m52/54

Prompt

RESEARCH QUESTION (plane continua / the HUG wall (W-PE, ACC-FLIP): in an (S)-instance with an isolated outchannel prime end theta_F, can small Theta-free crosscuts hug theta_F while shrinking onto a point of the other composant?). 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 reports api_opus_acc_flip, api_opus_composant_wall, api_opus_flip_mouth, b_opus_m2_quad, api_opus_pocket_force and the audited PROOF_L2_v2 Lemmas 4.2-4.6; single-agent statements, UNAUDITED unless a status says otherwise; INFERRED marks the spec author's own reading). - (S): X is a nondegenerate nonseparating plane continuum, U := C \ X (connected, unbounded), h: X -> X a homeomorphism with h^2 continuum-minimal (cm: no nonempty proper subcontinuum K with h^2(K) inside K). So h has no fixed point and no point of period 2; PPF below means h is periodic-point-free (the lab's usage, INFERRED; cm of h^2 gives periods 1 and 2). X is indecomposable. A composant is a union of all proper subcontinua containing a given point; the composants are dense, pairwise disjoint, and h permutes them; a proper subcontinuum lies in one composant. The class of (S)-instances may be empty (that is PER); the question concerns its members only. - Prime ends: psi: U u {infinity} -> D conformal, psi(infinity) = 0; prime ends = points of S^1. A crosscut q is an open arc in U whose closure is an arc with two distinct ends in X. Sh(q) is the bounded component of U \ q (the shadow), Out(q) the other; I_q in S^1 is the open arc with psi(Sh q) the cap over it (the ends of I_q are the corners of q). Im(zeta) = intersection over r of cl f(D cap B(zeta, r)) (f = psi^-1), the impression, a subcontinuum; Pr(zeta) the principal set. A prime end is full if Im(zeta) = X; Theta_X is the (closed) set of full prime ends; a gap is a component of S^1 \ Theta_X. A crosscut is Theta-free if cl I_q misses Theta_X. - Extensions and the outchannel (BFMOT Theorem 6.2.1, SOURCE; read in the lab as CP1): h has a perfect plane extension F (continuous, F|X = h, finitely many preimages of points off X, no preimage in X of points off X; not a homeomorphism) with a unique outchannel, a prime end theta_F with Pr(theta_F) = X (so Im(theta_F) = X and theta_F is full) and external ray R_t; likewise G for h^-1 with theta_G != theta_F (Cor P1, LAB). A crosscut q crosses R_t essentially (is an E_t-crosscut) iff theta_F is in I_q. There is delta_t > 0 with: every crosscut q, diam q < delta_t, has var(F, q) = -1 if q is an E_t-crosscut and var(F, q) = 0 otherwise (Lemma 4.2, SOURCE S14 read as CP1; by Lemma 4.3 var(F,q) is the winding-type intersection number I(beta, rho_v) of any short path beta in a small ball from h(a) to h(b) with an access rho_v starting at any v in q).- (2G), the window (a hypothesis of the lab's records, not proved): an open arc J containing theta_F and every arc I_q used below, with Theta_X cap cl J = {theta_F}. Gamma_+ and Gamma_- are the gaps of S^1 \ Theta_X on the two sides of theta_F (their near ends are theta_F); C_+ and C_- the composants with: for every zeta in Gamma_+ (resp. Gamma_-), Im(zeta) lies in C_+ (resp. C_-) (Lemma GAP-IMP, DERIVED here, below). SWAP (LAB, DERIVED): h(C_+) = C_- and h(C_-) = C_+; only h^2(C_+-) = C_+- is load-bearing. C_+ != C_- is part of the lab's frame (INFERRED: check). - Non-theta_F crosscut: a crosscut p with theta_F not in cl I_p. In the window, with I_p inside J, this is exactly Theta-free (Theta_X cap cl J = {theta_F}); a Theta-free crosscut with I_p in Gamma_+ has both ends in C_+ (GAP-IMP, ends are Pr of the corners, inside Im). - theta_F-HUG (the object under test). A sequence of crosscuts p_n with I_{p_n} inside Gamma_+ (or inside Gamma_-), theta_F not in cl I_{p_n}, with |I_{p_n}| -> 0 and cl I_{p_n} -> {theta_F} (Hausdorff), diam p_n -> 0, p_n -> x (a point of X), and x in the other composant (not C_+, resp. not C_-). The shadows Sh(p_n) are NOT assumed small in Euclidean diameter (they lie deep in every outchannel shadow Sh(d) of an E_t-crosscut d disjoint from the p_n, HUG-2G(iii) below). Read 'the shadows converge onto x' as: the arcs p_n converge to x (INFERRED: check).- SD (sheet duality, DERIVED cm-free, Sol-confirmed with minor repairs): for a closed round disc E with X not inside E and u, v in X cap int E, u and v lie in one component of X cap E iff phi_w(u) = phi_w(v) for every w in U \ E (sheets phi_w from the winding intersection I(., rho_w) of paths in int E; testing w within epsilon of the boundary of E suffices). B. LEMMAS (status; where used) Known (status as given; nothing here is VERIFIED by two audits). - Lemma GAP-IMP (DERIVED, api_opus_acc_flip; uses IMP*, CONFIRMED by a Sol audit: if cl I_c misses Theta_X then K_c := cl Sh(c) cap X is a proper subcontinuum): for every gap Gamma there is a composant C_Gamma containing Im(zeta) for all zeta in Gamma. - Lemma JUMP + Theorem ACC-FULL (DERIVED, UNAUDITED, audit owed; inputs GAP-IMP, LAND, ARC, upper semicontinuity of Im): let sigma be an arc from X to X in a closed disc, whose pieces (components of sigma \ X) are all Theta-free, with end labels in different composants. Then the label has a jump point t*, infinitely many pieces converge to t*, and with x = sigma(t*) the corner limits A(t*) satisfy: x is in Im(theta) for every theta in A(t*); A(t*) meets Theta_X; every sequence of pieces converging to t* whose label differs from comp(x) has all corner limits full; |I_p| -> 0 along them. So a composant jump of a theta-blind path is exactly a one-sided hug of a full prime end by small Theta-free crosscuts. - Cor HUG-2G (DERIVED, (2G)): every jump of a theta-blind path with all pieces in the window is a theta_F-hug from Gamma_+ or Gamma_- (pieces carry the label C_+- of their gap, differing from comp(x)). (iii) Depth: for any E_t-crosscut d disjoint from the hugging pieces p_n, eventually I_{p_n} is inside I_d, so p_n lies in Sh(d) (Lemma 4.5 nesting: c inside Sh(d) gives Sh(c) inside Sh(d)); the BULGE points of Sh(p_n) meeting the ray R_t are arbitrarily deep. - INVIS-Theta (DERIVED from the audited Lemma 4.4, Lemma 5.1 step 1 and Cor 4.6): if c is an E_t-crosscut and d a crosscut disjoint from c with v_c in Sh(d), then Sh(c) is inside Sh(d) and d is an E_t-crosscut (Cor 4.6). Consequence: for an arc sigma whose pieces are all non-theta_F and an E_t-crosscut c disjoint from sigma, I(sigma, rho_{v_c}) = 0. So no theta_F-test point sees a theta-blind path, and no law of the lab evaluates F at a point of Sh(p_n) lying on no E_t-crosscut disjoint from p_n. - Prop NO-LOCAL-INJ (DERIVED; Schoenflies and invariance of domain, then Cartwright-Littlewood or Bell 1978, RECALL, not quoted from a full text on disk): in (S) no perfect extension F is injective on any neighbourhood of X (else h extends to a plane homeomorphism, which has a fixed point). So there is no induced prime-end action of F. CHANNEL-INJ (is some F injective on a channel region f(D cap B(theta_F, r)) with image in U?) is OPEN. - Impression transport is vacuous (DERIVED): Im = X for every full prime end, including theta_F; a theta_F-hug has no landing point. - Prop DEC (b_opus_m2_quad, Tier 0 builder, UNAUDITED): the lab's laws split into a theta layer, a composant layer and a component layer; the only couplings need positive component joins or attributed mouths, which no law of (S) supplies. Record M2-quad passes every listed law at every generation. - NO-HUG is equivalent to ACC-FLIP (no theta-blind accumulation flip in a type-F cell), so a proof of NO-HUG is the missing planar line, not a reduction. C. THE TARGET In (S) with (2G) and theta_F != theta_G: prove or refute 'NO-HUG: no sequence of non-theta_F crosscuts p_n with I_{p_n} inside Gamma_+ (or Gamma_-), cl I_{p_n} -> {theta_F}, and p_n -> x with comp(x) != C_+ (resp. C_-) exists'. Acceptable outcomes: NO-HUG PROVED (each law used named, planarity marked); HUG CONSISTENT (explicit Tier 0 data or abstract configuration meeting every law above, the first missing law named; a finite chain is not a host); REDUCED to one named lemma L (gate G1-G4); PARTIAL. Not asked: PER, L2, the far-hug case (a full end other than theta_F), CHANNEL-INJ. D. ASK Part 1 (method). Name ONE method not on the dead-end list and say why it sees inside Sh(p_n) where INVIS-Theta, impression transport and S^1-level laws see nothing. Candidates, none claimed to work: (i) PAIR var(F, p_n) = 0 (Lemma 4.2: p_n is small and non-essential) with PPF and cm of h^2: F(p_n) is a small arc from h(a_n) to h(b_n), both ends in C_- = h(C_+) (SWAP), converging to h(x) not in C_-; a small crosscut q_n joining them is an E_t-crosscut (var -1) or not (var 0, Lemma 4.3); does either case iterate the hug to h^k(x) and contradict cm of h^2 at period 2; (ii) SD at the pieces: take a closed disc E around x with a_n, b_n in int E; if they lie in different components of X cap E (check by the composant law: they are in C_+, x is not), some far sheet w near the boundary of E has phi_w(a_n) != phi_w(b_n) although p_n is Theta-free; compare w with the BULGE points of Sh(p_n) and the zero variation of p_n; (iii) a Jordan or Janiszewski argument on p_n and a proper subcontinuum of X joining a_n to b_n inside K_{p_n}; (iv) your own. Part 2 (carry it out). Prove NO-HUG, or give the dichotomy 'NO-HUG, or the configuration C' with C excluded by a named planar law or realised, or reduce it to one named lemma L with gate G1-G4. Mark each use of planarity, of PPF, of cm of h^2, of (2G), and of the composant swap; say where the hypothesis that x is in the OTHER composant is used (it is what the controls below must break). State the result for h^2, not h^4. Part 3 (builder or decisive test). Try to realise a theta_F-hug against every law above: data (arcs p_n, ends and composant labels, their images under F, linked chords, far-sheet values at BULGE points), or name the first law (zero variation at p_n, INVIS-Theta, SD, SWAP, cm of h^2, PPF, composant law) that every candidate violates. Builder gate: name the TARGET law (B0); secure the background laws first (B1); test (B2); end with 'KILL: law | TARGET or BACKGROUND | mechanism | what a proof must use' (B3). What already failed (read DEADENDS first): a new method, even partial with an honest gap, beats a polished restatement. Take your time: check every step, hypothesis and quantifier. E. CONTROLS (all must hold) (C1) Knaster bucket handle K with a homeomorphism H (Barge-Martin, RECALL): chainable, Theta_X = {theta*} (one full end, one gap, 4th-kind end), and every chart flip from the endpoint composant to another composant hugs theta*: a hug exists and the action of H on prime ends exists and fixes theta*. It has a fixed point and a plane extension, so it fails PPF and NO-LOCAL-INJ. A proof of NO-HUG must therefore use PPF, cm of h^2 or the absence of a plane extension, named; (C2) two-endpoint Knaster, hugs of either end, same lesson; (C3) Lewis pseudo-arc: Theta_X = S^1, no Theta-free crosscut exists, (2G) fails and NO-HUG is vacuous, so it decides nothing; (C4) FLIP-S, S-TREE (locally connected trees, fixed point): impressions are points, Theta_X empty, one composant, no hug possible (vacuous), the mechanism is invisible in decomposable hosts; (C5) SWAP alone only permutes labels and passes every record (R1, M^U); say what a use of it adds. F. DEAD ENDS (not progress) (1) Induced prime-end action of F, G, F^2, G^2 and impression or principal-set transport: no action exists (NO-LOCAL-INJ), Im = X at every full end (vacuous), so nothing lands. (2) Nested comparison of shadows (Cor NO-COF, EXCURSION) and composant rank: blind. (3) S^1-level variation laws and the TRAVERSE covering of the circle: TRAVERSE fails for accumulation flips. (4) Pure composant bookkeeping and the labels C_+- under SWAP: records R1 and M^U-FLIP survive with forced hug data. (5) Attributed flip mouths: impossible (NM, BIN, Beurling), so the only theta-blind flip is the accumulation flip.(6) api_opus_hug_var and api_opus_pure_pocket are other lanes on the PAIRING idea; a pairing must prove more than 'zero variation at each p_n' (nothing, by INVIS-Theta). G. FORMAT Line 1 exactly: VERDICT: NO-HUG PROVED (theta_F-HUG impossible in (S); each law used named, the planar line marked) | HUG CONSISTENT (explicit Tier 0 data or abstract configuration meeting every listed law; first missing law named) | REDUCED (to one named lemma L, gate G1-G4 passed) | PARTIAL (one named sub-claim settled) | 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 F or G is a plane homeomorphism or has an induced prime-end action (NO-LOCAL-INJ); transporting impressions (Im theta_F = X, vacuous); composant bookkeeping alone; nested shadow comparison; claiming variation for a chord that crosses R_t essentially; using the Knaster or pseudo-arc controls as hosts; claiming ACC-FLIP, L2 or PER as proved; presenting NO-HUG as new when it is the restatement of ACC-FLIP (gate G3)