m21

Prompt

RESEARCH QUESTION (plane continua / intersection numbers of paths; WRAP-MPN of L-MPN under the extra hypothesis E_s = E_t: after the CP1 terms cancel, only the ray-detour terms D_k(t) of the base pull-back remain). 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 lab notes D2608, D2612 and the Opus lane wrapmpn_prove_opus_1102; every item marked DERIVED is the 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 (so U together with infinity is simply connected). 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); m_j := min_x d(x, h^j x) > 0 for j = 1, 2; m := min(m_1, m_2). F: C -> C is a perfect extension of h (continuous, closed, compact fibres). The lab target L2 is that (S) is impossible; we work inside a hypothetical (S). - 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). In (S), F has a unique outchannel E_t with external ray R_t, F^2 has a unique outchannel E_s with ray R_s, and Pr(E_t) = Pr(E_s) = X (SOURCE-lab, BFMOT as used in the lab's proof file PROOF_L2_v5). A crosscut q is an E_t-crosscut if a tail of R_t lies in Sh(q); then v_q is the first point of cl q met by R_t from infinity and rho_q the part of R_t from v_q to infinity (the access of q); s_q is the ray depth of v_q. q is deeper than c if q is inside Sh(c). - Intersection number: for a path beta missing v, 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 homotopy invariant rel endpoints in C \ {v}; (I5) two paths with the same endpoints in an open convex set missing v have the same I; (I6) additivity: for a ray split at a point, I(beta, rho) is the sum of the two parts. I_w(beta) denotes I against an access from w in U. Lemma SP (DERIVED): for paths whose endpoints lie in X, I_w depends only on w and is constant on the components of U minus the path. - Thresholds: r' <= m/6; 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); r_0 with F(B(y, r_0)) inside B(h(y), r_1). - The node. c = (a, b) is an E_t-crosscut with diam c < delta_0 and access rho_c from v_c. gamma := the segment [h(a), h(b)], inside B(h(a), r_1). An M-piece m is a component of gamma \ X that is not an E_t-crosscut; n_c(m) := I_{v_c}(F o m). L-MPN: n_c(m) = 0 for every M-piece with diam m > eta(c), eta(c) -> 0 as diam c -> 0. WRAP-MPN := the M-pieces with n_c(m) != 0. Fact T1: I_{v_c}(m) = 0 for an M-piece. - Hypothesis H-EQ (the object of this task, NOT known): E_s = E_t, i.e. R_s = R_t as a ray (so F and F^2 share the access rho_q of every E_t-crosscut q). B. LEMMAS (status; where used) Known (DERIVED by one agent, wrapmpn_prove_opus_1102 section 4, UNAUDITED, compressed here). - Base pull-back. For an M-piece m put x^+- := h^-1(m^+-) (both within omega_{h^-1}(omega_h(diam c)) of a) and tau := [x^-, x^+]; let t range over the pieces of tau (crosscuts near a). Then F o tau and m are paths from m^- to m^+ inside B(m^-, r_1), a disk that misses v_c. (PB0) n_c(m) = I_{v_c}(F^2 o tau). (PB1) sum over t of I_{v_c}(F o t) = I_{v_c}(m) = 0. (PB2) sum over t of I_{v_c}(F^2 o t) = n_c(m). Only finitely many terms are nonzero. (Proof: (I5) in the convex disks, plus a sup argument along the compact set of parameters of tau cap X, whose gap-jumps are the I-terms of the pieces.) - CP1 split (v5 Lemma 4.2 and its F^2 form; UNAUDITED). For a piece t oriented along tau, with eps(t) = +-1 the orientation sign: I_{v_t}(F o t) = -eps(t)[t is an E_t-crosscut] and I_{v_t}(F^2 o t) = -eps(t)[t is an E_s-crosscut], for every v_t in t. - Detour terms. D_k(t) := I_{v_c}(F^k o t) - I_{v_t}(F^k o t), k = 1, 2, where for a non-E_t piece v_t is any point of t. F(cl t) lies in B(m^-, r_1) and F^2(cl t) in B(h(m^-), r'), both far from a. - Lemma LV (DERIVED). If v_t and v_c lie in one component of U \ (cl B(m^-, r_1) u cl B(h(m^-), r')), then D_1(t) = D_2(t) = 0. Cor PB-LV: if every piece satisfies LV then n_c(m) = -sum eps(t)[t is an E_s-crosscut]; if moreover the E_s-pieces of tau are its E_t-pieces, n_c(m) = 0. - (INFERRED, to be checked in Part 1.) Under H-EQ the CP1 terms of PB1 and PB2 coincide, so subtracting gives n_c(m) = sum over ALL pieces t of tau of (D_2(t) - D_1(t)), where only pieces with D_k(t) != 0 contribute. C. THE TARGET Statement DETOUR. In (S) with H-EQ, for every M-piece m of gamma_c with diam m > eta(c), sum_t (D_2(t) - D_1(t)) = 0. Equivalent (INFERRED): n_c(m) = 0. By Lemma SP, D_k(t) is the difference of the degrees of F^k on Sh(t) at the two points v_c and v_t (Lemma DG), so D_k(t) != 0 forces v_c and v_t into different components of U \ F^k(cl t), whereas LV asks them into one component of U minus the two closed balls. The task concerns exactly the gap between these two conditions. D. ASK Part 1 (the reduction). Check the INFERRED identity: derive n_c(m) = sum (D_2 - D_1) from PB1, PB2 and the CP1 split under H-EQ; say whether the zero clause of CP1 is used for non-E_t pieces and whether the sum is finite. State which of PB, CP1 and H-EQ is used at each step. Part 2 (geometry of a nonzero detour). Characterise when D_k(t) != 0, in terms of R_t alone: v_c and v_t on R_t (for an E_t piece) or v_t arbitrary (for a non-E_t piece), the compact sets F(cl t) and F^2(cl t) in the two far balls, and the segment of R_t between the two points. Prove or refute: a nonzero D_k(t) forces the segment of R_t between v_t and v_c to enter the ball B_k and to cross F^k o t with nonzero algebraic number. Decide whether this is compatible with the depth order: is t shallower or deeper than c, and where is the entry point of R_t into B_k relative to s_t, s_c? Part 3 (decide DETOUR). Using Part 2, either prove DETOUR (for instance by showing that the E_t ray cannot make an excursion into the far balls between depths s_t and s_c without producing a nonzero term that is cancelled by another piece), or give a consistent configuration (X locally, h, F, a, c, m, a piece t, the ray R_t at three scales) with n_c(m) != 0, saying which parts of (S) you did not check (cm, indecomposability, planarity of the global X), or give the exact geometric criterion (CRITERION) and show it is not a restatement of n_c(m) != 0. Part 4 (only if Parts 1-3 are done). If you reduce to a lemma L it must pass the restatement gate below, and L must use H-EQ or the depth order at a named step. A lemma "the ray misses the balls between s_t and s_c" is LV itself and counts as a RESTATEMENT unless it comes with a reason that is strictly weaker than n_c(m) = 0. Each claim labelled PROVED, SOURCE (exact reference), LAB, DERIVED, CONJECTURE, OPEN or RESTATEMENT. UNKNOWN if unsure a cited result exists. Do not assume ND-M. E. CONTROLS (all must hold) (C1) Ambient control: if h were the restriction of a plane homeomorphism H with H(X) = X, prime-end transport would give n_c(m) = 0, but H has a fixed point (Cartwright-Littlewood, Bell, Kuperberg; RECALL) so (S) is empty; no step may use an ambient extension of h. (C2) Planar periodic control (by hand): 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, I = +-1; h has a fixed point, so this lies outside (S), and no argument may exclude it by a method that also excludes a hypothetical (S)-system; say what the detour terms are on this control. (C3) The detour terms are about the access, not about the endpoints: a proof must not replace I_{v_c} by a winding number about the point v_c alone (that is the convex-ball argument (I5), which gives zero for any path in a ball missing v_c and is wrong for I against a ray). (C4) Quantifier control: the thresholds r', r_1, r_0, delta_0 are fixed before c; eta(c) -> 0 is allowed to depend on (X, h, F) but not on m. F. DEAD ENDS (not progress) (1) Theorem WRAP alone (a necessary condition; it kills only the pocket germ). (2) Theorem BP and Cor BP2 as a proof: they need ND-M, which fails in a host with a second total channel. (3) The unconditional base pull-back: local visibility fails generically and the residue is RIG-type hole-gap data; this task is the case H-EQ, where the CP1 terms cancel exactly and only ray-detour terms remain, so no regress to hole-gap data is allowed unless it is derived from D_k. (4) The swept-angle cocycle (a coboundary; a restatement of L-MPN). (5) Prime-end transport for an ambient H (proves L-MPN only where (S) is empty). (6) The absolute threshold eta(c) = omega_h(diam c) (vacuous: it dominates the length of gamma). (7) The wall-limit step of the relative form (other task): do not use the limit wall Z_infty. 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: DETOUR-ZERO PROVED (n_c(m) = 0 under E_s = E_t; proof) | DETOUR-NONZERO CONSISTENT (explicit ray/ball configuration with n_c(m) != 0 meeting every listed law) | CRITERION (exact necessary and sufficient geometric condition on R_t for a nonzero detour term, proved) | PARTIAL (one named sub-claim settled, which) | 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 any density of a wall; using an extension of h to a plane homeomorphism; reading Lemma LV as a theorem when the ray enters the image balls; counting finitely many pieces as if the count made the terms vanish; reducing to a lemma that restates D_k(t) != 0; claiming L2 or PER or any downstream edge; using E_s = E_t to conclude anything outside this task