3 (counting). Use floor(1/kappa) in a way A164 did not: for fixed kappa there are at most N = floor(1/kappa) pieces per crosscut, each with its own wall, and the walls of disjoint pieces of one segment are nested or disjoint (Janiszewski, INFERRED). Decide whether the combinatorics of at most N nested/disjoint walls for each c_n forces the limit Z_infty to be a limit of walls of a bounded family, hence either invariant under the F-induced dynamics on the family or leaving it to a proper piece. If this gives nothing, say why. Part 4 (reduction, only if Parts 1-3 are done). If you reduce to a lemma L, it must pass the restatement gate: L strictly weaker than WALL-INV or WRAP-MPN-REL, stated with no occurrence of "bad sequence", n_c, or "piece", and L must use cm or the definition of E_t at a named step (G4). A lemma "the limit wall is h^2-invariant" is WALL-INV itself and is a RESTATEMENT unless it comes with a proof or with a weaker sufficient geometric condition on the walls. Each claim must be labelled PROVED, SOURCE, LAB, DERIVED, CONJECTURE, OPEN or RESTATEMENT. Do not assume ND-M or E_s = E_t. E. CONTROLS (all must hold) (C1) Ambient control: if h were the restriction of a plane homeomorphism H with H(X) = X, then L-MPN-REL would be provable by prime-end transport, but H has a fixed point (Cartwright-Littlewood, Bell, Kuperberg; RECALL) so (S) is empty. A proof must not use an extension of h to a plane homeomorphism. (C2) Planar periodic control: X = [-1,1] x {0}, h(x,0) = (-x,0), F(x,y) = (-x,y), m the upper half-circle of radius 1/2 through (-1/2,0) and (1/2,0), rho = {0} x [1/4, inf): F o m crosses rho once and I = +-1. h has a fixed point, so this is outside (S); no argument may exclude it by a method that also excludes a hypothetical (S)-system. (C3) Quantifier control: a proof that works with eta(c) = omega_h(diam c) in place of kappa Lgam is the vacuous form. (C4) Second-channel bay: for Z_m = X say what the argument does with a bay at the X-end of a second total channel E'. (C5) Invariance control: in the periodic control C2 the wall of the half-circle is [-1/2,1/2] x {0}, a proper h^2-invariant subcontinuum (type (i) of WALL-INV), which is exactly what cm forbids in (S); an argument for Part 1 must say how it separates this from an (S)-system, and what it does with a wall that is h-invariant but not h^2-minimal. F. DEAD ENDS (not progress) (a) Theorem WRAP alone (a necessary condition; it only removes pocket germs). (b) Theorem BP and Cor BP2 as a proof: they need ND-M, which fails with a second total channel. (c) The base pull-back identity: local visibility fails generically (regress to RIG-type gap data). (d) The swept-angle cocycle (a coboundary; restatement of L-MPN). (e) Prime-end transport (Lemma AMB): proves L-MPN only for ambient maps, where (S) is empty. (f) The absolute form eta(c) = omega_h(diam c) (vacuous). (g) A164: cm applied to Z_infty gives wall invariance (FAILED: cm says nothing unless h^2(Z_infty) is inside Z_infty is already known). (h) Counting floor(1/kappa) pieces to force n_c(m) = 0 (the count is finite but does not force vanishing). (i) Argon 55-57 style headlines "REDUCED to Lemma L" where L restates the node (Argon 58 and the gate classify them as RESTATEMENT). 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: WALL-INV PROVED (the limit wall is h^2-invariant or otherwise contradicts (S); proof) | WALL-INV FAILS (explicit configuration or local germ) | PARTIAL (one named sub-claim settled, which) | REDUCED (to ONE named lemma L strictly weaker, reduction proved, G1-G4 answered) | 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 ND-M or E_s = E_t; treating continuum-minimality as giving h^2(Z) inside Z for a limit wall (A164 showed it does not); letting eta(c) depend on omega_h so that it dominates diam gamma (vacuous); using an extension of h to a plane homeomorphism; declaring REDUCED with a lemma that restates WRAP-MPN-REL; counting floor(1/kappa) pieces as if the count forced n_c(m) = 0